An Efficient Numerical Approach for Nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin; Vedula, Prakash
2009-03-01
Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.
A quadrature based method of moments for nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin L.; Vedula, Prakash
2011-09-01
Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.
Study of Bunch Instabilities By the Nonlinear Vlasov-Fokker-Planck Equation
Warnock, Robert L.; /SLAC
2006-07-11
Instabilities of the bunch form in storage rings may be induced through the wake field arising from corrugations in the vacuum chamber, or from the wake and precursor fields due to coherent synchrotron radiation (CSR). For over forty years the linearized Vlasov equation has been applied to calculate the threshold in current for an instability, and the initial growth rate. Increasing interest in nonlinear aspects of the motion has led to numerical solutions of the nonlinear Vlasov equation, augmented with Fokker-Planck terms to describe incoherent synchrotron radiation in the case of electron storage rings. This opens the door to much deeper studies of coherent instabilities, revealing a rich variety of nonlinear phenomena. Recent work on this topic by the author and collaborators is reviewed.
Nonlinear Bayesian estimation via solution of the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Yoon, Jangho
2009-12-01
A general approach to optimal nonlinear filtering can be described by a recursive Bayesian approach. The key step in this approach is to determine the probability density function of the state vector conditioned on available measurements. However, an optimal solution to the Bayesian filtering problem can only be obtained exactly for a small class of problems such as linear and Gaussian cases. Therefore, in practice, approximate solutions, such as the extended Kalman filter, have been used. An optimal nonlinear filtering in a recursive Bayesian approach is a two-step process which consists of the prediction and the update process. In the update process, the priori conditional state probability density function (PDF) from the prediction process is updated through Bayes' rule using measurements from sensors. The prediction of conditional state PDF can be made by solving the Fokker-Planck equation (FPE) that governs the time-evolution the conditional state PDF. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with the exception of a few special cases. So far this estimation method has not been employed much in practice because of the high computational cost needed in solving the FPE numerically. In this dissertation, methods to improve the efficiency of the numerical method in solving the FPE are investigated to enhance the efficiency of the nonlinear filtering. Two finite difference methods, namely (i) the explicit forward method; and (ii) the alternating direction implicit (ADI) method, are used to solve the FPE numerically. Although the explicit forward method is much simpler to implement, the ADI method is preferred for its low computational cost. To reduce the computational cost further, as the first contribution of the dissertation, a moving domain scheme is developed to reduce the domain of integration required for solving the Fokker-Planck equation numerically. Simulation results show that the accuracy of the
Global existence for a nonlocal and nonlinear Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, Michael
2015-04-01
We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier, one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.
Fokker-Planck equation analysis of randomly excited nonlinear energy harvester
NASA Astrophysics Data System (ADS)
Kumar, P.; Narayanan, S.; Adhikari, S.; Friswell, M. I.
2014-03-01
The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker-Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.
Trigger, S. A.; Ebeling, W.; Heijst, G. J. F. van; Litinski, D.
2015-04-15
The problems of high linear conductivity in an electric field, as well as nonlinear conductivity, are considered for plasma-like systems. First, we recall several observations of nonlinear fast charge transport in dusty plasma, molecular chains, lattices, conducting polymers, and semiconductor layers. Exploring the role of noise we introduce the generalized Fokker-Planck equation. Second, one-dimensional models are considered on the basis of the Fokker-Planck equation with active and passive velocity-dependent friction including an external electrical field. On this basis, it is possible to find the linear and nonlinear conductivities for electrons and other charged particles in a homogeneous external field. It is shown that the velocity dependence of the friction coefficient can lead to an essential increase of the electron average velocity and the corresponding conductivity in comparison with the usual model of constant friction, which is described by the Drude-type conductivity. Applications including novel forms of controlled charge transfer and non-Ohmic conductance are discussed.
NASA Astrophysics Data System (ADS)
Nevolin, V. I.
2003-04-01
We present a method for analyzing the characteristics of nonlinear detectors using the algorithms of first-order nonlinear differential equations. This method is based on numerical solutions of the Fokker-Planck-Kolmogorov (FPK) equations in the form of series of functions over Hermite-Chebyshev polynomials for both nonlinear systems and their linear counterparts. The results of the solutions for the linear case are extended to nonlinear systems in a recurrent way.
Fokker-Planck equation in mirror research
Post, R.F.
1983-08-11
Open confinement systems based on the magnetic mirror principle depend on the maintenance of particle distributions that may deviate substantially from Maxwellian distributions. Mirror research has therefore from the beginning relied on theoretical predictions of non-equilibrium rate processes obtained from solutions to the Fokker-Planck equation. The F-P equation plays three roles: Design of experiments, creation of classical standards against which to compare experiment, and predictions concerning mirror based fusion power systems. Analytical and computational approaches to solving the F-P equation for mirror systems will be reviewed, together with results and examples that apply to specific mirror systems, such as the tandem mirror.
Fractional Fokker-Planck equation for fractal media.
Tarasov, Vasily E
2005-06-01
We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space. PMID:16035878
Spectral Decomposition of a Fokker-Planck Equation at Criticality
NASA Astrophysics Data System (ADS)
Bologna, M.; Beig, M. T.; Svenkeson, A.; Grigolini, P.; West, B. J.
2015-07-01
The mean field for a complex network consisting of a large but finite number of random two-state elements, , has been shown to satisfy a nonlinear Langevin equation. The noise intensity is inversely proportional to . In the limiting case , the solution to the Langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point. When , the inverse power law is truncated by an exponential decay with rate , the evaluation of which is the main purpose of this article. An analytic/numeric approach is used to obtain the lowest-order eigenvalues in the spectral decomposition of the solution to the corresponding Fokker-Planck equation and its equivalent Schrödinger equation representation.
Problems with the linear q-Fokker Planck equation
NASA Astrophysics Data System (ADS)
Yano, Ryosuke
2015-05-01
In this letter, we discuss the linear q-Fokker Planck equation, whose solution follows Tsallis distribution, from the viewpoint of kinetic theory. Using normal definitions of moments, we can expand the distribution function with infinite moments for 0 ⩽ q < 1, whereas we cannot expand the distribution function with infinite moments for 1 < q owing to emergences of characteristic points in moments. From Grad's 13 moment equations for the linear q-Fokker Planck equation, the dissipation rate of the heat flux via the linear q-Fokker Planck equation diverges at 0 ⩽ q < 2/3. In other words, the thermal conductivity, which defines the heat flux with the spatial gradient of the temperature and the thermal conductivity, which defines the heat flux with the spacial gradient of the density, jumps to zero at q = 2/3, discontinuously.
The Fokker-Planck Equation with Absorbing Boundary Conditions
NASA Astrophysics Data System (ADS)
Hwang, Hyung Ju; Jang, Juhi; Velázquez, Juan J. L.
2014-10-01
We study the initial-boundary value problem for the Fokker-Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker-Planck equation with absorbing boundary conditions, as well as the Hölder continuity of the solutions up to the singular set.
NASA Astrophysics Data System (ADS)
Ichiki, A.; Shiino, M.
2009-08-01
Phase transitions and effects of external noise on many-body systems are one of the main topics in physics. In mean-field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of Lévy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with Lévy noise have attracted much attention and behaviors of systems with double-well potential subjected to Lévy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an analytically solvable model to study the occurrence of phase transitions driven by Lévy stable noise.
New exact solutions to the Fokker Planck Kolmogorov equation
NASA Astrophysics Data System (ADS)
Ünal, Gazanfer; Sun, Jian-Qiao
2008-12-01
A generalized Liouville theorem has been proven for Itô systems. This allows us to show that the conserved quantities of the deterministic part of the Itô systems lead to the solution of the Fokker-Planck-Kolmogorov equation. The results have been applied to a stochastic 3-species Lotka Volterra system and the semi-classical Jaynes-Cummings system.
NASA Astrophysics Data System (ADS)
Che, Rui; Huang, Wen; Li, Yao; Tetali, Prasad
2016-08-01
In 2012, Chow, Huang, Li and Zhou [7] proposed the Fokker-Planck equations for the free energy on a finite graph, in which they showed that the corresponding Fokker-Planck equation is a nonlinear ODE defined on a Riemannian manifold of probability distributions. Different choices for inner products result in different Fokker-Planck equations. The unique global equilibrium of each equation is a Gibbs distribution. In this paper we proved that the exponential rate of convergence towards the global equilibrium of these Fokker-Planck equations. The rate is measured by both the decay of the L2 norm and that of the (relative) entropy. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [7]. The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [7].
Quantum Fokker-Planck-Kramers equation and entropy production
NASA Astrophysics Data System (ADS)
de Oliveira, Mário J.
2016-07-01
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.
State-space-split method for some generalized Fokker-Planck-Kolmogorov equations in high dimensions
NASA Astrophysics Data System (ADS)
Er, Guo-Kang; Iu, Vai Pan
2012-06-01
The state-space-split method for solving the Fokker-Planck-Kolmogorov equations in high dimensions is extended to solving the generalized Fokker-Planck-Kolmogorov equations in high dimensions for stochastic dynamical systems with a polynomial type of nonlinearity and excited by Poissonian white noise. The probabilistic solution of the motion of the stretched Euler-Bernoulli beam with cubic nonlinearity and excited by uniformly distributed Poissonian white noise is analyzed with the presented solution procedure. The numerical analysis shows that the results obtained with the state-space-split method together with the exponential polynomial closure method are close to those obtained with the Monte Carlo simulation when the relative value of the basic system relaxation time and the mean arrival time of the Poissonian impulse is in some limited range.
Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix
Schulze-Halberg, Axel
2012-10-15
We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schroedinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.
Fokker-Planck equation of Schramm-Loewner evolution.
Najafi, M N
2015-08-01
In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE(κ,ρc). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long- and short-time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE(κ,ρc) traces and show that it is related to the higher-order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented. PMID:26382350
Hypersonic expansion of the Fokker--Planck equation
Fernandez-Feria, R.
1989-02-01
A systematic study of the hypersonic limit of a heavy species diluted in a much lighter gas is made via the Fokker--Planck equation governing its velocity distribution function. In particular, two different hypersonic expansions of the Fokker--Planck equation are considered, differing from each other in the momentum equation of the heavy gas used as the basis of the expansion: in the first of them, the pressure tensor is neglected in that equation while, in the second expansion, the pressure tensor term is retained. The expansions are valid when the light gas Mach number is O(1) or larger and the difference between the mean velocities of light and heavy components is small compared to the light gas thermal speed. They can be applied away from regions where the spatial gradient of the distribution function is very large, but it is not restricted with respect to the temporal derivative of the distribution function. The hydrodynamic equations corresponding to the lowest order of both expansions constitute two different hypersonic closures of the moment equations. For the subsequent orders in the expansions, closed sets of moment equations (hydrodynamic equations) are given. Special emphasis is made on the order of magnitude of the errors of the lowest-order hydrodynamic quantities. It is shown that if the heat flux vanishes initially, these errors are smaller than one might have expected from the ordinary scaling of the hypersonic closure. Also it is found that the normal solution of both expansions is a Gaussian distribution at the lowest order.
Quantum Fokker-Planck-Kramers equation and entropy production.
de Oliveira, Mário J
2016-07-01
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance. PMID:27575097
Solution of the Fokker-Planck equation in a wind turbine array boundary layer
NASA Astrophysics Data System (ADS)
Melius, Matthew S.; Tutkun, Murat; Cal, Raúl Bayoán
2014-07-01
Hot-wire velocity signals from a model wind turbine array boundary layer flow wind tunnel experiment are analyzed. In confirming Markovian properties, a description of the evolution of the probability density function of velocity increments via the Fokker-Planck equation is attained. Solution of the Fokker-Planck equation is possible due to the direct computation of the drift and diffusion coefficients from the experimental measurement data which were acquired within the turbine canopy. A good agreement is observed in the probability density functions between the experimental data and numerical solutions resulting from the Fokker-Planck equation, especially in the far-wake region. The results serve as a tool for improved estimation of wind velocity within the array and provide evidence that the evolution of such a complex and turbulent flow is also governed by a Fokker-Planck equation at certain scales.
Fokker-Planck equation with arbitrary dc and ac fields: continued fraction method.
Lee, Chee Kong; Gong, Jiangbin
2011-07-01
The continued fraction method (CFM) is used to solve the Fokker-Planck equation with arbitrary dc and ac fields. With an appropriate choice of basis functions, the Fokker-Planck equation is converted into a set of linear algebraic equations with short-ranged coupling and then CFM is implemented to obtain numerical solutions with high efficiency. Both a proposed perturbative CFM and the numerically exact matrix CFM are used to study the nonlinear response of driven systems, with their results compared to assess the validity regime of the perturbative approach. The proposed perturbative CFM approach needs scalar quantities only and hence is more efficient within its validity regime. Two nonlinear systems of different nature are used as examples: molecular dipole (rotational Brownian motion) and particle in a periodic potential (translational Brownian motion). The associated full dynamics is presented in the compact form of hysteresis loops. It is observed that as the strength of an AC driving field increases, pronounced nonlinear effects are manifested in the deformation of the hysteresis loops. PMID:21867110
Transport in the spatially tempered, fractional Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Kullberg, A.; del-Castillo-Negrete, D.
2012-06-01
A study of truncated Lévy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, λ, and on the order of the fractional derivative in space, α. An expansion of the TFD operator for large λ is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit λ → ∞, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (λ > 0) truncations, and α ≠ 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit λ → ∞, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for α ≠ 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any λ > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with λ, as J ˜ λ-ζ, for α ⩾ 1.75. However, for α ⩽ 1.5, the steady-state current decays exponentially with λ, as J ˜ e-ξλ. In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the
Transport in the spatially tempered, fractional Fokker-Planck equation
Kullberg, A.; Del-Castillo-Negrete, Diego B
2012-01-01
A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, lambda, and on the order of the fractional derivative in space, alpha. An expansion of the TFD operator for large lambda is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit lambda -> infinity, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (lambda > 0) truncations, and alpha not equal 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit lambda -> infinity, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for alpha not equal 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any lambda > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with lambda, as J similar to lambda(-zeta), for alpha >= 1.75. However, for alpha <= 1.5, the steady-state current decays exponentially with lambda, as J similar to e(-xi lambda). In the presence of an asymmetry in the TFD operator, the tempering can lead
NASA Astrophysics Data System (ADS)
Alrachid, Houssam; Lelièvre, Tony; Talhouk, Raafat
2016-05-01
We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non-linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.
Fokker-Planck equations for charged-particle transport in random fields.
NASA Technical Reports Server (NTRS)
Jokipii, J. R.
1972-01-01
The Fokker-Planck equations for charged-particle dynamics are rederived, extending somewhat the elegant discussion of Hasselmann and Wibberenz. It is shown that the usual results are obtae and the conclusions in many cases are correct over a very broad range in energy. In particular, the rate for pitch-angle scattering may be accurately given down to energies much lower than previously thought. Recent claims that these Fokker-Planck equations are in general incorrect are thus shown to be in error.
Fokker-Planck equation in the presence of a uniform magnetic field
NASA Astrophysics Data System (ADS)
Dong, Chao; Zhang, Wenlu; Li, Ding
2016-08-01
The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.
NASA Technical Reports Server (NTRS)
Jokipii, J. R.
1973-01-01
A derivation of the Fokker-Planck equation, based on the central limit theorem, is presented which clearly illustrates the conditions for its validity. It is reiterated that previous use of the Fokker-Planck equation in cosmic-ray transport is correct. Higher-order effects associated with magnetic mirroring and field line random walk at low energies are discussed heuristically.
The Fokker-Planck equation for the radiation transfer in a strongly magnetized plasma
NASA Astrophysics Data System (ADS)
Bonazzola, S.
1982-04-01
The Fokker-Planck equation for the radiation transfer in a strongly magnetized plasma is obtained by means of an approximation. It is noted that this equation requires less computer time than the Monte Carlo method and that it allows the effect of the induced processes to be taken into account without difficulty.
Conservative differencing of the electron Fokker-Planck transport equation
Langdon, A.B.
1981-01-12
We need to extend the applicability and improve the accuracy of kinetic electron transport codes. In this paper, special attention is given to modelling of e-e collisions, including the dominant contributions arising from anisotropy. The electric field and spatial gradient terms are also considered. I construct finite-difference analogues to the Fokker-Planck integral-differential collision operator, which conserve the particle number, momentum and energy integrals (sums) regardless of the coarseness of the velocity zoning. Such properties are usually desirable, but are especially useful, for example, when there are spatial regions and/or time intervals in which the plasma is cool, so that the collision operator acts rapidly and the velocity distribution is poorly resolved, yet it is crucial that gross conservation properties be respected in hydro-transport applications, such as in the LASNEX code. Some points are raised concerning spatial differencing and time integration.
Fokker-Planck-Kolmogorov Equation for fBm: Derivation and Analytical Solutions
NASA Astrophysics Data System (ADS)
Ünal, Gazanfer
2007-04-01
The Fokker-Planck-Kolmogorov (FPK) equation for Itô systems with fractional Brownian motion (fBm) has been derived. The generalized Liouville theorem has been proved. Analytic solutions to the FPK have been obtained via conserved quantities of the deterministic part.
Analytical and Numerical Solutions of Generalized Fokker-Planck Equations - Final Report
Prinja, Anil K.
2000-12-31
The overall goal of this project was to develop advanced theoretical and numerical techniques to quantitatively describe the spreading of a collimated beam of charged particles in space, in angle, and in energy, as a result of small deflection, small energy transfer Coulomb collisions with the target nuclei and electrons. Such beams arise in several applications of great interest in nuclear engineering, and include electron and ion radiotherapy, ion beam modification of materials, accelerator transmutation of waste, and accelerator production of tritium, to name some important candidates. These applications present unique and difficult modeling challenges, but from the outset are amenable to the language of ''transport theory'', which is very familiar to nuclear engineers and considerably less-so to physicists and material scientists. Thus, our approach has been to adopt a fundamental description based on transport equations, but the forward peakedness associated with charged particle interactions precludes a direct application of solution methods developed for neutral particle transport. Unique problem formulations and solution techniques are necessary to describe the transport and interaction of charged particles. In particular, we have developed the Generalized Fokker-Planck (GFP) approach to describe the angular and radial spreading of a collimated beam and a renormalized transport model to describe the energy-loss straggling of an initially monoenergetic distribution. Both analytic and numerical solutions have been investigated and in particular novel finite element numerical methods have been developed. In the first phase of the project, asymptotic methods were used to develop closed form solutions to the GFP equation for different orders of expansion, and was described in a previous progress report. In this final report we present a detailed description of (i) a novel energy straggling model based on a Fokker-Planck approximation but which is adapted for a
A covariant Fokker-Planck equation for a simple gas from relativistic kinetic theory
Chacon-Acosta, Guillermo; Dagdug, Leonardo; Morales-Tecotl, Hugo A.
2010-12-14
A manifestly covariant Fokker-Planck differential equation is derived for the case of a relativistic simple gas by taking a small momentum transfer approximation within the collision integral of the relativistic Boltzmann equation. We follow closely previous work, with the main difference that we keep manifest covariance at every stage of the analysis. In addition, we use the covariant Juettner distribution function to find a relativistic generalization of the Einstein's fluctuation-dissipation relation.
Moment-Preserving SN Discretizations for the One-Dimensional Fokker-Planck Equation
Warsa, James S.; Prinja, Anil K.
2012-06-14
The Fokker-Planck equation: (1) Describes the transport and interactions of charged particles, (2) Many small-angle scattering collisions, (3) Asymptotic limit of the Boltzmann equation (Pomraning, 1992), and (4) The Boltzmann collision operator becomes the angular Laplacian. SN angular discretization: (1) Angular flux is collocated at the SN quadrature points, (2) The second-order derivatives in the Laplacian term must be discretized, and (3) Weighted finite-difference method preserves zeroth and first moments (Morel, 1985). Moment-preserving methods: (1) Collocate the Fokker-Planck operator at the SN quadrature points, (2) Develop several related and/or equivalent methods, and (3) Motivated by discretizations for the angular derivative appearing in the transport equation in one-dimensional spherical coordinates.
Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime
NASA Astrophysics Data System (ADS)
Liang, Jin-Rong; Wang, Jun; Lǔ, Long-Jin; Gu, Hui; Qiu, Wei-Yuan; Ren, Fu-Yao
2012-01-01
In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α, H ( t)= X H ( S α ( t)), 0< α, H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X α, H ( t) and the corresponding Black-Scholes formula for the fair prices of European option.
Danos, Rebecca J.; Fiege, Jason D.; Shalchi, Andreas E-mail: fiege@physics.umanitoba.ca
2013-07-20
We present numerical solutions to both the standard and modified two-dimensional Fokker-Planck equations with adiabatic focusing and isotropic pitch-angle scattering. With the numerical solution of the particle distribution function, we then discuss the related numerical issues, calculate the parallel diffusion coefficient using several different methods, and compare our numerical solutions for the parallel diffusion coefficient to the analytical forms derived earlier. We find the numerical solution to the diffusion coefficient for both the standard and modified Fokker-Planck equations agrees with that of Shalchi for the mean squared displacement method of computing the diffusion coefficient. However, we also show the numerical solution agrees with that of Litvinenko and Shalchi and Danos when calculating the diffusion coefficient via the velocity correlation function.
NASA Astrophysics Data System (ADS)
Sun, Yifei; Kumar, Mrinal
2015-05-01
In this paper, a tensor decomposition approach combined with Chebyshev spectral differentiation is presented to solve the high dimensional transient Fokker-Planck equations (FPE) arising in the simulation of polymeric fluids via multi-bead-spring (MBS) model. Generalizing the authors' previous work on the stationary FPE, the transient solution is obtained in a single CANDECOMP/PARAFAC decomposition (CPD) form for all times via the alternating least squares algorithm. This is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions, thereby decoupling it from them. As a result, the transient solution is obtained without resorting to expensive time stepping schemes. A new, relaxed approach for imposing the vanishing boundary conditions is proposed, improving the quality of the approximation. The asymptotic behavior of the temporal basis functions is studied. The proposed solver scales very well with the dimensionality of the MBS model. Numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix Riccati differential equation (for linear models) or Monte Carlo simulations (for nonlinear models).
Classical integrability for beta-ensembles and general Fokker-Planck equations
Rumanov, Igor
2015-01-15
Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here, we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g., there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system–a closed system of two nonlinear partial differential equations (PDEs) of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration. PMID:24329213
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
Bakhtiyari-Ramezani, M. Alinejad, N.; Mahmoodi, J.
2015-11-15
In the fusion devices, ions, H atoms, and H{sub 2} molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H{sub 2} molecules, and desorption of the recombined H{sub 2} molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.
Solving the Fokker-Planck equation with the finite-element method
Galán, Roberto F.; Ermentrout, G. Bard; Urban, Nathaniel N.
2008-01-01
We apply an efficient approach from computational engineering, the finite-element method, to numerically solve the Fokker-Planck equation in two dimensions. This approach permits us to find the solution to stochastic problems that cannot be solved analytically. We illustrate our strategy with an example from neuroscience that recently has attracted considerable attention - synchronization of neural oscillators. In particular, we show that resonators (type II neural oscillators) respond and synchronize more reliably when provided correlated stochastic inputs than do integrators (type I neural oscillators). This result is consistent with recent experimental and computational work. We briefly discuss its relevance for neuroscience. PMID:18233721
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
NASA Astrophysics Data System (ADS)
Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.
2015-11-01
In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.
Fokker Planck Rosenbluth-type equations for self-gravitating systems in the 1PN approximation
NASA Astrophysics Data System (ADS)
Ramos-Caro, Javier; González, Guillermo A.
2008-02-01
We present two formulations of Fokker Planck Rosenbluth-type (FPR) equations for many-particle self-gravitating systems, with first-order relativistic corrections in the post-Newtonian approach (1PN). The first starts from a covariant Fokker Planck equation for a simple gas, introduced recently by Chacón-Acosta and Kremer (2007 Phys. Rev. E 76 021201). The second derivation is based on the establishment of an 1PN-BBGKY hierarchy, developed systematically from the 1PN microscopic law of force and using the Klimontovich Dupree (KD) method. We close the hierarchy by the introduction of a two-point correlation function that describes adequately the relaxation process. This picture reveals an aspect that is not considered in the first formulation: the contribution of ternary correlation patterns to the diffusion coefficients, as a consequence of the nature of 1PN interaction. Both formulations can be considered as a generalization of the equation derived by Rezania and Sobouti (2000 Astron. Astrophys. 354 1110), to stellar systems where the relativistic effects of gravitation play a significant role.
Densmore, Jeffery D. Warsa, James S. Lowrie, Robert B. Morel, Jim E.
2009-09-01
The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.
Densmore, Jeffery D; Warsa, James S; Lowrie, Robert B; Morel, Jim E
2008-01-01
The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.
MODELING THE SUNSPOT NUMBER DISTRIBUTION WITH A FOKKER-PLANCK EQUATION
Noble, P. L.; Wheatland, M. S.
2011-05-01
Sunspot numbers exhibit large short-timescale (daily-monthly) variation in addition to longer-timescale variation due to solar cycles. A formal statistical framework is presented for estimating and forecasting randomness in sunspot numbers on top of deterministic (including chaotic) models for solar cycles. The Fokker-Planck approach formulated assumes a specified long-term or secular variation in sunspot number over an underlying solar cycle via a driver function. The model then describes the observed randomness in sunspot number on top of this driver function. We consider a simple harmonic choice for the driver function, but the approach is general and can easily be extended to include other drivers which account for underlying physical processes and/or empirical features of the sunspot numbers. The framework is consistent during both solar maximum and minimum, and requires no parameter restrictions to ensure non-negative sunspot numbers. Model parameters are estimated using statistically optimal techniques. The model agrees both qualitatively and quantitatively with monthly sunspot data even with the simplistic representation of the periodic solar cycle. This framework should be particularly useful for solar cycle forecasters and is complementary to existing modeling techniques. An analytic approximation for the Fokker-Planck equation is presented, which is analogous to the Euler approximation, which allows for efficient maximum likelihood estimation of large data sets and/or when using difficult to evaluate driver functions.
Yoon, E. S.; Chang, C. S.
2014-03-15
An approximate two-dimensional solver of the nonlinear Fokker-Planck-Landau collision operator has been developed using the assumption that the particle probability distribution function is independent of gyroangle in the limit of strong magnetic field. The isotropic one-dimensional scheme developed for nonlinear Fokker-Planck-Landau equation by Buet and Cordier [J. Comput. Phys. 179, 43 (2002)] and for linear Fokker-Planck-Landau equation by Chang and Cooper [J. Comput. Phys. 6, 1 (1970)] have been modified and extended to two-dimensional nonlinear equation. In addition, a method is suggested to apply the new velocity-grid based collision solver to Lagrangian particle-in-cell simulation by adjusting the weights of marker particles and is applied to a five dimensional particle-in-cell code to calculate the neoclassical ion thermal conductivity in a tokamak plasma. Error verifications show practical aspects of the present scheme for both grid-based and particle-based kinetic codes.
Analytical Solutions of the Fokker-Planck Equation for Generalized Morse and Hulthén Potentials
NASA Astrophysics Data System (ADS)
Anjos, R. C.; Freitas, G. B.; Coimbra-Araújo, C. H.
2016-01-01
In the present contribution we analytically calculate solutions of the transition probability of the Fokker-Planck equation (FPE) for both the generalized Morse potential and the Hulthén potential. The method is based on the formal analogy of the FPE with the Schrödinger equation using techniques from supersymmetric quantum mechanics.
Global well-posedness for the Fokker-Planck-Boltzmann equation in Besov-Chemin-Lerner type spaces
NASA Astrophysics Data System (ADS)
Liu, Zhengrong; Tang, Hao
2016-06-01
In this paper, motivated by [16], we use the Littlewood-Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker-Planck-Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov-Chemin-Lerner type spaces C ([ 0 , ∞) ; L˜>ξ 2 (B2,rs)) with 1 ≤ r ≤ 2 and s > 3 / 2 or s = 3 / 2 and r = 1. Besides, we also obtain the uniform stability of the dependence on the initial data.
Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
NASA Astrophysics Data System (ADS)
Ho, Choon-Lin
2011-04-01
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree ℓ = 1, 2, … , and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new Xℓ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker-Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.
Deterministic proton transport solving a one dimensional Fokker-Planck equation
Marr, D.; Prael, R.; Adams, K.; Alcouffe, R.
1997-10-01
The transport of protons through matter is characterized by many interactions which cause small deflections and slight energy losses. The few which are catastrophic or cause large angle scattering can be viewed as extinction for many applications. The transport of protons at this level of approximation can be described by a Fokker Planck Equation. This equation is solved using a deterministic multigroup differencing scheme with a highly resolved set of discrete ordinates centered around the beam direction which is adequate to properly account for deflections and energy losses due to multiple Coulomb scattering. Comparisons with LAHET for a large variety of problems ranging from 800 MeV protons on a copper step wedge to 10 GeV protons on a sandwich of material are presented. The good agreement with the Monte Carlo code shows that the solution method is robust and useful for approximate solutions of selected proton transport problems.
New Kinematic Model in comparing with Langevin equation and Fokker Planck Equation
NASA Astrophysics Data System (ADS)
Lee, Kyoung; Wang, Zhijian; Gardner, Robin
2010-03-01
An analytic approximate solution of New Kinematic Model with the boundary conditions is developed for the incompressible packing condition in Pebble Bed Reactors. It is based on velocity description of the packing density in the hopper. The packing structure can be presented with a jamming phenomenon from flow types. The gravity-driven macroscopic motions are governed not only by the geometry and external boundary conditions of silos and hoppers, but by flow prosperities of granular materials, such as friction, viscosity and porosity. The analytical formulas for the quasi-linear diffusion and convection coefficients of the velocity profile are obtained. Since it was found that the New Kinematic Model is dependent upon the granular packing density distribution, we are motivated to study the Langevin equation with friction under the influence of the Gravitational field. We also discuss the relation with the Fokker Planck Equation using Detailed balance and Metropolis-Hastings Algorithm. Markov chain Monte Carlo methods are shown to be a non-Maxwellian distribution function with the mean velocity of the field particles having an effective temperature.
Solution of the Fokker-Planck Equation with a Logarithmic Potential
NASA Astrophysics Data System (ADS)
Dechant, A.; Lutz, E.; Barkai, E.; Kessler, D. A.
2011-12-01
We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large | x| using the Fokker-Planck equation. An eigenfunction expansion shows that the Boltzmann equilibrium density does not fully describe the long-time limit of this problem. Instead this limit is characterized by an infinite covariant density. This non-normalizable density yields the mean square displacement of the particles, which for a certain range of parameters exhibits anomalous diffusion. In a symmetric potential with an asymmetric initial condition, the average position decays anomalously slowly. This problem also has applications outside the thermal context, as in the diffusion of the momenta of atoms in optical molasses.
NASA Astrophysics Data System (ADS)
Hager, Robert; Yoon, E. S.; Ku, S.; D'Azevedo, E. F.; Worley, P. H.; Chang, C. S.
2016-06-01
Fusion edge plasmas can be far from thermal equilibrium and require the use of a non-linear collision operator for accurate numerical simulations. In this article, the non-linear single-species Fokker-Planck-Landau collision operator developed by Yoon and Chang (2014) [9] is generalized to include multiple particle species. The finite volume discretization used in this work naturally yields exact conservation of mass, momentum, and energy. The implementation of this new non-linear Fokker-Planck-Landau operator in the gyrokinetic particle-in-cell codes XGC1 and XGCa is described and results of a verification study are discussed. Finally, the numerical techniques that make our non-linear collision operator viable on high-performance computing systems are described, including specialized load balancing algorithms and nested OpenMP parallelization. The collision operator's good weak and strong scaling behavior are shown.
Modelling of income distribution in the European Union with the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Jagielski, Maciej; Kutner, Ryszard
2013-05-01
Herein, we applied statistical physics to study incomes of three (low-, medium- and high-income) society classes instead of the two (low- and medium-income) classes studied so far. In the frame of the threshold nonlinear Langevin dynamics and its threshold Fokker-Planck counterpart, we derived a unified formula for description of income of all society classes, by way of example, of those of the European Union in years 2006 and 2008. Hence, the formula is more general than the well known formula of Yakovenko et al.. That is, our formula well describes not only two regions but simultaneously the third region in the plot of the complementary cumulative distribution function vs. an annual household income. Furthermore, the known stylised facts concerning this income are well described by our formula. Namely, the formula provides the Boltzmann-Gibbs income distribution function for the low-income society class and the weak Pareto law for the medium-income society class, as expected. Importantly, it predicts (to satisfactory approximation) the Zipf law for the high-income society class. Moreover, the region of medium-income society class is now distinctly reduced because the bottom of high-income society class is distinctly lowered. This reduction made, in fact, the medium-income society class an intermediate-income society class.
NASA Astrophysics Data System (ADS)
Jang, Seogjoo
2016-06-01
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.
Jang, Seogjoo
2016-06-01
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath. PMID:27276940
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as “Wigner space”), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations.
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material. PMID:25877565
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Tanimura, Yoshitaka
2015-04-01
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
NASA Astrophysics Data System (ADS)
Polson, James M.; Dunn, Taylor R.
2014-05-01
Brownian dynamics (BD) simulations are used to study the translocation dynamics of a coarse-grained polymer through a cylindrical nanopore. We consider the case of short polymers, with a polymer length, N, in the range N = 21-61. The rate of translocation is controlled by a tunable friction coefficient, γ0p, for monomers inside the nanopore. In the case of unforced translocation, the mean translocation time scales with polymer length as ⟨τ1⟩ ˜ (N - Np)α, where Np is the average number of monomers in the nanopore. The exponent approaches the value α = 2 when the pore friction is sufficiently high, in accord with the prediction for the case of the quasi-static regime where pore friction dominates. In the case of forced translocation, the polymer chain is stretched and compressed on the cis and trans sides, respectively, for low γ0p. However, the chain approaches conformational quasi-equilibrium for sufficiently large γ0p. In this limit the observed scaling of ⟨τ1⟩ with driving force and chain length supports the Fokker-Planck (FP) prediction that ⟨τ⟩ ∝ N/fd for sufficiently strong driving force. Monte Carlo simulations are used to calculate translocation free energy functions for the system. The free energies are used with the FP equation to calculate translocation time distributions. At sufficiently high γ0p, the predicted distributions are in excellent agreement with those calculated from the BD simulations. Thus, the FP equation provides a valid description of translocation dynamics for sufficiently high pore friction for the range of polymer lengths considered here. Increasing N will require a corresponding increase in pore friction to maintain the validity of the FP approach. Outside the regime of low N and high pore friction, the polymer is out of equilibrium, and the FP approach is not valid.
A data-driven alternative to the fractional Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Pressé, Steve
2015-07-01
Anomalous diffusion processes are ubiquitous in biology and arise in the transport of proteins, vesicles and other particles. Such anomalously diffusive behavior is attributed to a number of factors within the cell including heterogeneous environments, active transport processes and local trapping/binding. There are a number of microscopic principles—such as power law jump size and/or waiting time distributions—from which the fractional Fokker-Planck equation (FFPE) can be derived and used to provide mechanistic insight into the origins of anomalous diffusion. On the other hand, it is fair to ask if other microscopic principles could also have given rise to the evolution of an observed density profile that appears to be well fit by an FFPE. Here we discuss another possible mechanistic alternative that can give rise to densities like those generated by FFPEs. Rather than to fit a density (or concentration profile) using a solution to the spatial FFPE, we reconstruct the profile generated by an FFPE using a regular FPE with a spatial and time-dependent force. We focus on the special case of the spatial FFPE for superdiffusive processes. This special case is relevant to, for example, active transport in a biological context. We devise a prescription for extracting such forces on synthetically generated data and provide an interpretation to the forces extracted. In particular, the time-dependence of forces could tell us about ATP depletion or changes in the cell's metabolic activity. Modeling anomalous behavior with normal diffusion driven by these effective forces yields an alternative mechanistic picture that, ultimately, could help motivate future experiments.
NASA Astrophysics Data System (ADS)
Taitano, W. T.; Chacón, L.; Simakov, A. N.; Molvig, K.
2015-09-01
In this study, we demonstrate a fully implicit algorithm for the multi-species, multidimensional Rosenbluth-Fokker-Planck equation which is exactly mass-, momentum-, and energy-conserving, and which preserves positivity. Unlike most earlier studies, we base our development on the Rosenbluth (rather than Landau) form of the Fokker-Planck collision operator, which reduces complexity while allowing for an optimal fully implicit treatment. Our discrete conservation strategy employs nonlinear constraints that force the continuum symmetries of the collision operator to be satisfied upon discretization. We converge the resulting nonlinear system iteratively using Jacobian-free Newton-Krylov methods, effectively preconditioned with multigrid methods for efficiency. Single- and multi-species numerical examples demonstrate the advertised accuracy properties of the scheme, and the superior algorithmic performance of our approach. In particular, the discretization approach is numerically shown to be second-order accurate in time and velocity space and to exhibit manifestly positive entropy production. That is, H-theorem behavior is indicated for all the examples we have tested. The solution approach is demonstrated to scale optimally with respect to grid refinement (with CPU time growing linearly with the number of mesh points), and timestep (showing very weak dependence of CPU time with time-step size). As a result, the proposed algorithm delivers several orders-of-magnitude speedup vs. explicit algorithms.
NASA Astrophysics Data System (ADS)
Lukassen, Laura; Oberlack, Martin
2014-11-01
As described in literature, non-Brownian particles in shear flow show a diffusive behavior due to hydrodynamic interactions. This shear-induced diffusion differs from the well-known Brownian diffusion, as there is no separation of time scales. That means that the configuration of non-Brownian particles changes on the same time scale as the hydrodynamic velocity. This fact impedes the derivation of a Fokker-Planck equation describing non-Brownian particles in pure position space. In this context, we derived a new Fokker-Planck approach in coupled position-velocity space to assure the validity of the Markov process assumption which is violated in pure position space formulation (Lukassen, Oberlack, Phys. Rev. E 89, 2014). Here, we present a further validation of our new Fokker-Planck approach that allows us to establish a relation to a modified purely position space Fokker-Planck equation. This backward transformation exhibits additional correction terms when compared to other position space Fokker-Planck equations in that context known from literature. Our extended approach shall enable a better stochastic description of non-Brownian particle flows. The work of L. Lukassen is supported by the ``Excellence Initiative'' of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.
NASA Astrophysics Data System (ADS)
Palyulin, Vladimir V.; Chechkin, Aleksei V.; Metzler, Ralf
2014-11-01
Based on the space-fractional Fokker-Planck equation with a δ-sink term, we study the efficiency of random search processes based on Lévy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Lévy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Lévy flights, due to which Lévy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Lévy flights over the regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.
NASA Astrophysics Data System (ADS)
Yavorskij, V. A.; Andrushchenko, Zh. N.; Edenstrasser, J. W.; Goloborod'ko, V. Ya
1999-10-01
The five-dimensional (5D) drift kinetic Fokker-Planck equation for fast charged particles confined in a tokamak with a toroidal field (TF) ripple magnitude below the Goldston-White-Boozer stochasticity threshold is averaged over the banana and superbanana timescales. As a result, a three-dimensional (3D) Fokker-Planck equation in the constants of motion (COM) space describing the collisional transport of charged high-energy particles is obtained. Toroidally trapped particles with the toroidal precession being in resonance with the ripple perturbations are shown to yield the main contribution to the ripple induced transport. It is found that the rates of ripple superbanana diffusion and convection in the radial coordinate significantly exceed the corresponding rates of the bananas in the axisymmetric limit. The superbanana diffusion and convection shown to be dominant in the MeV energy range may be responsible for the loss of partially thermalized fusion products observed in the Tokamak fusion test reactor (TFTR) [S. J. Zweben, R. L. Boivin, C.-S. Chang et al., Nucl. Fusion 31, 2219 (1991); H. W. Herrmann, S. J. Zweben, D. S. Darrow et al., ibid. 37, 1437 (1997)].
Weibull Statistics for Upper Ocean Currents with the Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Chu, P. C.
2012-12-01
Upper oceans typically exhibit of a surface mixed layer with a thickness of a few to several hundred meters. This mixed layer is a key component in studies of climate, biological productivity and marine pollution. It is the link between the atmosphere and the deep ocean and directly affects the air-sea exchange of heat, momentum and gases. Vertically averaged horizontal currents across the mixed layer are driven by the residual between the Ekman transport and surface wind stress, and damped by the Rayleigh friction. A set of stochastic differential equations are derived for the two components of the current vector (u, v). The joint probability distribution function of (u, v) satisfies the Fokker-Planck equation (Chu, 2008, 2009), with the Weibull distribution as the solution for the current speed. To prove it, the PDF of the upper (0-50 m) tropical Pacific current speeds (w) was calculated from hourly ADCP data (1990-2007) at six stations for the Tropical Atmosphere Ocean project. In fact, it satisfies the two-parameter Weibull distribution reasonably well with different characteristics between El Nino and La Nina events: In the western Pacific, the PDF of w has a larger peakedness during the La Nina events than during the El Nino events; and vice versa in the eastern Pacific. However, the PDF of w for the lower layer (100-200 m) does not fit the Weibull distribution so well as the upper layer. This is due to the different stochastic differential equations between upper and lower layers in the tropical Pacific. For the upper layer, the stochastic differential equations, established on the base of the Ekman dynamics, have analytical solution, i.e., the Rayleigh distribution (simplest form of the Weibull distribution), for constant eddy viscosity K. Knowledge on PDF of w during the El Nino and La Nina events will improve the ensemble horizontal flux calculation, which contributes to the climate studies. Besides, the Weibull distribution is also identified from the
Asgarani, Somayeh
2015-02-01
A method of finding entropic form for a given stationary probability distribution and specified potential field is discussed, using the steady-state Fokker-Planck equation. As examples, starting with the Boltzmann and Tsallis distribution and knowing the force field, we obtain the Boltzmann-Gibbs and Tsallis entropies. Also, the associated entropy for the gamma probability distribution is found, which seems to be in the form of the gamma function. Moreover, the related Fokker-Planck equations are given for the Boltzmann, Tsallis, and gamma probability distributions. PMID:25768455
On the Derivation of a High-Velocity Tail from the Boltzmann-Fokker-Planck Equation for Shear Flow
NASA Astrophysics Data System (ADS)
Acedo, L.; Santos, A.; Bobylev, A. V.
2002-12-01
Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U x ( y)= ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f( r, v)= f( V), with V≡ v- U( r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K( θ)∝lim ∈→0 ∈ -2 δ( θ- ∈), where θ is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker-Planck operator. We have found analytically that for shear rates larger than a certain threshold value a th≃0.3520 ν (where ν is an average collision frequency and a th/ ν is the real root of the cubic equation 64 x 3+16 x 2+12 x-9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f( V; a)˜| V|-4- σ( a) Φ( ϕ; a), where ϕ≡tan V y / V x and the angular distribution function Φ( ϕ; a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition Φ( ϕ; a)= Φ( ϕ+ π; a) allows one to obtain the exponent σ( a) as a function of the shear rate. It diverges when a→ a th and tends to a minimum value σ min≃1.252 in the limit a→∞. As a consequence of this power-law decay for a> a th, all the velocity moments of a degree equal to or larger than 2+ σ( a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~ϕ( a), which rotates from ~ϕ=- π/4,3 π/4 when a→ a th to ~ϕ=0, π in the limit a→∞.
Numerical Methods for Nonlinear Fokker-Planck Collision Operator in TEMPEST
NASA Astrophysics Data System (ADS)
Kerbel, G.; Xiong, Z.
2006-10-01
Early implementations of Fokker-Planck collision operator and moment computations in TEMPEST used low order polynomial interpolation schemes to reuse conservative operators developed for speed/pitch-angle (v, θ) coordinates. When this approach proved to be too inaccurate we developed an alternative higher order interpolation scheme for the Rosenbluth potentials and a high order finite volume method in TEMPEST (,) coordinates. The collision operator is thus generated by using the expansion technique in (v, θ) coordinates for the diffusion coefficients only, and then the fluxes for the conservative differencing are computed directly in the TEMPEST (,) coordinates. Combined with a cut-cell treatment at the turning-point boundary, this new approach is shown to have much better accuracy and conservation properties.
NASA Astrophysics Data System (ADS)
Milovanov, Alexander V.
2001-04-01
The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 112 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given.
Milovanov, A V
2001-04-01
The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 11 / 2 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given. PMID:11308983
NASA Astrophysics Data System (ADS)
Herrmann, Michael; Niethammer, Barbara; Velázquez, Juan J. L.
2014-08-01
The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.
NASA Technical Reports Server (NTRS)
Ogallagher, J. J.; Maslyar, G. A., III
1975-01-01
A recently developed model predicts an energy dependent phase lag in the modulated cosmic ray density U(t) given by U(t) approximately equal to US (t - tau) where US is the solution to the Fokker-Planck equation under time independent conditions and tau is the average time spent by particles inside the modulating region. The delay times tau are functions of modulating parameters R (the radius of the modulating cavity), V (the solar wind velocity), and K (the effective average diffusion-coefficient which is a function of energy). This model is applied to predict the time evolution of the modulated cosmic ray proton spectrum over a simulated solar cycle. A modulation produced mostly by varying R over the solar cycle is less consistent with the observations.
Lo, Joseph; Shizgal, Bernie D
2006-11-21
Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered. PMID:17129090
NASA Astrophysics Data System (ADS)
He, S.; Ohara, N.
2015-12-01
Sub-grid spatial variability of snow still remains a challenge because of complicated snow accumulation and melt processes. In this study, the Fokker-Planck equation (FPE) was derived to simulate the probability distribution in two-dimensional probability space, snow depth versus snow density. This FPE describes the evolution of probability density function of the state variables throughout the snow season. The snow depth and snow density were selected as stochastic state variables while snow temperature was left deterministic. In this equation, the snowmelt and snow accumulation are treated as external sources of stochasticity. This means that both the mean and variance parts of these variables are taken into consideration in the FPE as advection convection and diffusion effects in the probability domain, respectively. The major challenge of the FPE is calculation of the covariance terms appeared in the diffusion terms due to limitation of the data. In this study, several possible simplifications of the FPE will be discussed. We will test the following hypotheses for the simplifications: (1) the convection correction term is small enough to be neglected compared to the mean convection term, (2) snowmelt and snow accumulation are independent each other, and (3) snow accumulation term can be evaluated by measured snow depth data. For the estimation of snowmelt terms, outputs of a distributed snow model may also be used to calculate the spatial and temporal distribution of snowmelt. Finally, the FPE will be solved with a numerical method in the probability domain of snow depth versus snow density.
NASA Astrophysics Data System (ADS)
Carlen, Eric A.; Maas, Jan
2014-11-01
Let denote the Clifford algebra over , which is the von Neumann algebra generated by n self-adjoint operators Q j , j = 1,…, n satisfying the canonical anticommutation relations, Q i Q j + Q j Q i = 2δ ij I, and let τ denote the normalized trace on . This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let denote the set of all positive operators such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space . The fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.
Fokker-Planck/Transport model for neutral beam driven tokamaks
Killeen, J.; Mirin, A.A.; McCoy, M.G.
1980-01-01
The application of nonlinear Fokker-Planck models to the study of beam-driven plasmas is briefly reviewed. This evolution of models has led to a Fokker-Planck/Transport (FPT) model for neutral-beam-driven Tokamaks, which is described in detail. The FPT code has been applied to the PLT, PDX, and TFTR Tokamaks, and some representative results are presented.
Effects of the Tempered Aging and the Corresponding Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Deng, Weihua; Wang, Wanli; Tian, Xinchun; Wu, Yujiang
2016-07-01
In the renewal processes, if the waiting time probability density function is a tempered power-law distribution, then the process displays a transition dynamics; and the transition time depends on the parameter λ of the exponential cutoff. In this paper, we discuss the aging effects of the renewal process with the tempered power-law waiting time distribution. By using the aging renewal theory, the p-th moment of the number of renewal events n_a(t_a, t) in the interval (t_a, t_a+t) is obtained for both the weakly and strongly aged systems; and the corresponding surviving probabilities are also investigated. We then further analyze the tempered aging continuous time random walk and its Einstein relation, and the mean square displacement is attained. Moreover, the tempered aging diffusion equation is derived.
Fokker-Planck formalism in magnetic resonance simulations.
Kuprov, Ilya
2016-09-01
This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use. PMID:27470597
Fokker-Planck formalism in magnetic resonance simulations
NASA Astrophysics Data System (ADS)
Kuprov, Ilya
2016-09-01
This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.
Fokker-Planck model of hydrodynamics.
Singh, S K; Ansumali, Santosh
2015-03-01
We present a phenomenological description of the hydrodynamics in terms of the Fokker-Planck (FP) equation for one-particle distribution function. Similar to the Boltzmann equation or the Bhatnager-Gross-Krook (BGK) model, this approach is thermodynamically consistent and has the H theorem. In this model, transport coefficients as well as the equation of state can be provided independently. This approach can be used as an alternate to BGK-based methods as well as the direct simulation Monte Carlo method for the gaseous flows. PMID:25871242
Sliusarenko, O. Yu.; Chechkin, A. V.; Slyusarenko, Yu. V.
2015-04-15
By generalizing Bogolyubov’s reduced description method, we suggest a formalism to derive kinetic equations for many-body dissipative systems in external stochastic field. As a starting point, we use a stochastic Liouville equation obtained from Hamilton’s equations taking dissipation and stochastic perturbations into account. The Liouville equation is then averaged over realizations of the stochastic field by an extension of the Furutsu-Novikov formula to the case of a non-Gaussian field. As the result, a generalization of the classical Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is derived. In order to get a kinetic equation for the single-particle distribution function, we use a regular cutoff procedure of the BBGKY hierarchy by assuming weak interaction between the particles and weak intensity of the field. Within this approximation, we get the corresponding Fokker-Planck equation for the system in a non-Gaussian stochastic field. Two particular cases are discussed by assuming either Gaussian statistics of external perturbation or homogeneity of the system.
ICPP: Numerical Fokker-Planck calculations in nonuniform grids
NASA Astrophysics Data System (ADS)
Bizarro, João P. S.
2000-10-01
The Fokker-Planck equation arises in a wide class of problems in plasma physics, so numerical schemes that provide efficient, accurate, and stable solutions to that equation are always welcome. One way to accomplish this is via nonuniform grids, which allow the use of different mesh sizes according to the real needs of the physical problem under consideration. The extension of the standard finite-difference approach to general nonuniform grids, taking into account proper weighting coefficients, has already been presented, and the results have been rather conclusive [J. P. S. Bizarro and P. Rodrigues, Nucl. Fusion Vol. 37, 1509 (1997)]. Besides reviewing what has been achieved with nonuniform grids, a numerical scheme that is accurate to second order (both in time step and mesh size) is here extended and detailed. Such an analysis is rigourous for one-dimensional Fokker-Planck equations, and is generalized to two-dimensional equations. The constraints on the design of the nonuniform grid are discussed, as well as the particle and energy conservation properties. The conditions under which the nonuniformity correction in the weighting coefficients is essential to secure physically meaningful solutions are also analyzed. The proposed scheme is shown to efficiently handle both linear and weakly nonlinear problems and, in addition, its ability to provide solutions to stronger nonlinear situations is demonstrated. Some particular problems in the field of plasma physics (e.g., Coulomb collisions, Compton scattering by an electronic population, and the rf heating and current drive of thermonuclear reactors) are solved in order to illustrate several features, most particularly the usefulness of nonuniform grids in reducing computational effort and in increasing accuracy.
Bounce-averaged Fokker-Planck code for stellarator transport
Mynick, H.E.; Hitchon, W.N.G.
1985-07-01
A computer code for solving the bounce-averaged Fokker-Planck equation appropriate to stellarator transport has been developed, and its first applications made. The code is much faster than the bounce-averaged Monte-Carlo codes, which up to now have provided the most efficient numerical means for studying stellarator transport. Moreover, because the connection to analytic kinetic theory of the Fokker-Planck approach is more direct than for the Monte-Carlo approach, a comparison of theory and numerical experiment is now possible at a considerably more detailed level than previously.
NASA Astrophysics Data System (ADS)
Karpov, S. A.; Potapenko, I. F.
2015-10-01
A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.
Karpov, S. A.; Potapenko, I. F.
2015-10-15
A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.
NASA Astrophysics Data System (ADS)
Bianucci, Marco
2015-05-01
In this paper using a projection approach and defining the adjoint-Lie time evolution of differential operators, that generalizes the ordinary time evolution of functions, we obtain a Fokker-Planck equation for the distribution function of a part of interest of a large class of dynamical systems. The main assumptions are the weak interaction between the part of interest and the rest of the system (typically non linear) and the average linear response to external perturbations of the irrelevant part. We do not use ad hoc statistical assumptions to introduce as given a priori phenomenological equilibrium or transport coefficients. The drift terms induced by the interaction with the irrelevant part is obtained with a procedure that is reminiscent of that developed some years ago by Bianucci and Grigolini (see for example (Bianucci et al 1995 Phys. Rev. E 51 3002)) to derive in a ‘genuine’ way thermodynamics and statistical mechanics of macroscopic variables of interest starting from microscopic dynamics. However here we stay in a more broad and formal context where the system of interest could be dissipative and the interaction between the two systems could be non Hamiltonian, thus the approach of the cited paper can not be used to obtain the diffusion part of the Fokker-Planck equation. To face the problem of dealing with the series of differential operators stemming from the projection approach applied to this general case, we introduce the formalism of the Lie derivative and the corresponding adjoint-Lie time evolution of differential operators. In this theoretical framework we are able to obtain well defined analytic functions both for the drift and the diffusion coefficients of the Fokker-Planck equation. We think that the basic elements of Lie algebra introduced in our projection approach can be useful to achieve even more general and more formally elegant results than those here presented. Thus we consider this paper as a first step of this formal path to
Prinja, A.K.
1995-08-01
We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S{sub N} angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes.
Fokker Planck theory for energetic electron deposition in laser fusion
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2014-10-01
We have developed a Fokker Planck model to calculate the transport and deposition of energetic electrons, produced for instance by the two plasmon decay instability at the quarter critical surface. In steady state, the Fokker Planck equation reduces to a single universal equation in energy and space, an equation which appears to be quite simple, but which has a rather unconventional boundary condition. The equation is equally valid in planar and spherical geometry, and it depends on only a single parameter, the charge state Z. Hence one can solve for a universal solution, valid for each Z. An asymptotic solution to this equation will be presented, which allows the heating of the main plasma to be calculated from a simple analytical expression. A more accurate solution in terms of a Bessel function expansion will also be presented. From this, one obtains a heating rate which can be simply incorporated into fluid simulations.
Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model
NASA Astrophysics Data System (ADS)
Wu, Kung-Chien
2015-09-01
In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543-1608, 2004) for Boltzmann equation, but the Fokker-Planck term in this paper creates some technical difficulties.
NASA Technical Reports Server (NTRS)
Englert, G. W.
1971-01-01
A model of the random walk is formulated to allow a simple computing procedure to replace the difficult problem of solution of the Fokker-Planck equation. The step sizes and probabilities of taking steps in the various directions are expressed in terms of Fokker-Planck coefficients. Application is made to many particle systems with Coulomb interactions. The relaxation of a highly peaked velocity distribution of particles to equilibrium conditions is illustrated.
NASA Astrophysics Data System (ADS)
Taitano, W. T.; Chacón, L.; Simakov, A. N.
2016-08-01
In this study, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth-Fokker-Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, vth (t), and explicitly considers the resulting inertial terms in the Fokker-Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker-Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic vth-ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. In particular, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.
Fokker-Planck modeling of current penetration during electron cyclotron current drive
Merkulov, A.; Westerhof, E.; Schueller, F. C.
2007-05-15
The current penetration during electron cyclotron current drive (ECCD) on the resistive time scale is studied with a Fokker-Planck simulation, which includes a model for the magnetic diffusion that determines the parallel electric field evolution. The existence of the synergy between the inductive electric field and EC driven current complicates the process of the current penetration and invalidates the standard method of calculation in which Ohm's law is simply approximated by j-j{sub cd}={sigma}E. Here it is proposed to obtain at every time step a self-consistent approximation to the plasma resistivity from the Fokker-Planck code, which is then used in a concurrent calculation of the magnetic diffusion equation in order to obtain the inductive electric field at the next time step. A series of Fokker-Planck calculations including a self-consistent evolution of the inductive electric field has been performed. Both the ECCD power and the electron density have been varied, thus varying the well known nonlinearity parameter for ECCD P{sub rf}[MW/m{sup -3}]/n{sub e}{sup 2}[10{sup 19} m{sup -3}] [R. W. Harvey et al., Phys. Rev. Lett 62, 426 (1989)]. This parameter turns out also to be a good predictor of the synergetic effects. The results are then compared with the standard method of calculations of the current penetration using a transport code. At low values of the Harvey parameter, the standard method is in quantitative agreement with Fokker-Planck calculations. However, at high values of the Harvey parameter, synergy between ECCD and E{sub parallel} is found. In the case of cocurrent drive, this synergy leads to the generation of large amounts of nonthermal electrons and a concomitant increase of the electrical conductivity and current penetration time. In the case of countercurrent drive, the ECCD efficiency is suppressed by the synergy with E{sub parallel} while only a small amount of nonthermal electrons is produced.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
Morel, J.E.
1987-01-01
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs.
Dimensional interpolation for nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Fred
2005-09-01
Dimensional interpolation has been used successfully by physicists and chemists to solve the Schroedinger equation for atoms and complex molecules. The same basic idea can be used to solve the Fokker-Planck equation for nonlinear filters. In particular, it is well known (by physicists) that two Schroedinger equations are equivalent to two Fokker-Planck equations. Moreover, we can avoid the Schroedinger equation altogether and use dimensional interpolation directly on the Fokker-Planck equation. Dimensional interpolation sounds like a crazy idea, but it works. We will attempt to make this paper accessible to normal engineers who do not have quantum mechanics for breakfast.
Fokker-Planck description of wealth dynamics and the origin of Pareto's law
NASA Astrophysics Data System (ADS)
Boghosian, Bruce
2014-05-01
The so-called "Yard-Sale Model" of wealth distribution posits that wealth is transferred between economic agents as a result of transactions whose size is proportional to the wealth of the less wealthy agent. In recent work [B. M. Boghosian, Phys. Rev. E89, 042804 (2014)], it was shown that this results in a Fokker-Planck equation governing the distribution of wealth. With the addition of a mechanism for wealth redistribution, it was further shown that this model results in stationary wealth distributions that are very similar in form to Pareto's well-known law. In this paper, a much simpler derivation of that Fokker-Planck equation is presented.
Fokker-Planck approach to the pulse packet propagation in synfire chain.
Câteau, H; Fukai, T
2001-01-01
We applied the Fokker-Planck method to the so-called 'synfire chain' network model and showed how a synchronous population spike (pulse packet) evolves to a narrow pulse packet (width < 1 ms) or fades away, depending on its initial size and width. The results of numerical integration of the Fokker-Planck equation are in good agreement with those of simulations on a network of leaky integrate-and-fire neurons. For a narrow input pulse packet, the integration of the Fokker-Planck equation requires careful numerical treatment. However, we can construct a precise analytical waveform of an output packet, which proves valid for narrow input pulse packets, from the stationary solution to the Fokker-Planck equation and a previously proposed approximate input-output relationship. Our methods enable us also to understand an essential role of the synaptic noise in the evolution, the peculiar temporal evolution of a broader pulse packets, and the irrelevance of the refractory period in determining the waveform of a pulse packet. Furthermore, we elucidate possible functional roles of multiple interactive pulse packets in spatiotemporal information processing, i.e. the association of information and the temporal competition. PMID:11665762
Two temperature gas equilibration model with a Fokker-Planck type collision operator
NASA Astrophysics Data System (ADS)
Méndez, A. R.; Chacón-Acosta, G.; García-Perciante, A. L.
2014-01-01
The equilibration process of a binary mixture of gases with two different temperatures is revisited using a Fokker-Planck type equation. The collision integral term of the Boltzmann equation is approximated by a Fokker-Planck differential collision operator by assuming that one of the constituents can be considered as a background gas in equilibrium while the other species diffuses through it. As a main result the coefficients of the linear term and of the first derivative are modified by the temperature and kinetic energy difference of the two species. These modifications are expected to influence the form of the solution for the distribution function and the corresponding transport equations. When temperatures are equal, the usual result of a Rayleigh gas is recovered.
Simulating transient dynamics of the time-dependent time fractional Fokker-Planck systems
NASA Astrophysics Data System (ADS)
Kang, Yan-Mei
2016-09-01
For a physically realistic type of time-dependent time fractional Fokker-Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker-Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.
Fokker Planck and Krook theory for energetic electron deposition in laser fusion
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2015-11-01
We have developed a Fokker Planck and Krook model to calculate the transport and deposition of energetic electrons, produced for instance by the two plasmon decay instability at the quarter critical surface of a laser produced plasma. In steady state, the Fokker Planck equation reduces to a single universal equation in energy and space, an equation which whose asymptotic solution we calculate. The Krook theory also gives rise to an analytic expression solution. From each, one can calculate the spatially dependent heating of the interior plasma, which can be implemented at each time step in a fluid simulation. The equation is equally valid in planar and spherical geometry, and it depends on only a single parameter, the charge state Z. Hence one can solve for a universal solution, valid for each Z. the two approaches will be compared and discussed. We look to cooperate with anyone having a more advanced simulation capability, Direct Simulation Monte Carlo or Fokker Planck, who is willing to test our results. Work supported by the NRL Laser fusion program, DOE- NNSA and ONR.
Current dependence of spin torque switching rate based on Fokker-Planck approach
Taniguchi, Tomohiro Imamura, Hiroshi
2014-05-07
The spin torque switching rate of an in-plane magnetized system in the presence of an applied field is derived by solving the Fokker-Planck equation. It is found that three scaling currents are necessary to describe the current dependence of the switching rate in the low-current limit. The dependences of these scaling currents on the applied field strength are also studied.
NASA Astrophysics Data System (ADS)
Frank, T. D.
2008-02-01
We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.
Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
NASA Astrophysics Data System (ADS)
Heinsalu, E.; Patriarca, M.; Goychuk, I.; Hänggi, P.
2007-09-01
We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.
The Fokker-Planck law of diffusion and pattern formation in heterogeneous environments.
Bengfort, Michael; Malchow, Horst; Hilker, Frank M
2016-09-01
We analyze the influence of spatially inhomogeneous diffusion on several common ecological problems. Diffusion is modeled with Fick's law and the Fokker-Planck law of diffusion. We discuss the differences between the two formalisms and when to use either the one or the other. In doing so, we start with a pure diffusion equation, then turn to a reaction-diffusion system with one logistically growing component which invades the spatial domain. We also look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator-prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in case of Fokker-Planck diffusion. PMID:26803768
Fokker Planck and Krook theory of energetic electron transport in a laser produced plasma
Manheimer, Wallace; Colombant, Denis
2015-09-15
Various laser plasma instabilities, such as the two plasma decay instability and the stimulated Raman scatter instability, produce large quantities of energetic electrons. How these electrons are transported and heat the plasma are crucial questions for laser fusion. This paper works out a Fokker Planck and Krook theory for such transport and heating. The result is a set of equations, for which one can find a simple asymptotic approximation for the solution, for the Fokker Planck case, and an exact solution for the Krook case. These solutions are evaluated and compared with one another. They give rise to expressions for the spatially dependent heating of the background plasma, as a function of the instantaneous laser and plasma parameters, in either planar or spherical geometry. These formulas are simple, universal (depending weakly only on the single parameter Z, the charge state), and can be easily be incorporated into a fluid simulation.
Fokker Planck and Krook theory of energetic electron transport in a laser produced plasma
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2015-09-01
Various laser plasma instabilities, such as the two plasma decay instability and the stimulated Raman scatter instability, produce large quantities of energetic electrons. How these electrons are transported and heat the plasma are crucial questions for laser fusion. This paper works out a Fokker Planck and Krook theory for such transport and heating. The result is a set of equations, for which one can find a simple asymptotic approximation for the solution, for the Fokker Planck case, and an exact solution for the Krook case. These solutions are evaluated and compared with one another. They give rise to expressions for the spatially dependent heating of the background plasma, as a function of the instantaneous laser and plasma parameters, in either planar or spherical geometry. These formulas are simple, universal (depending weakly only on the single parameter Z, the charge state), and can be easily be incorporated into a fluid simulation.
Fokker-Planck-DSMC algorithm for simulations of rarefied gas flows
NASA Astrophysics Data System (ADS)
Gorji, M. Hossein; Jenny, Patrick
2015-04-01
A Fokker-Planck based particle Monte Carlo algorithm was devised recently for simulations of rarefied gas flows by the authors [1-3]. The main motivation behind the Fokker-Planck (FP) model is computational efficiency, which could be gained due to the fact that the resulting stochastic processes are continuous in velocity space. This property of the model leads to simulations where the computational cost becomes independent of the Knudsen number (Kn) [3]. However, the Fokker-Planck model which can be seen as a diffusion approximation of the Boltzmann equation, becomes less accurate as Kn increases. In this study we propose a hybrid Fokker-Planck-Direct Simulation Monte Carlo (FP-DSMC) solution method, which is applicable for the whole range of Kn. The objective of this algorithm is to retain the efficiency of the FP scheme at low Kn (Kn ≪ 1) and to employ conventional DSMC at high Kn (Kn ≫ 1). Since the computational particles employed by the FP model represent the same data as in DSMC, the coupling between the two methods is straightforward. The new ingredient is a switching criterion which would ideally result in a hybrid scheme with the efficiency of the FP method and the accuracy of DSMC for the whole Kn-range. Here, we adopt the number of collisions in a given computational cell and for a given time step size as a decision criterion in order to switch between the FP model and DSMC. For assessment of the hybrid algorithm, different test cases including flow impingement and flow expansion through a slit were studied. Both accuracy and efficiency of the model are shown to be excellent for the presented test cases.
NASA Astrophysics Data System (ADS)
Tang, Xian-Zhu; Berk, H. L.; Guo, Zehua; McDevitt, C. J.
2014-03-01
Across a transition layer of disparate plasma temperatures, the high energy tail of the plasma distribution can have appreciable deviations from the local Maxwellian distribution due to the Knudson layer effect. The Fokker-Planck equation for the tail particle population can be simplified in a series of practically useful limiting cases. The first is the approximation of background Maxwellian distribution for linearizing the collision operator. The second is the supra-thermal particle speed ordering of vTi ≪ v ≪ vTe for the tail ions and vTi ≪ vTe ≪ v for the tail electrons. Keeping both the collisional drag and energy scattering is essential for the collision operator to produce a Maxwellian tail distribution. The Fokker-Planck model for following the tail ion distribution for a given background plasma profile is explicitly worked out for systems of one spatial dimension, in both slab and spherical geometry. A third simplification is an expansion of the tail particle distribution using the spherical harmonics, which are eigenfunctions of the pitch angle scattering operator. This produces a set of coupled Fokker-Planck equations that contain energy-dependent spatial diffusion terms in two coordinates (position and energy), which originate from pitch angle scattering in the original Fokker-Planck equation. It is shown that the well-known diffusive Fokker-Planck model is a poor approximation of the two-mode truncation model, which itself has fundamental deficiency compared with the three-mode truncation model. The cause is the lack of even-symmetry representation in pitch dependence in the two-mode truncation model.
Fokker-Planck description for the queue dynamics of large tick stocks
NASA Astrophysics Data System (ADS)
Garèche, A.; Disdier, G.; Kockelkoren, J.; Bouchaud, J.-P.
2013-09-01
Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. “Jump” events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.
Vlasov-Fokker-Planck Simulation of a Collisional Ion-Electron Shockwave
NASA Astrophysics Data System (ADS)
Taitano, William; Knoll, Dana; Prinja, Anil
2012-10-01
There has been recent increased interest in a range of kinetic plasma physics phenomena which may be important in simulating ICF pellet performance. [1] have numerically demonstrated the limitations of the classic Spitzer, Braginski fluid closures in collisional plasmas for shockwave problems. [1] has shown the importance of modeling kinetic effects for scale lengths of shockwave much larger than the ion collision mean free path. In [1], the ions were modeled kinetically using the Fokker-Planck approximation while the electrons were modeled as a fluid. An investigation of a full kinetic treatment of electron with collision is computationally intractable with standard explicit schemes due to collision CFL limitation that requires resolving the electron-electron collision timescale. [2] has developed a new, fully implicit and discretely consistent moment based accelerator method to solve the full ion-electron kinetic Vlasov-Ampere system. A similar moment based accelerator will be extended to a collisionless shock problem in order to accelerate the Fokker-Planck collision source in the kinetic equations. In the presentation, we provide some preliminary results. [4pt] [1] M. Casanova and O. Larroche, Phys. Rev. Let. 67-(16), 1991. [0pt] [2] W.T. Taitano et al. SISC in review.
A fractional Fokker-Planck model for anomalous diffusion
Anderson, Johan; Kim, Eun-jin; Moradi, Sara
2014-12-15
In this paper, we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.
Mapping the Monte Carlo scheme to Langevin dynamics: a Fokker-Planck approach.
Cheng, X Z; Jalil, M B A; Lee, Hwee Kuan; Okabe, Yutaka
2006-02-17
We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors alpha, unlike previous time-quantified MC methods which break down at low alpha, where precessional motion dominates. PMID:16606044
Mapping the Monte Carlo Scheme to Langevin Dynamics: A Fokker-Planck Approach
NASA Astrophysics Data System (ADS)
Cheng, X. Z.; Jalil, M. B.; Lee, Hwee Kuan; Okabe, Yutaka
2006-02-01
We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors α, unlike previous time-quantified MC methods which break down at low α, where precessional motion dominates.
A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas
Xiong, Z; Cohen, R H; Rognlien, T D; Xu, X Q
2007-04-18
A high-order finite volume algorithm is developed for the Fokker-Planck Operator (FPO) describing Coulomb collisions in strongly magnetized plasmas. The algorithm is based on a general fourth-order reconstruction scheme for an unstructured grid in the velocity space spanned by parallel velocity and magnetic moment. The method provides density conservation and high-order-accurate evaluation of the FPO independent of the choice of the velocity coordinates. As an example, a linearized FPO in constant-of-motion coordinates, i.e. the total energy and the magnetic moment, is developed using the present algorithm combined with a cut-cell merging procedure. Numerical tests include the Spitzer thermalization problem and the return to isotropy for distributions initialized with velocity space loss cones. Utilization of the method for a nonlinear FPO is straightforward but requires evaluation of the Rosenbluth potentials.
NASA Technical Reports Server (NTRS)
Hizanidis, Kyriakos
1989-01-01
The relativistic motion of electrons in an intense electromagnetic wave packet propagating obliquely to a uniform magnetic field is analytically studied on the basis of the Fokker-Planck-Kolmogorov (FPK) approach. The wavepacket consists of circularly polarized electron-cyclotron waves. The dynamical system in question is shown to be reducible to one with three degrees of freedom. Within the framework of the Hamiltonian analysis the nonlinear diffusion tensor is derived, and it is shown that this tensor can be separated into zeroth-, first-, and second-order parts with respect to the relative bandwidth. The zeroth-order part describes diffusive acceleration along lines of constant unperturbed Hamiltonian. The second-order part, which corresponds to the longest time scale, describes diffusion across those lines. A possible transport theory is outlined on the basis of this separation of the time scales.
Nonlinear Kramers equation associated with nonextensive statistical mechanics.
Mendes, G A; Ribeiro, M S; Mendes, R S; Lenzi, E K; Nobre, F D
2015-05-01
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results. PMID:26066118
A Fokker-Planck study of the eigenmodes in an unmagnetized pair plasma
Zhao Bin; Zheng Jian
2007-06-15
Linearized Fokker-Planck equations for unmagnetized pair plasmas are solved as an eigenvalue problem to investigate the sound waves and Langmuir waves. The frequencies and damping rates of sound waves and Langmuir waves as a function of k{lambda} and k{lambda}{sub D} are presented, where k is the wave number, {lambda} is the mean-free path, and {lambda}{sub D} is the Debye length. It is found that no electrostatic field is evolved in the process of sound wave. As a consequence, Landau damping is not relevant to the sound waves in a pair plasma. The damping mechanics of sound waves is fully governed by the Coulomb collisions. The valid regimes of fluid descriptions for the waves are also discussed by comparing with the computational results.
High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications
Duclous, Roland Dubroca, Bruno Filbet, Francis Tikhonchuk, Vladimir
2009-08-01
A high order, deterministic direct numerical method is proposed for the non-relativistic 2D{sub x}x3D{sub v} Vlasov-Maxwell system, coupled with Fokker-Planck-Landau collision operators. The magnetic field is perpendicular to the 2D{sub x} plane surface of computation, whereas the electric fields occur in this plane. Such a system is devoted to modelling of electron transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It is able to describe the kinetics of the plasma electrons in the nonlocal equilibrium regime, and permits to consider a large anisotropy degree of the distribution function. We develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Fast algorithms are employed, which makes this direct approach computationally affordable for simulations of hundreds of collisional times.
Compact Collision Kernels for Hard Sphere and Coulomb Cross Sections; Fokker-Planck Coefficients
Chang Yongbin; Shizgal, Bernie D.
2008-12-31
A compact collision kernel is derived for both hard sphere and Coulomb cross sections. The difference between hard sphere interaction and Coulomb interaction is characterized by a parameter {eta}. With this compact collision kernel, the calculation of Fokker-Planck coefficients can be done for both the Coulomb and hard sphere interactions. The results for arbitrary order Fokker-Planck coefficients are greatly simplified. An alternate form for the Coulomb logarithm is derived with concern to the temperature relaxation in a binary plasma.
Full linearized Fokker-Planck collisions in neoclassical transport simulations
NASA Astrophysics Data System (ADS)
Belli, E. A.; Candy, J.
2012-01-01
The complete linearized Fokker-Planck collision operator has been implemented in the drift-kinetic code NEO (Belli and Candy 2008 Plasma Phys. Control. Fusion 50 095010) for the calculation of neoclassical transport coefficients and flows. A key aspect of this work is the development of a fast numerical algorithm for treatment of the field particle operator. This Eulerian algorithm can accurately treat the disparate velocity scales that arise in the case of multi-species plasmas. Specifically, a Legendre series expansion in ξ (the cosine of the pitch angle) is combined with a novel Laguerre spectral method in energy to ameliorate the rapid numerical precision loss that occurs for traditional Laguerre spectral methods. We demonstrate the superiority of this approach to alternative spectral and finite-element schemes. The physical accuracy and limitations of more commonly used model collision operators, such as the Connor and Hirshman-Sigmar operators, are studied, and the effects on neoclassical impurity poloidal flows and neoclassical transport for experimental parameters are explored.
Híjar, Humberto
2015-02-01
We study the Brownian motion of a particle bound by a harmonic potential and immersed in a fluid with a uniform shear flow. We describe this problem first in terms of a linear Fokker-Planck equation which is solved to obtain the probability distribution function for finding the particle in a volume element of its associated phase space. We find the explicit form of this distribution in the stationary limit and use this result to show that both the equipartition law and the equation of state of the trapped particle are modified from their equilibrium form by terms increasing as the square of the imposed shear rate. Subsequently, we propose an alternative description of this problem in terms of a generalized Langevin equation that takes into account the effects of hydrodynamic correlations and sound propagation on the dynamics of the trapped particle. We show that these effects produce significant changes, manifested as long-time tails and resonant peaks, in the equilibrium and nonequilibrium correlation functions for the velocity of the Brownian particle. We implement numerical simulations based on molecular dynamics and multiparticle collision dynamics, and observe a very good quantitative agreement between the predictions of the model and the numerical results, thus suggesting that this kind of numerical simulations could be used as complement of current experimental techniques. PMID:25768490
Anomalous rotational relaxation: a fractional Fokker-Planck equation approach.
Aydiner, Ekrem
2005-04-01
In this study we have analytically obtained the relaxation function in terms of rotational correlation functions based on Brownian motion for complex disordered systems in a stochastic framework. We found out that the rotational relaxation function has a fractional form for complex disordered systems, which indicates that relaxation has nonexponential character and obeys the Kohlrausch-William-Watts law, following the Mittag-Leffler decay. PMID:15903722
Fokker-Planck description of the scattering of radio frequency waves at the plasma edge
Hizanidis, Kyriakos; Kominis, Yannis; Tsironis, Christos; Ram, Abhay K.
2010-02-15
In magnetic fusion devices, radio frequency (rf) waves in the electron cyclotron (EC) and lower hybrid (LH) range of frequencies are being commonly used to modify the plasma current profile. In ITER, EC waves are expected to stabilize the neoclassical tearing mode (NTM) by providing current in the island region [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)]. The appearance of NTMs severely limits the plasma pressure and leads to the degradation of plasma confinement. LH waves could be used in ITER to modify the current profile closer to the edge of the plasma. These rf waves propagate from the excitation structures to the core of the plasma through an edge region, which is characterized by turbulence--in particular, density fluctuations. These fluctuations, in the form of blobs, can modify the propagation properties of the waves by refraction. In this paper, the effect on rf due to randomly distributed blobs in the edge region is studied. The waves are represented as geometric optics rays and the refractive scattering from a distribution of blobs is formulated as a Fokker-Planck equation. The scattering can have two diffusive effects--one in real space and the other in wave vector space. The scattering can modify the trajectory of rays into the plasma and it can affect the wave vector spectrum. The refraction of EC waves, for example, could make them miss the intended target region where the NTMs occur. The broadening of the wave vector spectrum could broaden the wave generated current profile. The Fokker-Planck formalism for diffusion in real space and wave vector space is used to study the effect of density blobs on EC and LH waves in an ITER type of plasma environment. For EC waves the refractive effects become important since the distance of propagation from the edge to the core in ITER is of the order of a meter. The diffusion in wave vector space is small. For LH waves the refractive effects are insignificant but the diffusion in wave vector space is
Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment
NASA Astrophysics Data System (ADS)
Ge, Wenchao; Rodenburg, Brandon; Bhattacharya, M.
2016-08-01
Optically levitated nanoparticles have recently emerged as versatile platforms for investigating macroscopic quantum mechanics and enabling ultrasensitive metrology. In this paper we theoretically consider two damping regimes of an optically levitated nanoparticle cooled by cavityless parametric feedback. Our treatment is based on a generalized Fokker-Planck equation derived from the quantum master equation presented recently and shown to agree very well with experiment [B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, Quantum model of cooling and force sensing with an optically trapped nanoparticle, Optica 3, 318 (2016), 10.1364/OPTICA.3.000318]. For low damping, we find that the resulting Wigner function yields the single-peaked oscillator position distribution and recovers the appropriate energy distribution derived earlier using a classical theory and verified experimentally [J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state, Nat. Nano. 9, 358 (2014), 10.1038/nnano.2014.40]. For high damping, in contrast, we predict a double-peaked position distribution, which we trace to an underlying bistability induced by feedback. Unlike in cavity-based optomechanics, stochastic processes play a major role in determining the bistable behavior. To support our conclusions, we present analytical expressions as well as numerical simulations using the truncated Wigner function approach. Our work opens up the prospect of developing bistability-based devices, characterization of phase-space dynamics, and investigation of the quantum-classical transition using levitated nanoparticles.
Nonlinear Ehrenfest's urn model.
Casas, G A; Nobre, F D; Curado, E M F
2015-04-01
Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associated entropy is calculated, resulting in a sum of Boltzmann-Gibbs and Tsallis entropic forms. It is shown that the stationary state of the associated Fokker-Planck equation satisfies precisely the same equation obtained by extremization of the entropy. Moreover, the effects of the nonlinear contributions on the entropy production phenomenon are also analyzed. PMID:25974470
NASA Technical Reports Server (NTRS)
Khazanov, George V.; Khabibrakhmanov, Ildar K.; Glocer, Alex
2012-01-01
We present the results of a finite difference implementation of the kinetic Fokker-Planck model with an exact form of the nonlinear collisional operator, The model is time dependent and three-dimensional; one spatial dimension and two in velocity space. The spatial dimension is aligned with the local magnetic field, and the velocity space is defined by the magnitude of the velocity and the cosine of pitch angle. An important new feature of model, the concept of integration along the particle trajectories, is discussed in detail. Integration along the trajectories combined with the operator time splitting technique results in a solution scheme which accurately accounts for both the fast convection of the particles along the magnetic field lines and relatively slow collisional process. We present several tests of the model's performance and also discuss simulation results of the evolution of the plasma distribution for realistic conditions in Earth's plasmasphere under different scenarios.
NASA Astrophysics Data System (ADS)
Mohammadi, Masoumeh; Borzì, Alfio
2016-07-01
The Hermite spectral approximation of a hyperbolic Fokker-Planck (FP) optimality system arising in the control of an unbounded piecewise-deterministic process (PDP) is discussed. To control the probability density function (PDF) corresponding to the PDP process, an optimal control based on an FP strategy is considered. The resulting optimality system consists of a hyperbolic system with opposite-time orientation and an integral optimality condition equation. A Hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains. It is proven that the proposed scheme satisfies the conservativity requirement of the PDFs. The spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments.
Quasilinear simulation of auroral kilometric radiation by a relativistic Fokker-Planck code
Matsuda, Y.
1991-01-01
An intense terrestrial radiation called the auroral kilometric radiation (AKR) is believed to be generated by cyclotron maser instability. We study a quasilinear evolution of this instability by means of a two-dimensional relativistic Fokker-Planck code which treats waves and distributions self-consistently, including radiation loss and electron source and sink. We compare the distributions and wave amplitude with spacecraft observations to elucidate physical processes involved. 3 refs., 1 fig.
Variance reduction for Fokker-Planck based particle Monte Carlo schemes
NASA Astrophysics Data System (ADS)
Gorji, M. Hossein; Andric, Nemanja; Jenny, Patrick
2015-08-01
Recently, Fokker-Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1-3]. In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker-Planck model is considered. First, deviational based schemes were derived and reviewed, and it is shown that these deviational methods are not appropriate for practical Fokker-Planck based rarefied gas flow simulations. This is due to the fact that the deviational schemes considered in this study lead either to instabilities in the case of two-weight methods or to large statistical errors if the direct sampling method is applied. Motivated by this conclusion, we developed a novel scheme based on correlated stochastic processes. The main idea here is to synthesize an additional stochastic process with a known solution, which is simultaneously solved together with the main one. By correlating the two processes, the statistical errors can dramatically be reduced; especially for low Mach numbers. To assess the methods, homogeneous relaxation, planar Couette and lid-driven cavity flows were considered. For these test cases, it could be demonstrated that variance reduction based on parallel processes is very robust and effective.
NASA Astrophysics Data System (ADS)
Lin, XiaoHui; Zhang, ChiBin; Gu, Jun; Jiang, ShuYun; Yang, JueKuan
2014-11-01
A non-continuous electroosmotic flow model (PFP model) is built based on Poisson equation, Fokker-Planck equation and Navier-Stokse equation, and used to predict the DNA molecule translocation through nanopore. PFP model discards the continuum assumption of ion translocation and considers ions as discrete particles. In addition, this model includes the contributions of Coulomb electrostatic potential between ions, Brownian motion of ions and viscous friction to ion transportation. No ionic diffusion coefficient and other phenomenological parameters are needed in the PFP model. It is worth noting that the PFP model can describe non-equilibrium electroosmotic transportation of ions in a channel of a size comparable with the mean free path of ion. A modified clustering method is proposed for the numerical solution of PFP model, and ion current translocation through nanopore with a radius of 1 nm is simulated using the modified clustering method. The external electric field, wall charge density of nanopore, surface charge density of DNA, as well as ion average number density, influence the electroosmotic velocity profile of electrolyte solution, the velocity of DNA translocation through nanopore and ion current blockade. Results show that the ion average number density of electrolyte and surface charge density of nanopore have a significant effect on the translocation velocity of DNA and the ion current blockade. The translocation velocity of DNA is proportional to the surface charge density of nanopore, and is inversely proportional to ion average number density of electrolyte solution. Thus, the translocation velocity of DNAs can be controlled to improve the accuracy of sequencing by adjusting the external electric field, ion average number density of electrolyte and surface charge density of nanopore. Ion current decreases when the ion average number density is larger than the critical value and increases when the ion average number density is lower than the
NASA Astrophysics Data System (ADS)
Dominguez, Efrain; Rosmann, Thomas; Chavarro, John
2014-05-01
In order to extract the mathematical operators that rule complex system behavior, a numeric scheme of the multidimensional Fokker-Planck-Kolmogorov equation is proposed allowing, through conjugate gradient optimization, the identification of deterministic kernels for an observed complex system. This scheme is analyzed using a hydrological basin as example but can be used in many fields. It is assumed that there are observed input-output signals of the system and no especial assumptions about the system kernel are required. This approach can be used at different time resolutions and it is expected to be powerful enough to characterize hydrological variability at different time scales, even under no-stationary conditions. This inverse modeling scheme has three different identification methods, the first one is related to Langevin equations system types, thus random components are described, additively, as noises while in the second method they are represented by the noises intensities instead of noise processes itself. As a result of this inverse modeling approach, hydrological processes can be described as a combination of deterministic kernels and random processes and the system phase space dimensionality can be objectively established. In this work, proposed approach was used to study hydrological variability, trends and extremes at different time resolution.
NASA Astrophysics Data System (ADS)
Taguchi, Masayoshi
1982-05-01
The neoclassical transport coefficients from the plateau regime to the Pfirsch-Schlüter regime are calculated by using the linearized full Fokker-Planck collision operator. The inverse aspect ratio is assumed to be small and the mass ratio of the electron to the ion is much less than unity. The lowest-order terms of these ratios are retained in this calculation. The results obtained give correction to those of Rawls et al. who employed a model collision operator. In the Pfirsch-Schlüter regime, the results of this paper agree fairly well with those given by Hazeltine and Hinton.
Fokker-Planck Modelling of Delayed Loss of Charged Fusion Products in TFTR.
Edenstrasser, J.W.; Goloborod'ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A.; Zweben, S.
1998-08-01
The results of a Fokker-Planck simulation of the ripple-induced loss of charged fusion products in the Tokamak Fusion Test Reactor (TFTR) are presented. It is shown that the main features of the measured "delayed loss" of partially thermalized fusion products, such as the differences between deuterium-deuterium and deuterium-tritium discharges, the plasma current and major radius dependencies, etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR.
Modeling the dynamics of charged-particle beams using a Fokker-Planck approach
Bohn, C.L.; Delayen, J.R.
1994-10-01
At the previous Linac Conference, we introduced a semianalytic Fokker-Planck formalism for calculating the evolution of intense, nonrelativistic, mismatched beams propagating through focusing channels. We have since elaborated on its physical basis and greatly expanded its applicability. In this paper, we implement the model to study the dynamics of a circularly symmetric beam propagating through a linear focusing channel. An example is discussed for an ion beam and accelerator parameters which are representative of high-current spallation neutron sources in which space-charge forces are important. The example illustrates the dynamics of emittance growth and halo formation in the beam.
Coupled Ray-tracing and Fokker-Planck EBW Modeling for Spherical Tokamaks
NASA Astrophysics Data System (ADS)
Urban, J.; Decker, J.; Peysson, Y.; Preinhaelter, J.; Taylor, G.; Vahala, L.; Vahala, G.
2009-11-01
The AMR (Antenna—Mode-conversion—Ray-tracing) code [1, 2] has been recently coupled with the LUKE [3] Fokker-Planck code. This modeling suite is capable of complex simulations of electron Bernstein wave (EBW) emission, heating and current drive. We employ these codes to study EBW heating and current drive performance under spherical tokamak (ST) configurations—typical NSTX discharges are employed. EBW parameters, such as frequency, antenna position and direction, are varied and optimized for particular configurations and objectives. In this way, we show the versatility of EBWs.
NASA Technical Reports Server (NTRS)
Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1975-01-01
The Fokker-Planck coefficient for pitch-angle scattering, appropriate for cosmic rays in homogeneous stationary magnetic turbulence is computed without making any specific assumptions concerning the statistical symmetries of the random field. The Fokker-Planck coefficient obtained can be used to compute the parallel diffusion coefficient for high-energy cosmic rays propagating in the presence of strong turbulence, or for low-energy cosmic rays in the presence of weak turbulence. Because of the generality of magnetic turbulence allowed for in the analysis, special interplanetary magnetic field features, such as discontinuities or particular wave modes, can be included rigorously.
Sloan, D.P.
1983-05-01
Morel (1981) has developed multigroup Legendre cross sections suitable for input to standard discrete ordinates transport codes for performing charged-particle Fokker-Planck calculations in one-dimensional slab and spherical geometries. Since the Monte Carlo neutron transport code, MORSE, uses the same multigroup cross section data that discrete ordinates codes use, it was natural to consider whether Fokker-Planck calculations could be performed with MORSE. In order to extend the unique three-dimensional forward or adjoint capability of MORSE to Fokker-Planck calculations, the MORSE code was modified to correctly treat the delta-function scattering of the energy operator, and a new set of physically acceptable cross sections was derived to model the angular operator. Morel (1979) has also developed multigroup Legendre cross sections suitable for input to standard discrete ordinates codes for performing electron Boltzmann calculations. These electron cross sections may be treated in MORSE with the same methods developed to treat the Fokker-Planck cross sections. The large magnitude of the elastic scattering cross section, however, severely increases the computation or run time. It is well-known that approximate elastic cross sections are easily obtained by applying the extended transport (or delta function) correction to the Legendre coefficients of the exact cross section. An exact method for performing the extended transport cross section correction produces cross sections which are physically acceptable. Sample calculations using electron cross sections have demonstrated this new technique to be very effective in decreasing the large magnitude of the cross sections.
A Fokker-Planck code for laser plasma interaction in femtosecond-laser shock peening
NASA Astrophysics Data System (ADS)
Ren, Zhencheng; Wang, Guo-Xiang; Ye, Chang; Dong, Yalin
2016-03-01
A Fokker-Planck code is developed to simulate the laser-plasma interaction in the femtosecond-laser shock peening and forming processes. A numerical scheme dealing with high-energy concentration and its resulting steep gradient are presented, and the source code is provided as supplementary material for further usage. The breakdown of the classical heat transport theory is observed when the laser intensity increases. The difference in heat flow between the classical theory and simulation is presented. It is found that the classical heat transport theory overestimates heat flow by orders of magnitude during femtosecond-laser shock peening or forming. As a result, the electron pressure can be underestimated using the classical hydrodynamic code.
Conceptual foundation of the Fokker-Planck approach to space-charge effects
Bohn, C.L.
1996-07-01
An rms-mismatched beam can evolve rapidly to a configuration of quasiequilibrium under the influence of space-charge forces. As sit evolves, its emittance grows and a diffuse halo forms. The beam`s distribution function accounts for all the complicated dynamics. Unfortunately, the distribution function is difficult to calculate inasmuch as the physics lies at the interface between classical mechanics and thermodynamics. This paper presents the foundation for a statistical theory of the dynamics of nonequilibrium space-charge-dominated beams. Within certain approximations, the theory takes on a Fokker-Planck form. Key questions arise concerning the nature of the dynamical friction and diffusion in the beam`s phase space and of the quasiequilibrium configuration that ensues.
A method for the analysis of nonlinearities in aircraft dynamic response to atmospheric turbulence
NASA Technical Reports Server (NTRS)
Sidwell, K.
1976-01-01
An analytical method is developed which combines the equivalent linearization technique for the analysis of the response of nonlinear dynamic systems with the amplitude modulated random process (Press model) for atmospheric turbulence. The method is initially applied to a bilinear spring system. The analysis of the response shows good agreement with exact results obtained by the Fokker-Planck equation. The method is then applied to an example of control-surface displacement limiting in an aircraft with a pitch-hold autopilot.
Non-Markov stochastic processes satisfying equations usually associated with a Markov process
NASA Astrophysics Data System (ADS)
McCauley, J. L.
2012-04-01
There are non-Markov Ito processes that satisfy the Fokker-Planck, backward time Kolmogorov, and Chapman-Kolmogorov equations. These processes are non-Markov in that they may remember an initial condition formed at the start of the ensemble. Some may even admit 1-point densities that satisfy a nonlinear 1-point diffusion equation. However, these processes are linear, the Fokker-Planck equation for the conditional density (the 2-point density) is linear. The memory may be in the drift coefficient (representing a flow), in the diffusion coefficient, or in both. We illustrate the phenomena via exactly solvable examples. In the last section we show how such memory may appear in cooperative phenomena.
Nonstationary probability densities of a class of nonlinear system excited by external colored noise
NASA Astrophysics Data System (ADS)
Qi, LuYuan; Xu, Wei; Gu, XuDong
2012-03-01
This paper deals with the approximate nonstationary probability density of a class of nonlinear vibrating system excited by colored noise. First, the stochastic averaging method is adopted to obtain the averaged Itô equation for the amplitude of the system. The corresponding Fokker-Planck-Kolmogorov equation governing the evolutionary probability density function is deduced. Then, the approximate solution of the Fokker-Planck-Kolmogorov equation is derived by applying the Galerkin method. The solution is expressed as a sum of a series of expansion in terms of a set of proper basis functions with time-depended coefficients. Finally, an example is given to illustrate the proposed procedure. The validity of the proposed method is confirmed by Monte Carlo Simulation.
NASA Technical Reports Server (NTRS)
Fisk, L. A.; Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1973-01-01
For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximation yield the same integral for the Fokker-Planck coefficient for the pitch angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For large pitch angles these errors become significant, and for pitch angles of 90 deg the actual Fokker-Planck coefficient contains a delta function. The implications for these corrections relating cosmic ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.
NASA Technical Reports Server (NTRS)
Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1974-01-01
The Fokker-Planck coefficient for pitch angle scattering, appropriate for cosmic rays in homogeneous, stationary, magnetic turbulence, is computed from first principles. No assumptions are made concerning any special statistical symmetries the random field may have. This result can be used to compute the parallel diffusion coefficient for high energy cosmic rays moving in strong turbulence, or low energy cosmic rays moving in weak turbulence. Becuase of the generality of the magnetic turbulence which is allowed in this calculation, special interplanetary magnetic field features such as discontinuities, or particular wave modes, can be included rigorously. The reduction of this results to previously available expressions for the pitch angle scattering coefficient in random field models with special symmetries is discussed. The general existance of a Dirac delta function in the pitch angle scattering coefficient is demonstrated. It is proved that this delta function is the Fokker-Planck prediction for pitch angle scattering due to mirroring in the magnetic field.
Fokker-Planck simulation of runaway electron generation in disruptions with the hot-tail effect
NASA Astrophysics Data System (ADS)
Nuga, H.; Yagi, M.; Fukuyama, A.
2016-06-01
To study runaway electron generation in disruptions, we have extended the three-dimensional (two-dimensional in momentum space; one-dimensional in the radial direction) Fokker-Planck code, which describes the evolution of the relativistic momentum distribution function of electrons and the induced toroidal electric field in a self-consistent manner. A particular focus is placed on the hot-tail effect in two-dimensional momentum space. The effect appears if the drop of the background plasma temperature is sufficiently rapid compared with the electron-electron slowing down time for a few times of the pre-quench thermal velocity. It contributes to not only the enhancement of the primary runaway electron generation but also the broadening of the runaway electron distribution in the pitch angle direction. If the thermal energy loss during the major disruption is assumed to be isotropic, there are hot-tail electrons that have sufficiently large perpendicular momentum, and the runaway electron distribution becomes broader in the pitch angle direction. In addition, the pitch angle scattering also yields the broadening. Since the electric field is reduced due to the burst of runaway electron generation, the time required for accelerating electrons to the runaway region becomes longer. The longer acceleration period makes the pitch-angle scattering more effective.
Loading, absorption, and Fokker-Planck calculations for upcoming ICRF experiments on ATF
Shepard, T.D.; Carter, M.D.; Goulding, R.H.; Kwon, M.
1989-01-01
ICRF experiments on ATF at the 100-kW level are planned for the current 1989 operating period. These plans include the 2..omega../sub cH/ regime at f/sub RF/ = 28.88 MHz, D(H) at 14.44 MHz, and /sup 4/He(/sup 3/He) and D(/sup 3/He) at 9.63 MHz. ECH target plasmas have n/sub eO/ /approx lt/ 0.15 /times/ 10/sup 20/ m/sup /minus/3/ and B = 0.95 T. The density and temperature profiles obtained are broader than those from 1988, owing to recent field error corrections. The values used for target-plasma parameters in the calculations were taken from initial 1989 ATF data. Loading and absorption calculations have been performed using the 3D RF heating code ORION with a helically symmetric equilibrium, and Fokker-Planck calculations were performed using the steady-state code RFTRANS with two velocity dimensions and one spatial dimension. 6 refs., 3 figs.
A finite volume Fokker-Planck collision operator in constants-of-motion coordinates
NASA Astrophysics Data System (ADS)
Xiong, Z.; Xu, X. Q.; Cohen, B. I.; Cohen, R.; Dorr, M. R.; Hittinger, J. A.; Kerbel, G.; Nevins, W. M.; Rognlien, T.
2006-04-01
TEMPEST is a 5D gyrokinetic continuum code for edge plasmas. Constants of motion, namely, the total energy E and the magnetic moment μ, are chosen as coordinate s because of their advantage in minimizing numerical diffusion in advection operato rs. Most existing collision operators are written in other coordinates; using them by interpolating is shown to be less satisfactory in maintaining overall numerical accuracy and conservation. Here we develop a Fokker-Planck collision operator directly in (E,μ) space usin g a finite volume approach. The (E, μ) grid is Cartesian, and the turning point boundary represents a straight line cutting through the grid that separates the ph ysical and non-physical zones. The resulting cut-cells are treated by a cell-mergin g technique to ensure a complete particle conservation. A two dimensional fourth or der reconstruction scheme is devised to achieve good numerical accuracy with modest number of grid points. The new collision operator will be benchmarked by numerical examples.
Wang, Chi-Jen
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Bizarro, J.P.; Peysson, Y.; Bonoli, P.T.; Carrasco, J.; de Wit, T.D.; Fuchs, V.; Hoang, G.T.; Litaudon, X.; Moreau, D.; Pocheau, C.; Shkarofsky, I.P. )
1993-09-01
A detailed investigation is presented on the ability of combined ray-tracing and Fokker--Planck calculations to predict the hard x-ray (HXR) emission during lower-hybrid (LH) current drive in tokamaks when toroidally induced ray stochasticity is important. A large number of rays is used and the electron distribution function is obtained by self-consistently iterating the appropriate power deposition and Fokker--Planck calculations. It is shown that effects due to radial diffusion of suprathermal electrons and to radiation scattering by the inner wall can be significant. The experimentally observed features of the HXR emission are fairly well predicted, thus confirming that combined ray-tracing and Fokker--Planck codes are capable of correctly modeling the physics of LH current drive in tokamaks.
Heikkinen, J.A. ); Paettikangas, T.J.H. )
1994-09-01
The evolution of a one-dimensional velocity distribution is studied in the presence of a monochromatic large-amplitude periodic force which is turned on adiabatically. The periodic Vlasov-Poisson equations are solved in the presence of a linearized Fokker-Planck collision term. For a constant driving force, the system is found to approach, after transient oscillations, a steady state which is maintained by one wave at the driving frequency. This is in contrast to the result in the absence of collisions where the steady state tends to be supported by several waves. An analytical solution for the steady-state distribution function in the presence of a driven large-amplitude wave is obtained by a Hamiltonian approach. The distribution function is expanded in powers of a small parameter [Gamma] proportional to the collision strength. From the expansion, the zeroth order term is shown to give the space-averaged distribution function correct to first order in [Gamma]. Comparison with the results of the simulations and of the harmonics expansion method shows that the solution estimates the distribution with good accuracy. The plateau in the wave trapping regime is analyzed, and the current driven by the large-amplitude traveling wave is determined.
NASA Astrophysics Data System (ADS)
Joglekar, Archis; Thomas, Alexander; Read, Martin; Kingham, Robert
2014-10-01
Here, we present 2D numerical modeling of near critical density plasma using a fully implicit Vlasov-Fokker-Planck (VFP) code, IMPACTA, with the addition of a ray tracing package. In certain situations, such as those at the critical surface at the walls of a hohlraum, magnetic fields are generated through the crossed temperature and electron density gradients. Modeling shows 0.3 MG fields and the strong heating also results in magnetization of the plasma up to ωτ ~ 5 . In the case without magnetic field generation, the heat flows from the laser heating region are isotropic. Including magnetic fields causes the heat flow to form jets along the wall due to the Righi-Leduc effect. The heating of the wall region causes steeper temperature gradients. This serves as a positive feedback mechanism for the field generation rate resulting in nearly twice the amount of field generated in comparison to the case without magnetic fields over 1 ns. The heat conduction, field generation, and the calculation of other transport quantities, is performed ab-initio due to the nature of the VFP equation set. In order to determine the importance of the kinetic effects from IMPACTA, we perform direct comparison with a classical (Braginskii) transport code with hydrodynamic motion (CTC+). The authors would like to acknowledge DOE Grant #DESC0010621 and Advanced Research Computing, UM-AA.
Tzoufras, M.; Tableman, A.; Tsung, F. S.; Mori, W. B.; Bell, A. R.
2013-05-15
To study the kinetic physics of High-Energy-Density Laboratory Plasmas, we have developed the parallel relativistic 2D3P Vlasov-Fokker-Planck code Oshun. The numerical scheme uses a Cartesian mesh in configuration-space and incorporates a spherical harmonic expansion of the electron distribution function in momentum-space. The expansion is truncated such that the necessary angular resolution of the distribution function is retained for a given problem. Finite collisionality causes rapid decay of the high-order harmonics, thereby providing a natural truncation mechanism for the expansion. The code has both fully explicit and implicit field-solvers and employs a linearized Fokker-Planck collision operator. Oshun has been benchmarked against well-known problems, in the highly kinetic limit to model collisionless relativistic instabilities, and in the hydrodynamic limit to recover transport coefficients. The performance of the code, its applicability, and its limitations are discussed in the context of simple problems with relevance to inertial fusion energy.
NASA Astrophysics Data System (ADS)
Colombant, Denis; Manheimer, Wallace
2008-11-01
The Krook model described in the previous talk has been incorporated into a fluid simulation. These fluid simulations are then compared with Fokker Planck simulations and also with a recent NRL Nike experiment. We also examine several other models for electron energy transport that have been used in laser fusion research. As regards comparison with Fokker Planck simulation, the Krook model gives better agreement than the other models, especially in the time asymptotic limit. As regards the NRL experiment, all models except one give reasonable agreement.
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Mischler, S.; Mouhot, C.
2016-08-01
The aim of the present paper is twofold: 1. We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the "mild perturbation" part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.
Fokker-Planck/Ray Tracing for Electron Bernstein and Fast Wave Modeling in Support of NSTX
Harvey, R. W.
2009-11-12
This DOE grant supported fusion energy research, a potential long-term solution to the world's energy needs. Magnetic fusion, exemplified by confinement of very hot ionized gases, i.e., plasmas, in donut-shaped tokamak vessels is a leading approach for this energy source. Thus far, a mixture of hydrogen isotopes has produced 10's of megawatts of fusion power for seconds in a tokamak reactor at Princeton Plasma Physics Laboratory in New Jersey. The research grant under consideration, ER54684, uses computer models to aid in understanding and projecting efficacy of heating and current drive sources in the National Spherical Torus Experiment, a tokamak variant, at PPPL. The NSTX experiment explores the physics of very tight aspect ratio, almost spherical tokamaks, aiming at producing steady-state fusion plasmas. The current drive is an integral part of the steady-state concept, maintaining the magnetic geometry in the steady-state tokamak. CompX further developed and applied models for radiofrequency (rf) heating and current drive for applications to NSTX. These models build on a 30 year development of rf ray tracing (the all-frequencies GENRAY code) and higher dimensional Fokker-Planck rf-collisional modeling (the 3D collisional-quasilinear CQL3D code) at CompX. Two mainline current-drive rf modes are proposed for injection into NSTX: (1) electron Bernstein wave (EBW), and (2) high harmonic fast wave (HHFW) modes. Both these current drive systems provide a means for the rf to access the especially high density plasma--termed high beta plasma--compared to the strength of the required magnetic fields. The CompX studies entailed detailed modeling of the EBW to calculate the efficiency of the current drive system, and to determine its range of flexibility for driving current at spatial locations in the plasma cross-section. The ray tracing showed penetration into NSTX bulk plasma, relatively efficient current drive, but a limited ability to produce current over the whole
The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models
NASA Astrophysics Data System (ADS)
Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.
1997-05-01
We report a new measurement of the velocity dispersion profile within 1' (3 pc) of the center of the globular cluster M15 (NGC 7078), using long-slit spectra from the 4.2 m William Herschel Telescope at La Palma Observatory. We obtained spatially resolved spectra for a total of 23 slit positions during two observing runs. During each run, a set of parallel slit positions was used to map out the central region of the cluster; the position angle used during the second run was orthogonal to that used for the first. The spectra are centered in wavelength near the Ca II infrared triplet at 8650 Å, with a spectral range of about 450 Å. We determined radial velocities by cross-correlation techniques for 131 cluster members. A total of 32 stars were observed more than once. Internal and external comparisons indicate a velocity accuracy of about 4 km s-1. The velocity dispersion profile rises from about σ = 7.2 +/- 1.4 km s-1 near 1' from the center of the cluster to σ = 13.9 +/- 1.8 km s-1 at 20". Inside of 20", the dispersion remains approximately constant at about 10.2 +/- 1.4 km s-1 with no evidence for a sharp rise near the center. This last result stands in contrast with that of Peterson, Seitzer, & Cudworth who found a central velocity dispersion of 25 +/- 7 km s-1, based on a line-broadening measurement. Our velocity dispersion profile is in good agreement with those determined in the recent studies of Gebhardt et al. and Dubath & Meylan. We have developed a new set of Fokker-Planck models and have fitted these to the surface brightness and velocity dispersion profiles of M15. We also use the two measured millisecond pulsar accelerations as constraints. The best-fitting model has a mass function slope of x = 0.9 (where 1.35 is the slope of the Salpeter mass function) and a total mass of 4.9 × 105 M⊙. This model contains approximately 104 neutron stars (3% of the total mass), the majority of which lie within 6" (0.2 pc) of the cluster center. Since the
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
NASA Technical Reports Server (NTRS)
Hizanidis, Kyriakos; Vlahos, L.; Polymilis, C.
1989-01-01
The relativistic motion of an ensemble of electrons in an intense monochromatic electromagnetic wave propagating obliquely in a uniform external magnetic field is studied. The problem is formulated from the viewpoint of Hamiltonian theory and the Fokker-Planck-Kolmogorov approach analyzed by Hizanidis (1989), leading to a one-dimensional diffusive acceleration along paths of constant zeroth-order generalized Hamiltonian. For values of the wave amplitude and the propagating angle inside the analytically predicted stochastic region, the numerical results suggest that the diffusion probes proceeds in stages. In the first stage, the electrons are accelerated to relatively high energies by sampling the first few overlapping resonances one by one. During that stage, the ensemble-average square deviation of the variable involved scales quadratically with time. During the second stage, they scale linearly with time. For much longer times, deviation from linear scaling slowly sets in.
Neoclassical Transport Including Collisional Nonlinearity
Candy, J.; Belli, E. A.
2011-06-10
In the standard {delta}f theory of neoclassical transport, the zeroth-order (Maxwellian) solution is obtained analytically via the solution of a nonlinear equation. The first-order correction {delta}f is subsequently computed as the solution of a linear, inhomogeneous equation that includes the linearized Fokker-Planck collision operator. This equation admits analytic solutions only in extreme asymptotic limits (banana, plateau, Pfirsch-Schlueter), and so must be solved numerically for realistic plasma parameters. Recently, numerical codes have appeared which attempt to compute the total distribution f more accurately than in the standard ordering by retaining some nonlinear terms related to finite-orbit width, while simultaneously reusing some form of the linearized collision operator. In this work we show that higher-order corrections to the distribution function may be unphysical if collisional nonlinearities are ignored.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Physics-based computational complexity of nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Frederick E.; Huang, Jim
2004-08-01
Our theory is based on the mapping between two Fokker-Planck equations and two Schroedinger equations (see [1] & [2]), which is well known in physics, but which has not been exploited in filtering theory. This theory expands Brockett's Lie algebra homomorphism conjecture for characterizing finite dimensional filters. In particular, the Schroedinger equation generates a group, whereas the Zakai equation (as well as the Fokker-Planck equation) does not, owing to the lack of a smooth inverse. Simple non-pathological low-dimensional linear-Gaussian timeinvariant counterexamples show that Brockett's conjecture does not reliably predict when a nonlinear filtering problem will have an exact finite dimensional solution. That is, there are manifestly finite dimensional filters for estimation problems with infinite dimensional Lie algebras. There are three reasons that the Lie algebraic approach as originally formulated by Brockett is incomplete: (1) the Zakai equation does not generate a group; (2) Lie algebras are coordinate free, whereas separation of variables in PDEs is not coordinate free, and (3) Brockett's theory aims to characterize finite dimensional filters for any initial condition of the Zakai equation, whereas SOV for PDEs generally depends on the initial condition. We will attempt to make this paper accessible to normal engineers who do not have Lie algebras for breakfast.
Comparison of Finite Element Non-Linear Beam Random Response with Experimental Results
NASA Astrophysics Data System (ADS)
Chen, R. R.; Mei, C.; Wolfe, HF
1996-09-01
A finite element formulation combined with the equivalent linearization technique and normal mode method is developed for the non-linear random response of beams subjected to acoustic and thermal loads applied simultaneously. To validate the present formulation and solution procedure, results are compared with the classical continuum solution and the Fokker-Planck-Kolmogorov equation solution. Comparison is also made with experimental data for a pre-stretched clamped beam. Random responses of thermally buckled simply supported beam, clamped beam and simply supported-clamped beam are presented. The comparison of the present simultaneously loaded response with the existing sequentially loaded results shows a significant difference between them.
NASA Astrophysics Data System (ADS)
Chen, Ruixi; Mei, Chuh
1993-04-01
A finite element formulation combined with the equivalent linearization technique and the normal mode method is developed for the study of nonlinear random response of beams subjected to simultaneously applied acoustic and thermal loads. Examples include thermally buckled random response of simply supported beam, clamped-clamped beam and simply supported-clamped beam. To compare and validate the present formulation, results are compared with the solutions from existing sequential load method, and significant difference has been found. Results by classical continuum solution and the solution of Fokker-Planck-Kolmogorov equation are also derived and obtained for comparison.
Keanini, R.G.
2011-04-15
Research Highlights: > Systematic approach for physically probing nonlinear and random evolution problems. > Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. > Organization of near-molecular scale vorticity mediated by hydrodynamic modes. > Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion
NASA Astrophysics Data System (ADS)
Gnesin, Silvano; Goodman, Timothy; Coda, Stefano; Decker, Joan; Peysson, Yves
2010-11-01
The TCV tokamak is equipped with nine electron cyclotron (EC) wave gyrotron/launcher systems: six 0.5 MW in the 2nd harmonic X-mode (X2) and three 0.5 MW in the 3rd harmonic X-mode (X3). TCV experiments have been expressly devised to study the X2/X3 interplay, especially through the dynamics and transport properties of the suprathermal electron population generated primarily by X2 and its influence on the X3 wave absorption. Fokker Planck modeling of X2/X3 TCV experiments with the quasilinear fully relativistic LUKE code, coupled with the C3PO ray-tracing module and the R5X2 bremsstrahlung module, is presented here. Two series of experiments are discussed: 1) X2/X3 synergy when both X2 (82.7 GHz) and X3 (118 GHz) waves are injected into the plasma and 2) X2/X3 synergetic absorption at the same frequency (82.7 GHz). The role of suprathermal electron transport has been investigated by comparing the bremsstrahlung emission measured by a hard X-ray camera with the simulated signal.
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method
NASA Astrophysics Data System (ADS)
Er, GuoKang; Iu, VaiPan
2011-09-01
The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.
Mesh-free adjoint methods for nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Fred
2005-09-01
We apply a new industrial strength numerical approximation, called the "mesh-free adjoint method", to solve the nonlinear filtering problem. This algorithm exploits the smoothness of the problem, unlike particle filters, and hence we expect that mesh-free adjoints are superior to particle filters for many practical applications. The nonlinear filter problem is equivalent to solving the Fokker-Planck equation in real time. The key idea is to use a good adaptive non-uniform quantization of state space to approximate the solution of the Fokker-Planck equation. In particular, the adjoint method computes the location of the nodes in state space to minimize errors in the final answer. This use of an adjoint is analogous to optimal control algorithms, but it is more interesting. The adjoint method is also analogous to importance sampling in particle filters, but it is better for four reasons: (1) it exploits the smoothness of the problem; (2) it explicitly minimizes the errors in the relevant functional; (3) it explicitly models the dynamics in state space; and (4) it can be used to compute a corrected value for the desired functional using the residuals. We will attempt to make this paper accessible to normal engineers who do not have PDEs for breakfast.
NASA Astrophysics Data System (ADS)
Green, P. L.; Worden, K.; Atallah, K.; Sims, N. D.
2012-09-01
This work is concerned with the performance of a single degree of freedom electromagnetic energy harvester when subjected to a broadband white noise base acceleration. First, using the Fokker-Planck-Kolmogorov equation, it is shown that Duffing-type nonlinearities can be used to reduce the size of energy harvesting devices without affecting their power output. This is then verified using the technique of Equivalent Linearisation. Second, it is shown analytically that the optimum load resistance of the device is different to that which is dictated by the principle of impedance matching. This result is then verified experimentally.
Optimal feedback control of strongly non-linear systems excited by bounded noise
NASA Astrophysics Data System (ADS)
Zhu, W. Q.; Huang, Z. L.; Ko, J. M.; Ni, Y. Q.
2004-07-01
A strategy for non-linear stochastic optimal control of strongly non-linear systems subject to external and/or parametric excitations of bounded noise is proposed. A stochastic averaging procedure for strongly non-linear systems under external and/or parametric excitations of bounded noise is first developed. Then, the dynamical programming equation for non-linear stochastic optimal control of the system is derived from the averaged Itô equations by using the stochastic dynamical programming principle and solved to yield the optimal control law. The Fokker-Planck-Kolmogorov equation associated with the fully completed averaged Itô equations is solved to give the response of optimally controlled system. The application and effectiveness of the proposed control strategy are illustrated with the control of cable vibration in cable-stayed bridges and the feedback stabilization of the cable under parametric excitation of bounded noise.
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
NASA Astrophysics Data System (ADS)
Kazakevičius, R.; Ruseckas, J.
2015-11-01
Subdiffusive behavior of one-dimensional stochastic systems can be described by time-subordinated Langevin equations. The corresponding probability density satisfies the time-fractional Fokker-Planck equations. In the homogeneous systems the power spectral density of the signals generated by such Langevin equations has power-law dependency on the frequency with the exponent smaller than 1. In this paper we consider nonhomogeneous systems and show that in such systems the power spectral density can have power-law behavior with the exponent equal to or larger than 1 in a wide range of intermediate frequencies.
Electron dynamics with radiation and nonlinear wigglers
Jowett, J.M.
1986-06-01
The physics of electron motion in storage rings is described by supplementing the Hamiltonian equations of motion with fluctuating radiation reaction forces to describe the effects of synchrotron radiation. This leads to a description of radiation damping and quantum diffusion in single-particle phase-space by means of Fokker-Planck equations. For practical purposes, most storage rings remain in the regime of linear damping and diffusion; this is discussed in some detail with examples, concentrating on longitudinal phase space. However special devices such as nonlinear wigglers may permit the new generation of very large rings to go beyond this into regimes of nonlinear damping. It is shown how a special combined-function wiggler can be used to modify the energy distribution and current profile of electron bunches.
NASA Astrophysics Data System (ADS)
Barrett, John W.; Süli, Endre
2016-07-01
We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Ω in Rd, d = 2, for the density ρ, the velocity u ˜ and the pressure p of the fluid, with an equation of state of the form p (ρ) =cpργ, where cp is a positive constant and γ > 1. The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Süli (2016) [9], which established the existence of global-in-time weak solutions to the system for d ∈ { 2 , 3 } and γ >3/2, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d = 2 and γ > 1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density ρ0 ∈L∞ (Ω) for the continuity equation; a square-integrable initial velocity datum u˜0 for the Navier-Stokes momentum equation; and a nonnegative initial probability density function ψ0
Isostable reduction with applications to time-dependent partial differential equations.
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system. PMID:27575127
Isostable reduction with applications to time-dependent partial differential equations
NASA Astrophysics Data System (ADS)
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.
Nonlinear filtering for spacecraft attitude estimation
NASA Technical Reports Server (NTRS)
Vathsal, S.
1986-01-01
Nonlinear filtering techniques are applied to spacecraft attitude estimation using quaternion parameterization for the attitude kinematics. By replacing the angular velocity vector by the gyro output vector, a state dependent noise vector is introduced in the seven-dimensional system equations. The resulting conditional probability density function from the Ito differential rule is governed by the Fokker Planck partial differential equation which is approximated by the second order mean and covariance differential equations. In order to minimize computer loading, the covariance propagation is carried out in six-dimensional state space using a matrix transformation. The star tracker data is used to update the covariance matrix in the seven-dimensional space. The algorithm is simulated for an earth pointing spacecraft mission, using Monte Carlo samples of gyro and star measurements. The performance of the second order filter is compared with the extended Kalman Filter through several simulation runs and drift rates have been identified.
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
Coupled Particle Transport and Pattern Formation in a Nonlinear Leaky-Box Model
NASA Technical Reports Server (NTRS)
Barghouty, A. F.; El-Nemr, K. W.; Baird, J. K.
2009-01-01
Effects of particle-particle coupling on particle characteristics in nonlinear leaky-box type descriptions of the acceleration and transport of energetic particles in space plasmas are examined in the framework of a simple two-particle model based on the Fokker-Planck equation in momentum space. In this model, the two particles are assumed coupled via a common nonlinear source term. In analogy with a prototypical mathematical system of diffusion-driven instability, this work demonstrates that steady-state patterns with strong dependence on the magnetic turbulence but a rather weak one on the coupled particles attributes can emerge in solutions of a nonlinearly coupled leaky-box model. The insight gained from this simple model may be of wider use and significance to nonlinearly coupled leaky-box type descriptions in general.
The probabilistic solution of stochastic oscillators with even nonlinearity under poisson excitation
NASA Astrophysics Data System (ADS)
Guo, Siu-Siu; Er, Guo-Kang
2012-06-01
The probabilistic solutions of nonlinear stochastic oscillators with even nonlinearity driven by Poisson white noise are investigated in this paper. The stationary probability density function (PDF) of the oscillator responses governed by the reduced Fokker-Planck-Kolmogorov equation is obtained with exponentialpolynomial closure (EPC) method. Different types of nonlinear oscillators are considered. Monte Carlo simulation is conducted to examine the effectiveness and accuracy of the EPC method in this case. It is found that the PDF solutions obtained with EPC agree well with those obtained with Monte Carlo simulation, especially in the tail regions of the PDFs of oscillator responses. Numerical analysis shows that the mean of displacement is nonzero and the PDF of displacement is nonsymmetric about its mean when there is even nonlinearity in displacement in the oscillator. Numerical analysis further shows that the mean of velocity always equals zero and the PDF of velocity is symmetrically distributed about its mean.
On stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems.
Zhu, Wei-qiu; Ying, Zu-guang
2004-11-01
A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed. The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy. PMID:15495321
A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems
NASA Astrophysics Data System (ADS)
Ying, Z. G.; Zhu, W. Q.
2008-02-01
A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems is proposed. The optimal control force consists of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi-Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimate errors of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.
Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems
NASA Astrophysics Data System (ADS)
Feng, Ju; Zhu, Weiqiu; Ying, Zuguang
2010-01-01
The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The response of the controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation and the Riccati equation for the estimated error of system states. As an example to illustrate the procedure and effectiveness of the proposed method, the stochastic optimal control problem of a partially observable two-degree-of-freedom quasi-integrable Hamiltonian system is worked out in detail.
External and parametric random excitation of non-linear offshore systems
Thampi, S.K.
1989-01-01
The development of accurate response prediction methods for nonlinear offshore structures is addressed in this study. The Markov approach is adopted for this purpose and the solution methods are illustrated through applications to deepwater offshore systems which include an oceanographic buoy, fixed jacked structures, marine riser systems and a guyed offshore platform. Gaussian and non-Gaussian response predictions for single and multiple degree of freedom systems are presented and discussed at length. The major difficulties associated with Markov methods in dealing with practical systems are the requirements of white noise excitation and the solution of the Fokker-Planck-Kolmogorov equation. These problems are addressed through the development of dimensionless shaping filters to produce realistic excitation and the use of moment equations to compute response statistics. The application of these techniques to non-linear systems requires additional closure approximations. The solutions are compared with those from linear spectral analysis, stochastic averaging and time domain simulations.
On the conditions for the onset of nonlinear chirping structures in NSTX
NASA Astrophysics Data System (ADS)
Duarte, Vinicius; Podesta, Mario; Berk, Herbert; Gorelenkov, Nikolai
2015-11-01
The nonlinear dynamics of phase space structures is a topic of interest in tokamak physics in connection with fast ion loss mechanisms. The onset of phase-space holes and clumps has been theoretically shown to be associated with an explosive solution of an integro-differential, nonlocal cubic equation that governs the early mode amplitude evolution in the weakly nonlinear regime. The existence and stability of the solutions of the cubic equation have been theoretically studied as a function of Fokker-Planck coefficients for the idealized case of a single resonant point of a localized mode. From realistic computations of NSTX mode structures and resonant surfaces, we calculate effective pitch angle scattering and slowing-down (drag) collisional coefficients and analyze NSTX discharges for different cases with respect to chirping experimental observation. Those results are confronted to the theory that predicts the parameters region that allow for chirping to take place.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
A finite element method for the statistics of non-linear random vibration
NASA Astrophysics Data System (ADS)
Langley, R. S.
1985-07-01
The transitional probability density function for the random response of a certain class of non-linear system satisfies the Fokker-Planck-Kolmogorov equation. This paper concerns the numerical solution of the stationary form of this equation, yielding the stationary probability density function of response. The weighted residual statement for the problem is integrated by parts to yield the weak form of the equations, which are then solved by the finite element method. The method is applied to a Duffing oscillator and good agreement is found with the exact result, and the method is compared favourably with a Galerkin solution method given by Bhandari and Sherrer [1]. Also, the method is applied to the ship rolling problem and good agreement is found with an approximate analytical result due to Roberts [2].
NASA Astrophysics Data System (ADS)
Han, Hua; Ding, Yongsheng; Hao, Kuangrong; Hu, Liangjian
2013-07-01
In this article, we first introduce the problem of state estimation of jump Markov nonlinear systems (JMNSs). Since the density evolution method for predictor equations satisfies Fokker-Planck-Kolmogorov equation (FPKE) in Bayes estimation, the FPKE in conjunction with Bayes' conditional density update formula can provide optimal estimation for a general continuous-discrete nonlinear filtering problem. It is well known that the analytical solution of the FPKE and Bayes' formula is extremely difficult to obtain except a few special cases. Hence, we try to design a particle filter to achieve Bayes estimation of the JMNSs. In order to test the viability of our algorithm, we apply it to multiple targets tracking in video surveillance. Before starting simulation, we introduce the 'birth' and 'death' description of targets, targets' transitional probability model, and observation probability. The experiment results show good performance of our proposed filter for multiple targets tracking.
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
NASA Astrophysics Data System (ADS)
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Nonlinear SCHRÖDINGER-PAULI Equations
NASA Astrophysics Data System (ADS)
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
NASA Astrophysics Data System (ADS)
Jin, X. L.; Huang, Z. L.
The nonstationary probability densities of system responses are obtained for nonlinear multi-degree-of-freedom systems subject to stochastic parametric and external excitations. First, the stochastic averaging method is used to obtain the averaged Itô equation for amplitude envelopes of the system response. Then, the corresponding Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude envelopes is deduced. By applying the Galerkin method, the nonstationary probability density can be expressed as a series expansion in terms of a set of orthogonal base functions with time-dependent coefficients. Finally, the nonstationary probability densities for the amplitude response, as well as those for the state-space response, are solved approximately. To illustrate the applicability, the proposed method is applied to a two-degree-of-freedom van der Pol oscillator subject to external excitations of Gaussian white noises.
NASA Astrophysics Data System (ADS)
Westerhof, E.; Pratt, J.; Ayten, B.
2015-03-01
In the presence of electron cyclotron current drive (ECCD), the Ohm's law of single fluid magnetohydrodynamics (MHD) is modified as E + v × B = η(J - JECCD). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm's law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands.
Effect of background plasma nonlinearities on dissipation processes in plasmas
NASA Astrophysics Data System (ADS)
Nekrasov, F. M.; Elfimov, A. G.; de Azevedo, C. A.; de Assis, A. S.
1999-01-01
The Coulomb collision effect on the bounce-resonance dissipation is considered for toroidal magnetized plasmas. The solution of the Vlasov equation with a simplified Fokker-Planck collision operator is presented. The parallel components of the dielectric tensor are obtained. A collisionless limit of wave dissipation is found.
NASA Astrophysics Data System (ADS)
Tanimura, Yoshitaka; Steffen, Thomas
2000-12-01
The relaxation processes in a quantum system nonlinearly coupled to a harmonic Gaussian-Markovian heat bath are investigated by the quantum Fokker-Planck equation in the hierarchy form. This model describes frequency fluctuations in the quantum system with an arbitrary correlation time and thus bridges the gap between the Brownian oscillator model and the stochastic model by Anderson and Kubo. The effects of the finite correlation time and the system-bath coupling strength are studied for a harmonic model system by numerically integrating the equation of motion. The one-time correlation function of the system coordinate, which is measured in conventional Raman and infrared absorption experiments, already reflects the inhomogeneous character of the relaxation process. The finite correlation time of the frequency fluctuations, however, is directly evident only in the two- and three-time correlation function as probed by multidimensional spectroscopic techniques such as the Raman echo and the fifth-order 2D Raman experiment.
Vibrational spectroscopy of a harmonic oscillator system nonlinearly coupled to a heat bath
NASA Astrophysics Data System (ADS)
Kato, Tsuyoshi; Tanimura, Yoshitaka
2002-10-01
Vibrational relaxation of a harmonic oscillator nonlinearly coupled to a heat bath is investigated by the Gaussian-Markovian quantum Fokker-Planck equation approach. The system-bath interaction is assumed to be linear in the bath coordinate, but linear plus square in the system coordinate modeling the elastic and inelastic relaxation mechanisms. Interplay of the two relaxation processes induced by the linear-linear and square-linear interactions in Raman or infrared spectra is discussed for various system-bath couplings, temperatures, and correlation times for the bath fluctuations. The one-quantum coherence state created through the interaction with the pump laser pulse relaxes through different pathways in accordance with the mechanisms of the system-bath interactions. Relations between the present theory, Redfield theory, and stochastic theory are also discussed.
An exact solution to a certain non-linear random vibration problem
NASA Astrophysics Data System (ADS)
Dimentberg, M. F.
A single-degree-of-freedom system with a special type of nonlinear damping and both external and parametric white-noise excitations is considered. For the special case, when the intensities of coordinates and velocity modulation satisfy a certain condition an exact analytical solution is obtained to the corresponding stationary Fokker-Planck-Kolmogorov equation yielding an expression for joint probability density of coordinate and velocity. This solution is analyzed particularly in connection with stochastic stability problem for the corresponding linear system; certain implications are illustrated for the system, which is stable with respect to probability but unstable in the mean square. The solution obtained may be used to check different approximate methods for analysis of systems with randomly varying parameters.
Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states
2011-01-01
Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter. AMS Subject Classification: 35K60, 82C31, 92B20. PMID:22657097
NASA Astrophysics Data System (ADS)
Shinozaki, Takashi; Okada, Masato; Reyes, Alex D.; Câteau, Hideyuki
2010-01-01
Intermingled neural connections apparent in the brain make us wonder what controls the traffic of propagating activity in the brain to secure signal transmission without harmful crosstalk. Here, we reveal that inhibitory input but not excitatory input works as a particularly useful traffic controller because it controls the degree of synchrony of population firing of neurons as well as controlling the size of the population firing bidirectionally. Our dynamical system analysis reveals that the synchrony enhancement depends crucially on the nonlinear membrane potential dynamics and a hidden slow dynamical variable. Our electrophysiological study with rodent slice preparations show that the phenomenon happens in real neurons. Furthermore, our analysis with the Fokker-Planck equations demonstrates the phenomenon in a semianalytical manner.
Harvey, R. W.; Petrov, Yu. V.
2013-12-03
Within the US Department of Energy/Office of Fusion Energy magnetic fusion research program, there is an important whole-plasma-modeling need for a radio-frequency/neutral-beam-injection (RF/NBI) transport-oriented finite-difference Fokker-Planck (FP) code with combined capabilities for 4D (2R2V) geometry near the fusion plasma periphery, and computationally less demanding 3D (1R2V) bounce-averaged capabilities for plasma in the core of fusion devices. Demonstration of proof-of-principle achievement of this goal has been carried out in research carried out under Phase I of the SBIR award. Two DOE-sponsored codes, the CQL3D bounce-average Fokker-Planck code in which CompX has specialized, and the COGENT 4D, plasma edge-oriented Fokker-Planck code which has been constructed by Lawrence Livermore National Laboratory and Lawrence Berkeley Laboratory scientists, where coupled. Coupling was achieved by using CQL3D calculated velocity distributions including an energetic tail resulting from NBI, as boundary conditions for the COGENT code over the two-dimensional velocity space on a spatial interface (flux) surface at a given radius near the plasma periphery. The finite-orbit-width fast ions from the CQL3D distributions penetrated into the peripheral plasma modeled by the COGENT code. This combined code demonstrates the feasibility of the proposed 3D/4D code. By combining these codes, the greatest computational efficiency is achieved subject to present modeling needs in toroidally symmetric magnetic fusion devices. The more efficient 3D code can be used in its regions of applicability, coupled to the more computationally demanding 4D code in higher collisionality edge plasma regions where that extended capability is necessary for accurate representation of the plasma. More efficient code leads to greater use and utility of the model. An ancillary aim of the project is to make the combined 3D/4D code user friendly. Achievement of full-coupling of these two Fokker-Planck
NASA Astrophysics Data System (ADS)
Frank, T. D.
2007-01-01
We present a generalized Kramers Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker Planck equation derived earlier in the literature is a special case of the proposed Kramers Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed.
Generalization of the diffusion equation by using the maximum entropy principle
NASA Astrophysics Data System (ADS)
Jumarie, Guy
1985-06-01
By using the so-called maximum entropy principle in information theory, one derives a generalization of the Fokker-Planck-Kolmogorov equation which applies when the n first transition moments of the process are proportional to Δt, while the other ones can be neglected.
Engineering integrable nonautonomous nonlinear Schroedinger equations
He Xugang; Zhao Dun; Li Lin; Luo Honggang
2009-05-15
We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schroedinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
Stochastic inflation and nonlinear gravity
NASA Astrophysics Data System (ADS)
Salopek, D. S.; Bond, J. R.
1991-02-01
We show how nonlinear effects of the metric and scalar fields may be included in stochastic inflation. Our formalism can be applied to non-Gaussian fluctuation models for galaxy formation. Fluctuations with wavelengths larger than the horizon length are governed by a network of Langevin equations for the physical fields. Stochastic noise terms arise from quantum fluctuations that are assumed to become classical at horizon crossing and that then contribute to the background. Using Hamilton-Jacobi methods, we solve the Arnowitt-Deser-Misner constraint equations which allows us to separate the growing modes from the decaying ones in the drift phase following each stochastic impulse. We argue that the most reasonable choice of time hypersurfaces for the Langevin system during inflation is T=ln(Ha), where H and a are the local values of the Hubble parameter and the scale factor, since T is the natural time for evolving the short-wavelength scalar field fluctuations in an inhomogeneous background. We derive a Fokker-Planck equation which describes how the probability distribution of scalar field values at a given spatial point evolves in T. Analytic Green's-function solutions obtained for a single scalar field self-interacting through an exponential potential are used to demonstrate (1) if the initial condition of the Hubble parameter is chosen to be consistent with microwave-background limits, H(φ0)/mρ<~10-4, then the fluctuations obey Gaussian statistics to a high precision, independent of the time hypersurface choice and operator-ordering ambiguities in the Fokker-Planck equation, and (2) for scales much larger than our present observable patch of the Universe, the distribution is non-Gaussian, with a tail extending to large energy densities; although there are no observable manifestations, it does show eternal inflation. Lattice simulations of our Langevin network for the exponential potential demonstrate how spatial correlations are incorporated. An initially
Spurious Solutions Of Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Yee, H. C.; Sweby, P. K.; Griffiths, D. F.
1992-01-01
Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.
Numerical methods for high-dimensional probability density function equations
NASA Astrophysics Data System (ADS)
Cho, H.; Venturi, D.; Karniadakis, G. E.
2016-01-01
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
Nonlinear quantum equations: Classical field theory
Rego-Monteiro, M. A.; Nobre, F. D.
2013-10-15
An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schrödinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q→ 1. The main characteristic of this field theory consists on the fact that besides the usual Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 4}, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
The quasicontinuum Fokker-Plank equation
Alexander, Francis J
2008-01-01
We present a regularized Fokker-Planck equation with more accurate short-time and high-frequency behavior for continuous-time, discrete-state systems. The regularization preserves crucial aspects of state-space discreteness lost in the standard Kramers-Moyal expansion. We apply the method to a simple example of biochemical reaction kinetics and to a two-dimensional symmetric random walk, and suggest its application to more complex systerns.
A Stochastic Differential Equation Approach To Multiphase Flow In Porous Media
NASA Astrophysics Data System (ADS)
Dean, D.; Russell, T.
2003-12-01
The motivation for using stochastic differential equations in multiphase flow systems stems from our work in developing an upscaling methodology for single phase flow. The long term goals of this project include: I. Extending this work to a nonlinear upscaling methodology II. Developing a macro-scale stochastic theory of multiphase flow and transport that accounts for micro-scale heterogeneities and interfaces. In this talk, we present a stochastic differential equation approach to multiphase flow, a typical example of which is flow in the unsaturated domain. Specifically, a two phase problem is studied which consists of a wetting phase and a non-wetting phase. The approach given results in a nonlinear stochastic differential equation describing the position of the non-wetting phase fluid particle. Our fundamental assumption is that the flow of fluid particles is described by a stochastic process and that the positions of the fluid particles over time are governed by the law of the process. It is this law which we seek to determine. The nonlinearity in the stochastic differential equation arises because both the drift and diffusion coefficients depend on the volumetric fraction of the phase which in turn depends on the position of the fluid particles in the experimental domain. The concept of a fluid particle is central to the development of the model described in this talk. Expressions for both saturation and volumetric fraction are developed using the fluid particle concept. Darcy's law and the continuity equation are then used to derive a Fokker-Planck equation using these expressions. The Ito calculus is then applied to derive a stochastic differential equation for the non-wetting phase. This equation has both drift and diffusion terms which depend on the volumetric fraction of the non-wetting phase. Standard stochastic theories based on the Ito calculus and the Wiener process and the equivalent Fokker-Planck PDE's are typically used to model dispersion
Markovian master equation for nonlinear systems
NASA Astrophysics Data System (ADS)
de los Santos-Sánchez, O.; Récamier, J.; Jáuregui, R.
2015-06-01
Within the f-deformed oscillator formalism, we derive a Markovian master equation for the description of the damped dynamics of nonlinear systems that interact with their environment. The applicability of this treatment to the particular case of a Morse-like oscillator interacting with a thermal field is illustrated, and the decay of quantum coherence in such a system is analyzed in terms of the evolution on phase space of its nonlinear coherent states via the Wigner function.
Algorithms For Integrating Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
Numerical Methods for Stochastic Partial Differential Equations
Sharp, D.H.; Habib, S.; Mineev, M.B.
1999-07-08
This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
Quantum nonlinear Schrodinger equation on a lattice
Bogolyubov, N.M.; Korepin, V.E.
1986-09-01
A local Hamiltonian is constructed for the nonlinear Schrodinger equation on a lattice in both the classical and the quantum variants. This Hamiltonian is an explicit elementary function of the local Bose fields. The lattice model possesses the same structure of the action-angle variables as the continuous model.
NASA Astrophysics Data System (ADS)
Thomson, Mark J.; McKellar, Bruce H. J.
1991-04-01
A simple, non-linear generalization of the MSW equation is presented and its analytic solution is outlined. The orbits of the polarization vector are shown to be periodic, and to lie on a sphere. Their non-trivial flow patterns fall into two topological categories, the more complex of which can become chaotic if perturbed.
Two coupled nonlinear cavities in a driven-dissipative environment
NASA Astrophysics Data System (ADS)
Cao, Bin; Mahmud, Khan; Hafezi, Mohammad
We investigate two coupled nonlinear cavities that are driven coherently in a dissipative environment. This is the simplest setting containing a good number of features of an array of coupled cavity quantum simulator with Kerr nonlinearity which gives rise to many strongly correlated phases. We find analytical solution for the steady state using the generalized P representation and expressing the master equation in the form of Fokker-Planck equation. A comparison shows a good match of the analytical and numerical solutions across different regimes. We investigate the quantum correlations in the steady state by solving the full master equation numerically, analyzing its second-order coherence, entanglment entropy and Liouvillian gap as a function of drive and detuning. This gives us insights into the nature of bistability and how the tunneling-induced bistability emerges in coupled cavities when going beyond a single cavity. We can understand much of the semiclassical physics in terms of the underlying phase space dynamics of a driven and damped classical pendulum. Furthermore, in the semiclassical analysis, we find steady state solutions with different number density in the two wells that can be considered an analog of double well self-trapped states.
Non-equilibrium effects upon the non-Markovian Caldeira-Leggett quantum master equation
Bolivar, A.O.
2011-05-15
Highlights: > Classical Brownian motion described by a non-Markovian Fokker-Planck equation. > Quantization process. > Quantum Brownian motion described by a non-Markovian Caldeira-Leggett equation. > A non-equilibrium quantum thermal force is predicted. - Abstract: We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker-Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira-Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.
Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Mihaila, Bogdan; Saxena, Avadh
2010-09-01
We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction g{2}/k+1(ΨΨ){k+1} , as well as a vector-vector self interaction g{2}/k+1(Ψγ{μ}ΨΨγ{μ}Ψ){1/2(k+1)} . We find the exact analytic form for solitary waves for arbitrary k and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the 1/2m correction to the NLSE, valid when |ω-m|<2m , where ω is the frequency of the solitary wave in the rest frame. We discuss the stability and blowup of solitary waves assuming the modified NLSE is valid and find that they should be stable for k<2 . PMID:21230200
Taming the nonlinearity of the Einstein equation.
Harte, Abraham I
2014-12-31
Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically interesting cases where metrics become approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations that directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well. PMID:25615299
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
NASA Technical Reports Server (NTRS)
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
Forces Associated with Nonlinear Nonholonomic Constraint Equations
NASA Technical Reports Server (NTRS)
Roithmayr, Carlos M.; Hodges, Dewey H.
2010-01-01
A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.
Transport equations for low-energy solar particles in evolving interplanetary magnetic fields
NASA Technical Reports Server (NTRS)
Ng, C. K.
1988-01-01
Two new forms of a simplified Fokker-Planck equation are derived for the transport of low-energy solar energetic particles in an evolving interplanetary magnetic field, carried by a variable radial solar wind. An idealized solution suggests that the 'invariant' anisotropy direction reported by Allum et al. (1974) may be explained within the conventional theoretical framework. The equations may be used to relate studies of solar particle propagation to solar wind transients, and vice versa.
Nonlocal growth equations-a test case for dynamic renormalization group analysis
NASA Astrophysics Data System (ADS)
Schwartz, Moshe; Katzav, Eytan
2003-12-01
In this paper we discuss nonlocal growth equations such as the generalization of the Kardar-Parisi-Zhang (KPZ) equation that includes long-range interactions, also known as the Nonlocal-Kardar-Parisi-Zhang (NKPZ) equation, and the nonlocal version of the molecular-beam-epitaxy (NMBE) equation. We show that the steady-state strong coupling solution for nonlocal models such as NKPZ and NMBE can be obtained exactly in one dimension for some special cases, using the Fokker-Planck form of these equations. The exact results we derive do not agree with previous results obtained by Dynamic Renormalization Group (DRG) analysis. This discrepancy is important because DRG is a common method used extensively to deal with nonlinear field equations. While difficulties with this method for d>1 has been realized in the past, it has been believed so far that DRG is still safe in one dimension. Our result shows differently. The reasons for the failure of DRG to recover the exact one-dimensional results are also discussed.
Dark soliton solutions of (N+1)-dimensional nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Demiray, Seyma Tuluce; Bulut, Hasan
2016-06-01
In this study, we investigate exact solutions of (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation by using generalized Kudryashov method. (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation can be returned to nonlinear ordinary differential equation by suitable transformation. Then, generalized Kudryashov method has been used to seek exact solutions of the (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation. Also, we obtain dark soliton solutions for these (N+1)-dimensional nonlinear evolution equations. Finally, we denote that this method can be applied to solve other nonlinear evolution equations.
Using the scalable nonlinear equations solvers package
Gropp, W.D.; McInnes, L.C.; Smith, B.F.
1995-02-01
SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of large-scale systems of nonlinear equations on both uniprocessors and parallel architectures. SNES also contains a component for the solution of unconstrained minimization problems, called SUMS (Scalable Unconstrained Minimization Solvers). Newton-like methods, which are known for their efficiency and robustness, constitute the core of the package. As part of the multilevel PETSc library, SNES incorporates many features and options from other parts of PETSc. In keeping with the spirit of the PETSc library, the nonlinear solution routines are data-structure-neutral, making them flexible and easily extensible. This users guide contains a detailed description of uniprocessor usage of SNES, with some added comments regarding multiprocessor usage. At this time the parallel version is undergoing refinement and extension, as we work toward a common interface for the uniprocessor and parallel cases. Thus, forthcoming versions of the software will contain additional features, and changes to parallel interface may result at any time. The new parallel version will employ the MPI (Message Passing Interface) standard for interprocessor communication. Since most of these details will be hidden, users will need to perform only minimal message-passing programming.
NASA Astrophysics Data System (ADS)
Er, Guo-Kang
2014-04-01
In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker-Planck-Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler-Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.
The beam equation with nonlinear memory
NASA Astrophysics Data System (ADS)
D'Abbicco, Marcello; Lucente, Sandra
2016-06-01
In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., {u_{tt}+Δ^2u = F(t, u)}, where F = intlimits0tf(t - s)N(u)(s, x) {d}s, quad N(u)≈ |u|^p. For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., {F = N(u)}.
Exact and explicit solitary wave solutions to some nonlinear equations
Jiefang Zhang
1996-08-01
Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative {Phi}{sup 4}-model equation, the generalized Fisher equation, and the elastic-medium wave equation.
FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations
NASA Astrophysics Data System (ADS)
Ibragimov, N. H.; Torrisi, M.; Tracinà, R.
2010-11-01
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.
Solution spectrum of nonlinear diffusion equations
Ulmer, W.
1992-08-01
The stationary version of the nonlinear diffusion equation -{partial_derivative}c/{partial_derivative}t+D{Delta}c=A{sub 1}c-A{sub 2}c{sup 2} can be solved with the ansatz c={summation}{sub p=1}{sup {infinity}} A{sub p}(cosh kx){sup -p}, inducing a band structure with regard to the ratio {lambda}{sub 1}/{lambda}{sub 2}. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c={summation}{sub p,q=1}{sup infinity}A{sub pa}(cosh kx){sup -p}[(cosh {alpha}t){sup -q-1} sinh {alpha}t+b(cosh {alpha}t){sup -q}] represents a solution spectrum of the time-dependent nonlinear equations, and the stationary version can be found from the asymptotic behaviour of the expansion. The solutions can be associated with reactive processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheriods. 33 refs., 1 tab.
Nonlinear scattering term in the gyrokinetic Vlasov equation
Wang, Shaojie
2013-08-15
Nonlinear scattering term is found from the nonlinear gyrokinetic equation by decoupling the perturbed gyrocenter motion from the unperturbed motion. The gyro-center distribution function is determined by the well-understood unperturbed motion, with the effects of fields perturbation included in the nonlinear scattering term, which explicitly reveals the nonlinear stochastic dissipation on the time scale longer than the wave correlation time.
Westerhof, E. Pratt, J.
2014-10-15
In the presence of electron cyclotron current drive (ECCD), the Ohm's law of single fluid magnetohydrodynamics is modified as E + v × B = η(J – J{sub EC}). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm's law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands. The new model contains the model put forward by Giruzzi et al. [Nucl. Fusion 39, 107 (1999)] in one of its limits.
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail. PMID:23509385
Forced nonlinear Schrödinger equation with arbitrary nonlinearity
NASA Astrophysics Data System (ADS)
Cooper, Fred; Khare, Avinash; Quintero, Niurka R.; Mertens, Franz G.; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction (g2)/(κ+1)(ψψ)κ+1 in the presence of the external forcing terms of the form re-i(kx+θ)-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where vk=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq˙(t)<0, where p(t) is the normalized canonical momentum p(t)=(1)/(M(t))(∂L)/(∂q˙), and q˙(t) is the solitary wave velocity. Here M(t)=∫dxψ(x,t)ψ(x,t). Stability is also studied using a “phase portrait” of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.
Castellano, Claudio; Muñoz, Miguel A; Pastor-Satorras, Romualdo
2009-10-01
We introduce a nonlinear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have a unanimous opinion, still a voter can flip its state with probability epsilon . We solve the model on a fully connected network (i.e., in mean field) and compute the exit probability as well as the average time to reach consensus by employing the backward Fokker-Planck formalism and scaling arguments. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ( Z2-symmetric) absorbing states. In particular, by deriving explicitly the coefficients of such a Langevin equation as a function of the microscopic flipping probabilities, we find that in mean field the q-voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other: (i) a unique (generalized-voter-like) transition, (ii) a series of two consecutive transitions, one (Ising-like) in which the Z2 symmetry is broken and a separate one (in the directed-percolation class) in which the system falls into an absorbing state, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a type of ordering dynamics emerges, is rationalized and found to be specific of mean field, i.e., fluctuations are explicitly shown to wash it out in spatially extended systems. PMID:19905295
Capillary waves in the subcritical nonlinear Schroedinger equation
Kozyreff, G.
2010-01-15
We expand recent results on the nonlinear Schroedinger equation with cubic-quintic nonlinearity to show that some solutions are described by the Bernoulli equation in the presence of surface tension. As a consequence, capillary waves are predicted and found numerically at the interface between regions of large and low amplitude.
Forced nonlinear Schrödinger equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Quintero, Niurka R; Mertens, Franz G; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave. PMID:22680598
Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2015-10-01
The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis. PMID:27347461
Stochastic differential equations for non-linear hydrodynamics
NASA Astrophysics Data System (ADS)
Español, Pep
1998-02-01
We formulate the stochastic differential equations for non-linear hydrodynamic fluctuations. The equations incorporate the random forces through a random stres tensor and random heat flux as in the Landau and Lifshitz theory. However, the equations are non-linear and the random forces are non-Gaussian. We provide explicit expressions for these random quantities in terms of the well-defined increments of the Wienner process.
Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation
Ismail, M. S.
2010-09-30
The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.
Analytic solutions of a general nonlinear functional equations near resonance
NASA Astrophysics Data System (ADS)
Xu, Bing; Zhang, Weinian
2006-05-01
Existence of analytic solutions of a general class of nonlinear functional equations is discussed. This general class includes some specific functional equations studied recently. Moreover, we can generalize this problem to finding analytic solutions of a general class of iterative equations.
NASA Astrophysics Data System (ADS)
Kato, Tsuyoshi; Tanimura, Yoshitaka
2004-01-01
Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.
Kato, Tsuyoshi; Tanimura, Yoshitaka
2004-01-01
Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model. PMID:15267286
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
A quasi-linear kinetic equation for cosmic rays in the interplanetary medium
NASA Technical Reports Server (NTRS)
Luhmann, J. G.
1976-01-01
A kinetic equation for interplanetary cosmic rays is set up with the aid of weak-plasma-turbulence theory for an idealized radially symmetric model of the interplanetary magnetic field. As a starting point, this treatment invokes the Vlasov equation instead of the traditional Fokker-Planck equation. Quasi-linear theory is applied to obtain a momentum diffusion equation for the heliocentric frame of reference which describes the interaction of cosmic rays with convecting magnetic irregularities in the solar-wind plasma. Under restricted conditions, the well-known equation of solar modulation can be obtained from this kinetic equation.
NA Nonlinear Equation-of-state Inversion
NASA Astrophysics Data System (ADS)
Jackson, I.; Kennett, B. L.
2008-12-01
A fully non-linear inversion scheme is introduced for the determination of the parameters controlling the equation-of-state and elasticity of mineral phases using the thermodynamically consistent finite-strain formulation introduced by Stixrude & Lithgow-Bertelloni (2005). This inversion exploits a directed search in an eight-dimensional parameter space using the Neighbourhood Algorithm (NA) of Sambridge (1999) to search for the minimum of an objective function representing the misfit to multiple data sets that constrain different aspects of the mineral behaviour. No derivatives are employed and the progress towards the minimum builds on the accumulated information on the character of the parameter space acquired as the inversion progresses. When only a limited range of experimental information is available there is a strong possibility of multiple minima in the objective function, which can pose problems for conventional iterative least-squares or other gradient methods. The addition of many different styles of data tends to produce a better defined minimum. The influence of different data types can be readily assessed by allowing differential weighting. The new procedure is illustrated by application to MgO, for which extensive experimental data are available. These include the variation of relative volume V with temperature T and pressure P from both static and shock-compression experiments, acoustic measurements of compressional and shear (and hence bulk) moduli, and calorimetric determinations of entropy as a function of temperature at atmospheric pressure. Preliminary NA modeling highlighted tensions between marginally incompatible subsets of data. We therefore excluded one-atmosphere V(T) data for T ≥ 1800 K for which the quasi-harmonic approximation is inadequate (Wu et al., 2008) along with elastic moduli derived from Brillouin spectroscopy under conditions (P ≥ 14 GPa) where significant departures from hydrostatic conditions are expected. With these
Nonlinear modes of the tensor Dirac equation and CPT violation
NASA Technical Reports Server (NTRS)
Reifler, Frank J.; Morris, Randall D.
1993-01-01
Recently, it has been shown that Dirac's bispinor equation can be expressed, in an equivalent tensor form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was also shown that the free tensor Dirac equation is a completely integrable Hamiltonian system with Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. The Yang-Mills equation for a finite coupling constant is investigated. It is shown that the nonlinear Yang-Mills equation has exact plane wave solutions in one-to-one correspondence with the plane wave solutions of Dirac's bispinor equation. The theory of nonlinear dispersive waves is applied to establish the existence of wave packets. The CPT violation of these nonlinear wave packets, which could lead to new observable effects consistent with current experimental bounds, is investigated.
Fractional kinetic equation for Hamiltonian chaos
NASA Astrophysics Data System (ADS)
Zaslavsky, G. M.
1994-09-01
Hamiltonian chaotic dynamics of particles (or passive particles in fluids) can be described by a fractional generalization of the Fokker-Planck-Kolmogorov equation (FFPK) which is defined by two fractional critical exponents (α, β) responsible for the space and time derivatives of the distribution function correspondingly. A renormalization method has been proposed to determine (α, β) from the first principles (ie. from the Hamiltonian). The anomalous transport exponent μ is derived as μ = β/α or μ = β/2α for the first order mean displacement in self-similar transport.
Painlevé equations--nonlinear special functions
NASA Astrophysics Data System (ADS)
Clarkson, Peter A.
2003-04-01
The six Painlevé equations (PI-PVI) were first discovered about a hundred years ago by Painlevé and his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painlevé equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painlevé equations may be thought of a nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations. In this paper, I discuss some of the remarkable properties which the Painlevé equations possess including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is used to illustrate these properties and some of the applications of PII are also discussed.
An integrable shallow water equation with linear and nonlinear dispersion.
Dullin, H R; Gottwald, G A; Holm, D D
2001-11-01
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases. PMID:11690414
Solution Methods for Certain Evolution Equations
NASA Astrophysics Data System (ADS)
Vega-Guzman, Jose Manuel
Solution methods for certain linear and nonlinear evolution equations are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic equation considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution equation. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck equation, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type equation. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic equations is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers equation are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value
Oscillation theorems for second order nonlinear forced differential equations.
Salhin, Ambarka A; Din, Ummul Khair Salma; Ahmad, Rokiah Rozita; Noorani, Mohd Salmi Md
2014-01-01
In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature. PMID:25077054
Multivariate Padé Approximations For Solving Nonlinear Diffusion Equations
NASA Astrophysics Data System (ADS)
Turut, V.
2015-11-01
In this paper, multivariate Padé approximation is applied to power series solutions of nonlinear diffusion equations. As it is seen from tables, multivariate Padé approximation (MPA) gives reliable solutions and numerical results.
Late-time attractor for the cubic nonlinear wave equation
Szpak, Nikodem
2010-08-15
We apply our recently developed scaling technique for obtaining late-time asymptotics to the cubic nonlinear wave equation and explain the appearance and approach to the two-parameter attractor found recently by Bizon and Zenginoglu.
Nonlinear ordinary differential equations: A discussion on symmetries and singularities
NASA Astrophysics Data System (ADS)
Paliathanasis, Andronikos; Leach, P. G. L.
2016-06-01
Two essential methods, the symmetry analysis and the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations is the purpose of this work. The main similarities and the differences of these two different methods are discussed.
Approximation of the Duffing oscillator frequency response function using the FPK equation
NASA Astrophysics Data System (ADS)
Cross, E. J.; Worden, K.
2011-02-01
Although a great deal of work has been carried out on nonlinear structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Perhaps the most fundamental of these quantities is the frequency response function (FRF). A number of years ago, Yar and Hammond took an interesting approach to estimating the FRF of a Duffing oscillator system which was based on an approximate solution of the Fokker-Planck-Kolmogorov equation. Despite reproducing the general features of the statistical linearisation estimate, the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally. The current paper simply extends the work of Yar and Hammond to a higher-order of approximation and is thus able to show the existence of a third 'harmonic' in the FRF. A comparison is made with previous work where an approximation to the FRF was computed using the Volterra series.
Acceleration of High Energy Cosmic Rays in the Nonlinear Shock Precursor
NASA Astrophysics Data System (ADS)
Derzhinsky, F.; Diamond, P. H.; Malkov, M. A.
2006-10-01
The problem of understanding acceleration of very energetic cosmic rays to energies above the 'knee' in the spectrum at 10^15-10^16eV remains one of the great challenges in modern physics. Recently, we have proposed a new approach to understanding high energy acceleration, based on exploiting scattering of cosmic rays by inhomogenities in the compressive nonlinear shock precursor, rather than by scattering across the main shock, as is conventionally assumed. We extend that theory by proposing a mechanism for the generation of mesoscale magnetic fields (krg<1, where rg is the cosmic ray gyroradius). The mechanism is the decay or modulational instability of resonantly generated Alfven waves scattering off ambient density perturbations in the precursors. Such perturbations can be produced by Drury instability. This mechanism leads to the generation of longer wavelength Alfven waves, thus enabling the confinement of higher energy particles. A simplified version of the theory, cast in the form of a Fokker-Planck equation for the Alfven population, will also be presented. This process also limits field generation on rg scales.
Kinetic effects on Alfven wave nonlinearity. II - The modified nonlinear wave equation
NASA Technical Reports Server (NTRS)
Spangler, Steven R.
1990-01-01
A previously developed Vlasov theory is used here to study the role of resonant particle and other kinetic effects on Alfven wave nonlinearity. A hybrid fluid-Vlasov equation approach is used to obtain a modified version of the derivative nonlinear Schroedinger equation. The differences between a scalar model for the plasma pressure and a tensor model are discussed. The susceptibilty of the modified nonlinear wave equation to modulational instability is studied. The modulational instability normally associated with the derivative nonlinear Schroedinger equation will, under most circumstances, be restricted to left circularly polarized waves. The nonlocal term in the modified nonlinear wave equation engenders a new modulational instability that is independent of beta and the sense of circular polarization. This new instability may explain the occurrence of wave packet steepening for all values of the plasma beta in the vicinity of the earth's bow shock.
Invariant tori for a class of nonlinear evolution equations
Kolesov, A Yu; Rozov, N Kh
2013-06-30
The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.
Lattice Boltzmann model for generalized nonlinear wave equations
NASA Astrophysics Data System (ADS)
Lai, Huilin; Ma, Changfeng
2011-10-01
In this paper, a lattice Boltzmann model is developed to solve a class of the nonlinear wave equations. Through selecting equilibrium distribution function and an amending function properly, the governing evolution equation can be recovered correctly according to our proposed scheme, in which the Chapman-Enskog expansion is employed. We validate the algorithm on some problems where analytic solutions are available, including the second-order telegraph equation, the nonlinear Klein-Gordon equation, and the damped, driven sine-Gordon equation. It is found that the numerical results agree well with the analytic solutions, which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems.
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
NASA Astrophysics Data System (ADS)
Abdel-Salam, Emad A.-B.; Hassan, Gamal F.
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
NASA Astrophysics Data System (ADS)
Emad A-B., Abdel-Salam; Gamal, F. Hassan
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag-Leffler function methods. The obtained results recover the well-know solutions when α = 1.
Derivation of an Applied Nonlinear Schroedinger Equation.
Pitts, Todd Alan; Laine, Mark Richard; Schwarz, Jens; Rambo, Patrick K.; Karelitz, David B.
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
Nonlinear flap-lag axial equations of a rotating beam
NASA Technical Reports Server (NTRS)
Kaza, K. R. V.; Kvaternik, R. G.
1977-01-01
It is possible to identify essentially four approaches by which analysts have established either the linear or nonlinear governing equations of motion for a particular problem related to the dynamics of rotating elastic bodies. The approaches include the effective applied load artifice in combination with a variational principle and the use of Newton's second law, written as D'Alembert's principle, applied to the deformed configuration. A third approach is a variational method in which nonlinear strain-displacement relations and a first-degree displacement field are used. The method introduced by Vigneron (1975) for deriving the linear flap-lag equations of a rotating beam constitutes the fourth approach. The reported investigation shows that all four approaches make use of the geometric nonlinear theory of elasticity. An alternative method for deriving the nonlinear coupled flap-lag-axial equations of motion is also discussed.
Entropy and convexity for nonlinear partial differential equations
Ball, John M.; Chen, Gui-Qiang G.
2013-01-01
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768
The numerical dynamic for highly nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
The Jeffcott equations in nonlinear rotordynamics
NASA Technical Reports Server (NTRS)
Zalik, R. A.
1987-01-01
The Jeffcott equations are a system of coupled differential equations representing the behavior of a rotating shaft. This is a simple model which allows investigation of the basic dynamic behavior of rotating machinery. Nolinearities can be introduced by taking into consideration deadband, side force, and rubbing, among others. The properties of the solutions of the Jeffcott equations with deadband are studied. In particular, it is shown how bounds for the solution of these equations can be obtained from bounds for the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density (PSD) plots. This study offers insight into a possible detection method to determine pump stability margins during flight and hot fire tests, and was motivated by the need to explain a phenomenon observed in the development phase of the cryogenic pumps of the Space Shuttle, during hot fire ground testing; namely, the appearance of vibrations at frequencies that could not be accounted for by means of linear models.
Relations between nonlinear Riccati equations and other equations in fundamental physics
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Generalized nonlinear Proca equation and its free-particle solutions
NASA Astrophysics Data System (ADS)
Nobre, F. D.; Plastino, A. R.
2016-06-01
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schrödinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ ^{μ }(ěc {x},t), involves an additional field Φ ^{μ }(ěc {x},t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E2 = p2c2 + m2c4 for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed.
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations
Baranwal, Vipul K.; Pandey, Ram K.
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0, γ1, γ2,… and auxiliary functions H0(x), H1(x), H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
An iterative method for systems of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Scroggs, Jeffrey S.
1989-01-01
An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.
Liapunov functions for non-linear difference equation stability analysis.
NASA Technical Reports Server (NTRS)
Park, K. E.; Kinnen, E.
1972-01-01
Liapunov functions to determine the stability of non-linear autonomous difference equations can be developed through the use of auxiliary exact difference equations. For this purpose definitions are introduced for the gradient of an implicit function of a discrete variable, a principal sum, a definite sum and an exact difference equation, and a theorem for exactness of a difference form is proved. Examples illustrate the procedure.
Option pricing formulas and nonlinear filtering: a Feynman path integral perspective
NASA Astrophysics Data System (ADS)
Balaji, Bhashyam
2013-05-01
Many areas of engineering and applied science require the solution of certain parabolic partial differential equa tions, such as the Fokker-Planck and Kolmogorov equations. The fundamental solution, or the Green's function, for such PDEs can be written in terms of the Feynman path integral (FPI). The partial differential equation arising in the valuing of options is the Kolmogorov backward equation that is referred to as the Black-Scholes equation. The utility of this is demonstrated and numerical examples that illustrate the high accuracy of option price calculation even when using a fairly coarse grid.
Nonlinear Resonance and Duffing's Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…
Nonlinear Resonance and Duffing's Spring Equation II
ERIC Educational Resources Information Center
Fay, T. H.; Joubert, Stephan V.
2007-01-01
The paper discusses the boundary in the frequency-amplitude plane for boundedness of solutions to the forced spring Duffing type equation x[umlaut] + x + [epsilon]x[cubed] = F cos[omega]t. For fixed initial conditions and for representative fixed values of the parameter [epsilon], the results are reported of a systematic numerical investigation…
Non-Linear Spring Equations and Stability
ERIC Educational Resources Information Center
Fay, Temple H.; Joubert, Stephan V.
2009-01-01
We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…
Noise in Nonlinear Dynamical Systems 3 Volume Paperback Set
NASA Astrophysics Data System (ADS)
Moss, Frank; McClintock, P. V. E.
2011-11-01
Volume 1: List of contributors; Preface; Introduction to volume one; 1. Noise-activated escape from metastable states: an historical view Rolf Landauer; 2. Some Markov methods in the theory of stochastic processes in non-linear dynamical systems R. L. Stratonovich; 3. Langevin equations with coloured noise J. M. Sancho and M. San Miguel; 4. First passage time problems for non-Markovian processes Katja Lindenberg, Bruce J. West and Jaume Masoliver; 5. The projection approach to the Fokker-Planck equation: applications to phenomenological stochastic equations with coloured noises Paolo Grigolini; 6. Methods for solving Fokker-Planck equations with applications to bistable and periodic potentials H. Risken and H. D. Vollmer; 7. Macroscopic potentials, bifurcations and noise in dissipative systems Robert Graham; 8. Transition phenomena in multidimensional systems - models of evolution W. Ebeling and L. Schimansky-Geier; 9. Coloured noise in continuous dynamical systems: a functional calculus approach Peter Hanggi; Appendix. On the statistical treatment of dynamical systems L. Pontryagin, A. Andronov and A. Vitt; Index. Volume 2: List of contributors; Preface; Introduction to volume two; 1. Stochastic processes in quantum mechanical settings Ronald F. Fox; 2. Self-diffusion in non-Markovian condensed-matter systems Toyonori Munakata; 3. Escape from the underdamped potential well M. Buttiker; 4. Effect of noise on discrete dynamical systems with multiple attractors Edgar Knobloch and Jeffrey B. Weiss; 5. Discrete dynamics perturbed by weak noise Peter Talkner and Peter Hanggi; 6. Bifurcation behaviour under modulated control parameters M. Lucke; 7. Period doubling bifurcations: what good are they? Kurt Wiesenfeld; 8. Noise-induced transitions Werner Horsthemke and Rene Lefever; 9. Mechanisms for noise-induced transitions in chemical systems Raymond Kapral and Edward Celarier; 10. State selection dynamics in symmetry-breaking transitions Dilip K. Kondepudi; 11. Noise in a
A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Yaşar, Emrullah; San, Sait
2016-05-01
In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov's new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer-Chree (PC) equation and the Kaup-Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.
NASA Astrophysics Data System (ADS)
Baskonus, Haci Mehmet; Bulut, Hasan
2015-10-01
In this paper, a new computational algorithm called the "Improved Bernoulli sub-equation function method" has been proposed. This algorithm is based on the Bernoulli Sub-ODE method. Firstly, the nonlinear evaluation equations used for representing various physical phenomena are converted into ordinary differential equations by using various wave transformations. In this way, nonlinearity is preserved and represent nonlinear physical problems. The nonlinearity of physical problems together with the derivations is seen as the secret key to solve the general structure of problems. The proposed analytical schema, which is newly submitted to the literature, has been expressed comprehensively in this paper. The analytical solutions, application results, and comparisons are presented by plotting the two and three dimensional surfaces of analytical solutions obtained by using the methods proposed for some important nonlinear physical problems. Finally, a conclusion has been presented by mentioning the important discoveries in this study.
Evolution equation for non-linear cosmological perturbations
Brustein, Ram; Riotto, Antonio E-mail: Antonio.Riotto@cern.ch
2011-11-01
We present a novel approach, based entirely on the gravitational potential, for studying the evolution of non-linear cosmological matter perturbations. Starting from the perturbed Einstein equations, we integrate out the non-relativistic degrees of freedom of the cosmic fluid and obtain a single closed equation for the gravitational potential. We then verify the validity of the new equation by comparing its approximate solutions to known results in the theory of non-linear cosmological perturbations. First, we show explicitly that the perturbative solution of our equation matches the standard perturbative solutions. Next, using the mean field approximation to the equation, we show that its solution reproduces in a simple way the exponential suppression of the non-linear propagator on small scales due to the velocity dispersion. Our approach can therefore reproduce the main features of the renormalized perturbation theory and (time)-renormalization group approaches to the study of non-linear cosmological perturbations, with some possibly important differences. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Wackerbauer, R. ); Huebler, A. . Center for Complex Systems Research); Mayer-Kress, G. California Univ., Santa Cruz, CA . Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation
NASA Astrophysics Data System (ADS)
Doha, Eid; Bhrawy, Ali; Abdelkawy, Mohamed; Hafez, Ramy
2014-02-01
This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.
NASA Astrophysics Data System (ADS)
Tchinang Tchameu, J. D.; Togueu Motcheyo, A. B.; Tchawoua, C.
2016-09-01
The discrete multi-rogue waves (DMRW) as solution of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearities is studied numerically. These biological rogue waves represent the complex probability amplitude of finding an amide-I vibrational quantum at a site. We observe that the growth in the higher order saturable nonlinearity implies the formation of DMRW including an increase in the short-living DMRW and a decrease in amplitude of the long-living DMRW.
Stochasticity in numerical solutions of the nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Shen, Mei-Mei; Nicholson, D. R.
1987-01-01
The cubically nonlinear Schroedinger equation is an important model of nonlinear phenomena in fluids and plasmas. Numerical solutions in a spatially periodic system commonly involve truncation to a finite number of Fourier modes. These solutions are found to be stochastic in the sense that the largest Liapunov exponent is positive. As the number of modes is increased, the size of this exponent appears to converge to zero, in agreement with the recent demonstration of the integrability of the spatially periodic case.
The Buoyancy Budget With a Nonlinear Equation of State
NASA Astrophysics Data System (ADS)
Hieronymus, M. H.; Nycander, J.
2012-12-01
There has been a number of studies focusing on different aspects of having a nonlinear equation of state for seawater. Amongst other things it has been shown that the nonlinear equation of state has implications for the oceanic energy budget and that nonlinear processes can be a significant source of dense water production. This presentation will focus on the oceanic buoyancy budget. The nonlinear equation of state of seawater can introduce a sink or source of buoyancy when water parcels of unequal salinities and temperatures are mixed. A common example is the process known as cabbeling, which is responsible for forming a water mass that is denser than the original constituents in a mixture of two water masses with equal densities but different salinities and temperatures. This presentation will contain quantitative estimates of these nonlinear effects on the buoyancy budget of the global ocean. Because of these nonlinear effects there is a net sink of buoyancy in the oceans interior and the size of this sink can be determined from the buoyancy fluxes at the ocean boundaries. These boundary buoyancy fluxes are calculated using two surface heat flux climatologies one based on in situ measurements, the other on a reanalysis and in both cases using a nonlinear equation of state. The presentation also treats the buoyancy budget in the State of the art ocean model Nucleus for European Modelling of the Ocean (NEMO) and the results from NEMO are seen to be in good agreement with the buoyancy budgets based on the heat flux climatologies. Using the ocean model is a good complement to the surface flux climatologies, because in NEMO the buoyancy fluxes can be evaluated at all vertical model levels. This means that the vertical distribution of the buoyancy sink can be looked into. The results from NEMO shows that in large parts of the ocean the nonlinear buoyancy sink is the largest contribution to the buoyancy budget.
Nonlinear generalized master equations and accounting for initial correlations
NASA Astrophysics Data System (ADS)
Los, V. F.
2009-08-01
We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can
Decay and stability for nonlinear hyperbolic equations
NASA Astrophysics Data System (ADS)
Marcati, Pierangelo
This paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under some restrictions, sufficient conditions for precompactness of the orbits and decay of solutions are given. An existence and uniqueness theorem is proved in the Appendix. Some examples are given.
Cylindrical nonlinear Schroedinger equation versus cylindrical Korteweg-de Vries equation
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2008-10-15
A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) equations and the one of cylindrical Korteweg-de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS equation can be associated with non-stationary soliton-like solutions of cKdV equation.
Yang Xiao; Du Dianlou
2010-08-15
The Poisson structure on C{sup N}xR{sup N} is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
NASA Astrophysics Data System (ADS)
Yang, Xiao; Du, Dianlou
2010-08-01
The Poisson structure on CN×RN is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1988-01-01
An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.
Gelin, Maxim F
2014-12-01
We consider a classical point particle bilinearly coupled to a harmonic bath. Assuming that the evolution of the particle is monitored on a timescale which is longer than the characteristic bath correlation time, we derive the Markovian master equation for the probability density of the particle. The relaxation operator of this master equation is evaluated analytically, without invoking the perturbation theory and the approximation of weak system-bath coupling. When the bath correlation time tends to zero, the Fokker-Planck equation is recovered. For a finite bath correlation time, the relaxation operator contains contributions of all orders in the system-bath coupling. PMID:25481131
On the Dirichlet problem for a nonlinear elliptic equation
NASA Astrophysics Data System (ADS)
Egorov, Yu V.
2015-04-01
We prove the existence of an infinite set of solutions to the Dirichlet problem for a nonlinear elliptic equation of the second order. Such a problem for a nonlinear elliptic equation with Laplace operator was studied earlier by Krasnosel'skii, Bahri, Berestycki, Lions, Rabinowitz, Struwe and others. We study the spectrum of this problem and prove the weak convergence to 0 of the sequence of normed eigenfunctions. Moreover, we obtain some estimates for the 'Fourier coefficients' of functions in W^1p,0(Ω). This allows us to improve the preceding results. Bibliography: 8 titles.
Stochastic Calculus and Differential Equations for Physics and Finance
NASA Astrophysics Data System (ADS)
McCauley, Joseph L.
2013-02-01
1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.
Multiply scaled constrained nonlinear equation solvers. [for nonlinear heat conduction problems
NASA Technical Reports Server (NTRS)
Padovan, Joe; Krishna, Lala
1986-01-01
To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.
Bogoliubov equations and functional mechanics
NASA Astrophysics Data System (ADS)
Volovich, I. V.
2010-09-01
The functional classical mechanics based on the probability approach, where a particle is described not by a trajectory in the phase space but by a probability distribution, was recently proposed for solving the irreversibility problem, i.e., the problem of matching the time reversibility of microscopic dynamics equations and the irreversibility of macrosystem dynamics. In the framework of functional mechanics, we derive Bogoliubov-Boltzmann-type equations for finitely many particles. We show that a closed equation for a one-particle distribution function can be rigorously derived in functional mechanics without any additional assumptions required in the Bogoliubov method. We consider the possibility of using diffusion processes and the Fokker-Planck-Kolmogorov equation to describe isolated particles.
Intermittency and solitons in the driven dissipative nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Moon, H. T.; Goldman, M. V.
1984-01-01
The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
Conservation laws of inviscid Burgers equation with nonlinear damping
NASA Astrophysics Data System (ADS)
Abdulwahhab, Muhammad Alim
2014-06-01
In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).
Optimization of a finite difference method for nonlinear wave equations
NASA Astrophysics Data System (ADS)
Chen, Miaochao
2013-07-01
Wave equations have important fluid dynamics background, which are extensively used in many fields, such as aviation, meteorology, maritime, water conservancy, etc. This paper is devoted to the explicit difference method for nonlinear wave equations. Firstly, a three-level and explicit difference scheme is derived. It is shown that the explicit difference scheme is uniquely solvable and convergent. Moreover, a numerical experiment is conducted to illustrate the theoretical results of the presented method.
Ikeda, Tatsushi; Ito, Hironobu; Tanimura, Yoshitaka
2015-06-01
We explore and describe the roles of inter-molecular vibrations employing a Brownian oscillator (BO) model with linear-linear (LL) and square-linear (SL) system-bath interactions, which we use to analyze two-dimensional (2D) THz-Raman spectra obtained by means of molecular dynamics (MD) simulations. In addition to linear infrared absorption (1D IR), we calculated 2D Raman-THz-THz, THz-Raman-THz, and THz-THz-Raman signals for liquid formamide, water, and methanol using an equilibrium non-equilibrium hybrid MD simulation. The calculated 1D IR and 2D THz-Raman signals are compared with results obtained from the LL+SL BO model applied through use of hierarchal Fokker-Planck equations with non-perturbative and non-Markovian noise. We find that all of the qualitative features of the 2D profiles of the signals obtained from the MD simulations are reproduced with the LL+SL BO model, indicating that this model captures the essential features of the inter-molecular motion. We analyze the fitted 2D profiles in terms of anharmonicity, nonlinear polarizability, and dephasing time. The origins of the echo peaks of the librational motion and the elongated peaks parallel to the probe direction are elucidated using optical Liouville paths. PMID:26049441
Topological horseshoes in travelling waves of discretized nonlinear wave equations
Chen, Yi-Chiuan; Chen, Shyan-Shiou; Yuan, Juan-Ming
2014-04-15
Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.
Tensor methods for large sparse systems of nonlinear equations
Bouaricha, A.; Schnabel, R.B.
1996-12-31
This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time.
Bounded and periodic solutions of nonlinear functional differential equations
Slyusarchuk, Vasilii E
2012-05-31
Conditions for the existence of bounded and periodic solutions of the nonlinear functional differential equation d{sup m}x(t)/dt{sup m} + (Fx)(t) = h(t), t element of R, are presented, involving local linear approximations to the operator F. Bibliography: 23 titles.
Maximum Likelihood Estimation of Nonlinear Structural Equation Models.
ERIC Educational Resources Information Center
Lee, Sik-Yum; Zhu, Hong-Tu
2002-01-01
Developed an EM type algorithm for maximum likelihood estimation of a general nonlinear structural equation model in which the E-step is completed by a Metropolis-Hastings algorithm. Illustrated the methodology with results from a simulation study and two real examples using data from previous studies. (SLD)
Model Comparison of Nonlinear Structural Equation Models with Fixed Covariates.
ERIC Educational Resources Information Center
Lee, Sik-Yum; Song, Xin-Yuan
2003-01-01
Proposed a new nonlinear structural equation model with fixed covariates to deal with some complicated substantive theory and developed a Bayesian path sampling procedure for model comparison. Illustrated the approach with an illustrative example using data from an international study. (SLD)
Local Influence Analysis of Nonlinear Structural Equation Models
ERIC Educational Resources Information Center
Lee, Sik-Yum; Tang, Nian-Sheng
2004-01-01
By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis…
Painleve analysis for a nonlinear Schroedinger equation in three dimensions
Chowdhury, A.R.; Chanda, P.K.
1987-09-01
A Painleve analysis is performed for the nonlinear Schroedinger equation in (2 + 1) dimensions following the methodology of Weiss et al. simplified in the sense of Kruskal. At least for one branch it is found that the required number of arbitrary functions (as demanded by the Cauchy-Kovalevskaya theorem) exists, signalling complete integrability.
Forced oscillations of nonlinear damped equation of suspended string
NASA Astrophysics Data System (ADS)
Yamaguchi, Masaru; Nagai, Tohru; Matsukane, Katsuya
2008-06-01
We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.
Optimal analytic method for the nonlinear Hasegawa-Mima equation
NASA Astrophysics Data System (ADS)
Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle
2014-05-01
The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2009-11-10
A review of the recent studies on the correspondence between a wide family of the generalized nonlinear Schroedinger equations and a wide family of the generalized Korteweg-de Vries equations is presented. It was constructed some years ago within the framework of a recently-developed approach based on the Madelung's fluid representation of the generalized nonlinear Schroedinger equation. The present analysis extends the former approach, developed for nonlinear Schroedinger equation with a nonlinear term proportional to a multiplicative operator, to the cases of derivative operators and the ones corresponding to cylindrical nonlinear Schroedinger equations.
Shock-wave structure using nonlinear model Boltzmann equations.
NASA Technical Reports Server (NTRS)
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
Transformation matrices between non-linear and linear differential equations
NASA Technical Reports Server (NTRS)
Sartain, R. L.
1983-01-01
In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.
New forms of two-particle and one-particle kinetic equations
NASA Astrophysics Data System (ADS)
Saveliev, V. L.; Yonemura, S.
2012-11-01
Pair collisions are the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was done by Ludwig Boltzmann we wrote the kinetic equation for two-particle distribution function of molecules in gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. We developed a new technique for factorization of the scattering operator on the bases of right inverses to the Casimir operator of the group of rotations. We exactly transformed the Boltzmann collision integral to the Landau-Fokker-Planck like form.
Dou, Wenjie; Subotnik, Joseph E
2016-01-14
A broadened classical master equation (BCME) is proposed for modeling nonadiabatic dynamics for molecules near metal surfaces over a wide range of parameter values and with arbitrary initial conditions. Compared with a standard classical master equation-which is valid in the limit of weak molecule-metal couplings-this BCME should be valid for both weak and strong molecule-metal couplings. (The BCME can be mapped to a Fokker-Planck equation that captures level broadening correctly.) Finally, our BCME can be solved with a simple surface hopping algorithm; numerical tests of equilibrium and dynamical observables look very promising. PMID:26772563
Phase space lattices and integrable nonlinear wave equations
NASA Astrophysics Data System (ADS)
Tracy, Eugene; Zobin, Nahum
2003-10-01
Nonlinear wave equations in fluids and plasmas that are integrable by Inverse Scattering Theory (IST), such as the Korteweg-deVries and nonlinear Schrodinger equations, are known to be infinite-dimensional Hamiltonian systems [1]. These are of interest physically because they predict new phenomena not present in linear wave theories, such as solitons and rogue waves. The IST method provides solutions of these equations in terms of a special class of functions called Riemann theta functions. The usual approach to the theory of theta functions tends to obscure the underlying phase space structure. A theory due to Mumford and Igusa [2], however shows that the theta functions arise naturally in the study of phase space lattices. We will describe this theory, as well as potential applications to nonlinear signal processing and the statistical theory of nonlinear waves. 1] , S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of solitons: the inverse scattering method (Consultants Bureau, New York, 1984). 2] D. Mumford, Tata lectures on theta, Vols. I-III (Birkhauser); J. Igusa, Theta functions (Springer-Verlag, New York, 1972).
An adaptive grid algorithm for one-dimensional nonlinear equations
NASA Technical Reports Server (NTRS)
Gutierrez, William E.; Hills, Richard G.
1990-01-01
Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and
Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two
NASA Astrophysics Data System (ADS)
Chiron, David; Scheid, Claire
2016-02-01
We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and "one-dimensional spreading" phenomena.
Stochastic differential equation model to Prendiville processes
Granita; Bahar, Arifah
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Surh, M P; Sturgeon, J B; Wolfer, W G
2005-01-03
We have recently discovered an error in our void nucleation code used in three prior publications [1-3]. A term was omitted in the model for vacancy re-emission that (especially at high temperature) affects void nucleation and growth during irradiation as well as void annealing and Ostwald ripening of the size distribution after irradiation. The omission was not immediately detected because the calculations predict reasonable void densities and swelling behaviors when compared to experiment at low irradiation temperatures, where void swelling is prominent. (Comparable neutron irradiation experiments are less prevalent at higher temperatures, e.g., > 500 C.)
Golush, W.G.
1994-12-31
Nonlinear equations are expressed using a new OMNI statement FORM NLE. This allows OMNI Constructs, Classes, Tables, and New Variables to be used in nonlinear equations. The interface passes the nonlinear equations and symbolic derivatives to a general nonlinear solver. After optimization, the row and column activities of the solution are written to an OMNI Standard Solution File. Reports are written from this file using the OMNI FORM LINE report writer. The interface will be illustrated with an example of a nonlinear model written in OMNI and solved using the MINOS nonlinear solver.
Multi-soliton rational solutions for some nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Osman, Mohamed S.
2016-01-01
The Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota's method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.
Numerical solution of control problems governed by nonlinear differential equations
Heinkenschloss, M.
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Modified non-linear Burgers' equations and cosmic ray shocks
NASA Technical Reports Server (NTRS)
Zank, G. P.; Webb, G. M.; Mckenzie, J. F.
1988-01-01
A reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.
Solving nonlinear evolution equation system using two different methods
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Bekir, Ahmet; Ozer, Mehmet N.
2015-12-01
This paper deals with constructing more general exact solutions of the coupled Higgs equation by using the (G0/G, 1/G)-expansion and (1/G0)-expansion methods. The obtained solutions are expressed by three types of functions: hyperbolic, trigonometric and rational functions with free parameters. It has been shown that the suggested methods are productive and will be used to solve nonlinear partial differential equations in applied mathematics and engineering. Throughout the paper, all the calculations are made with the aid of the Maple software.
Quadratic nonlinear Klein-Gordon equation in one dimension
NASA Astrophysics Data System (ADS)
Hayashi, Nakao; Naumkin, Pavel I.
2012-10-01
We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v - vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah ["Normal forms and quadratic nonlinear Klein-Gordon equations," Commun. Pure Appl. Math. 38, 685-696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort ["Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi-linéaire á données petites en dimension 1," Ann. Sci. Ec. Normale Super. 34(4), 1-61 (2001)].
Numerical study of fractional nonlinear Schrödinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Unitary qubit extremely parallelized algorithms for coupled nonlinear Schrodinger equations
NASA Astrophysics Data System (ADS)
Oganesov, Armen; Flint, Chris; Vahala, George; Vahala, Linda; Yepez, Jeffrey; Soe, Min
2015-11-01
The nonlinear Schrodinger equation (NLS) is a ubiquitous equation occurring in plasma physics, nonlinear optics and in Bose Einstein condensates. Viewed from the BEC standpoint of phase transitions, the wave function is the order parameter and topological defects in that manifold are simply the vortices, which for a scalar NLS have quantized circulation. In multi-species NLS the topological nature of the vortices are radically different with some classes of vortices no longer having quantized circulation as in classical turbulence. Moreover, some of the vortex equivalence classes need no longer be Abelian. This strongly effects the permitted vortex reconnections. The effect of these structures on the spectral properties of the ensuing turbulence will be investigated. Our 3D algorithm is based on a novel unitary qubit lattice scheme that is ideally parallelized - tested up to 780 000 cores on Mira. This scheme is mesoscopic (like lattice Boltzmann), but fully unitary (unlike LB). Supported by NSF, DoD.
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Nonlinear Generalized Hydrodynamic Wave Equations in Strongly Coupled Dusty Plasmas
Veeresha, B. M.; Sen, A.; Kaw, P. K.
2008-09-07
A set of nonlinear equations for the study of low frequency waves in a strongly coupled dusty plasma medium is derived using the phenomenological generalized hydrodynamic (GH) model and is used to study the modulational stability of dust acoustic waves to parallel perturbations. Dust compressibility contributions arising from strong Coulomb coupling effects are found to introduce significant modifications in the threshold and range of the instability domain.
Fast neural solution of a nonlinear wave equation
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad; Barhen, Jacob
1992-01-01
A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.
Approximate solutions for non-linear iterative fractional differential equations
NASA Astrophysics Data System (ADS)
Damag, Faten H.; Kiliçman, Adem; Ibrahim, Rabha W.
2016-06-01
This paper establishes approximate solution for non-linear iterative fractional differential equations: d/γv (s ) d sγ =ℵ (s ,v ,v (v )), where γ ∈ (0, 1], s ∈ I := [0, 1]. Our method is based on some convergence tools for analytic solution in a connected region. We show that the suggested solution is unique and convergent by some well known geometric functions.
Inhomogeneous critical nonlinear Schroedinger equations with a harmonic potential
Cao Daomin; Han Pigong
2010-04-15
In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schroedinger equation with a harmonic potential: i{partial_derivative}{sub t}u=-div(f(x){nabla}u)+|x|{sup 2}u-k(x)|u|{sup 4/N}u, x is an element of R{sup N}, N{>=}1, which models the remarkable Bose-Einstein condensation. We discuss the existence and nonexistence results and investigate the limiting profile of blow-up solutions with critical mass.
Parallel iterative methods for sparse linear and nonlinear equations
NASA Technical Reports Server (NTRS)
Saad, Youcef
1989-01-01
As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.
NASA Astrophysics Data System (ADS)
Tamizhmani, K. M.; Krishnakumar, K.; Leach, P. G. L.
2015-11-01
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to change from a nonlinearity with an arbitrary exponent to a nonlinearity with a specific numerical exponent.
Improved algorithm for solving nonlinear parabolized stability equations
NASA Astrophysics Data System (ADS)
Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng
2016-08-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).
NASA Astrophysics Data System (ADS)
Xie, Xi-Yang; Tian, Bo; Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-01
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable-coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
NASA Astrophysics Data System (ADS)
Olivares-Rivas, Wilmer; Colmenares, Pedro J.
2016-09-01
The non-static generalized Langevin equation and its corresponding Fokker-Planck equation for the position of a viscous fluid particle were solved in closed form for a time dependent external force. Its solution for a constant external force was obtained analytically. The non-Markovian stochastic differential equation, associated to the dynamics of the position under a colored noise, was then applied to the description of the dynamics and persistence time of particles constrained within absorbing barriers. Comparisons with molecular dynamics were very satisfactory.
Equations for Nonlinear MHD Convection in Shearless Magnetic Systems
Pastukhov, V.P.
2005-07-15
A closed set of reduced dynamic equations is derived that describe nonlinear low-frequency flute MHD convection and resulting nondiffusive transport processes in weakly dissipative plasmas with closed or open magnetic field lines. The equations obtained make it possible to self-consistently simulate transport processes and the establishment of the self-consistent plasma temperature and density profiles for a large class of axisymmetric nonparaxial shearless magnetic devices: levitated dipole configurations, mirror systems, compact tori, etc. Reduced equations that are suitable for modeling the long-term evolution of the plasma on time scales comparable to the plasma lifetime are derived by the method of the adiabatic separation of fast and slow motions.
Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas
Artun, M.; Tang, W.M.
1994-03-01
The influence of sheared equilibrium flows on the confinement properties of tokamak plasmas is a topic of much current interest. A proper theoretical foundation for the systematic kinetic analysis of this important problem has been provided here by presented the derivation of a set of nonlinear electromagnetic gyrokinetic equations applicable to low frequency microinstabilities in a rotating axisymmetric plasma. The subsonic rotation velocity considered is in the direction of symmetry with the angular rotation frequency being a function of the equilibrium magnetic flux surface. In accordance with experimental observations, the rotation profile is chosen to scale with the ion temperature. The results obtained represent the shear flow generalization of the earlier analysis by Frieman and Chen where such flows were not taken into account. In order to make it readily applicable to gyrokinetic particle simulations, this set of equations is cast in a phase-space-conserving continuity equation form.
Multipulses of Nonlinearly Coupled Schrödinger Equations
NASA Astrophysics Data System (ADS)
Yew, Alice C.
2001-06-01
The capacity of coupled nonlinear Schrödinger (NLS) equations to support multipulse solutions (multibump solitary-waves) is investigated. A detailed analysis is undertaken for a system of quadratically coupled equations that describe the phenomena of second harmonic generation and parametric wave interaction in non-centrosymmetric optical materials. Utilising the framework of homoclinic bifurcation theory, and employing a Lyapunov-Schmidt reduction method developed by Hale, Lin, and Sandstede, a novel mechanism for the generation of multipulses is identified, which arises from a resonant semi-simple eigenvalue configuration of the linearised steady-state equations. Conditions for the existence of multipulses, as well as a description of their geometry, are derived from the analysis.
From Lévy flights to the fractional kinetic equation for dynamical chaos
NASA Astrophysics Data System (ADS)
Zaslavsky, G. M.
Chaotic dynamics of Hamiltonian systems can be described by the random process which resembles the Lévy-type flights and trappings in the phase space of a system. The probability distribution function satisfies the fractional in space and time generalization of the Fokker-Planck-Kolmogorov equation. Orders of the fractional derivatives in space and time can be connected to the Pesin's dimensions of the trajectories. A new look on the problem of Maxwell's Demon is discussed in the context of the anomalous ("strange") kinetics.
Solovchuk, Maxim; Sheu, Tony W H; Thiriet, Marc
2013-11-01
This study investigates the influence of blood flow on temperature distribution during high-intensity focused ultrasound (HIFU) ablation of liver tumors. A three-dimensional acoustic-thermal-hydrodynamic coupling model is developed to compute the temperature field in the hepatic cancerous region. The model is based on the nonlinear Westervelt equation, bioheat equations for the perfused tissue and blood flow domains. The nonlinear Navier-Stokes equations are employed to describe the flow in large blood vessels. The effect of acoustic streaming is also taken into account in the present HIFU simulation study. A simulation of the Westervelt equation requires a prohibitively large amount of computer resources. Therefore a sixth-order accurate acoustic scheme in three-point stencil was developed for effectively solving the nonlinear wave equation. Results show that focused ultrasound beam with the peak intensity 2470 W/cm(2) can induce acoustic streaming velocities up to 75 cm/s in the vessel with a diameter of 3 mm. The predicted temperature difference for the cases considered with and without acoustic streaming effect is 13.5 °C or 81% on the blood vessel wall for the vein. Tumor necrosis was studied in a region close to major vessels. The theoretical feasibility to safely necrotize the tumors close to major hepatic arteries and veins was shown. PMID:24180802
The method of patches for solving stiff nonlinear differential equations
NASA Astrophysics Data System (ADS)
Brydon, David Van George, Jr.
1998-12-01
This dissertation describes a new method for solving very stiff sets of ordinary differential equations. The basic idea is to replace the original nonlinear equations with a set of equally stiff equations that are piecewise linear, and therefore can be solved exactly. We demonstrate the value of the method on small systems of equations for which some other methods are inefficient or produce spurious solutions, estimate error bounds, and discuss extensions of the method to larger systems of equations and to partial differential equations. Putzer's method is developed in a novel way for efficient and accurate solution of dx/dt = Ax+b. The physical problem of interest is spatial pattern formation in open reaction-diffusion chemical systems, as studied in the experiments of Kyoung Lee, Harry Swinney, et al. I develop a new experiment model that agrees reasonably well with experimental results. I solve the model, applying the new method to the two-variable Gaspar- Showalter chemical kinetics in two space dimensions. Because of time and computer limitations, only preliminary pattern-formation results are achieved and reported.
The truncation model of the derivative nonlinear Schroedinger equation
Sanchez-Arriaga, G.; Hada, T.; Nariyuki, Y.
2009-04-15
The derivative nonlinear Schroedinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.
Theoretical and numerical studies of nonlinear shell equations
NASA Astrophysics Data System (ADS)
Hermann, M.; Kaiser, D.; Schröder, M.
1999-07-01
We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T( x, λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M. These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.
On the nonlinear Schrodinger equation with nonzero boundary conditions
NASA Astrophysics Data System (ADS)
Fagerstrom, Emily
This thesis is concerned with the study of the nonlinear Schrodinger (NLS) equation, which is important both from a physical and a mathematical point of view. In physics, it is a universal model for the evolutions of weakly nonlinear dispersive wave trains. As such it appears in many physical contexts, such as optics, acoustics, plasmas, biology, etc. Mathematically, it is a completely integrable, infinite-dimensional Hamiltonian system, and possesses a surprisingly rich structure. This equation has been extensively studied in the last 50 years, but many important questions are still open. In particular, this thesis contains the following original contributions: NLS with real spectral singularities. First, the focusing NLS equation is considered with decaying initial conditions. This situation has been studied extensively before, but the assumption is almost always made that the scattering coefficients have no real zeros, and thus the scattering data had no poles on the real axis. However, it is easy to produce example potentials with this behavior. For example, by modifying parameters in Satsuma-Yajima's sech potential, or by choosing a "box" potential with a particular area, one can obtain corresponding scattering entries with real zeros. The inverse scattering transform can be implemented by formulating the modified Jost eigenfunctions and the scattering data as a Riemann Hilbert problem. But it can also be formulated by using integral kernels. Doing so produces the Gelf'and-Levitan-Marchenko (GLM) equations. Solving these integral equations requires integrating an expression containing the reflection coefficient over the real axis. Under the usual assumption, the reflection coefficient has no poles on the real axis. In general, the integration contour cannot be deformed to avoid poles, because the reflection coefficient may not admit analytic extension off the real axis. Here it is shown that the GLM equations may be (uniquely) solved using a principal value
Vortex Solutions of the Defocusing Discrete Nonlinear Schroedinger Equation
Cuevas, J.; Kevrekidis, P. G.; Law, K. J. H.
2009-09-09
We consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing DNLS equation, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization-destabilization windows for any finite lattice.
Pseudorecurrence and chaos of cubic-quintic nonlinear Schroedinger equation
Zhou, C.; Lai, C.H.
1996-12-01
Recurrence, pseudorecurrence, and chaotic solutions for a continuum Hamiltonian system in which there exist spatial patterns of solitary wave structures are investigated using the nonlinear Schrodinger equation (NSE) with cubic and quintic terms. The theoretical analyses indicate that there may exist Birkhoff`s recurrence for the arbitrary parameter values. The numerical experiments show that there may be Fermi-Pasta-Ulam (FPU) recurrence, pseudorecurrence, and chaos when different initial conditions are chosen. The fact that the system energy is effectively shared by finite Fourier modes suggests that it may be possible to describe the continuum system in terms of some effective degrees of freedom.
Nonlinear damping model for flexible structures. Ph.D. Thesis
NASA Technical Reports Server (NTRS)
Zang, Weijian
1990-01-01
The study of nonlinear damping problem of flexible structures is addressed. Both passive and active damping, both finite dimensional and infinite dimensional models are studied. In the first part, the spectral density and the correlation function of a single DOF nonlinear damping model is investigated. A formula for the spectral density is established with O(Gamma(sub 2)) accuracy based upon Fokker-Planck technique and perturbation. The spectral density depends upon certain first order statistics which could be obtained if the stationary density is known. A method is proposed to find the approximate stationary density explicitly. In the second part, the spectral density of a multi-DOF nonlinear damping model is investigated. In the third part, energy type nonlinear damping model in an infinite dimensional setting is studied.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
Khare, Avinash; Saxena, Avadh
2014-03-15
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1982-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
NASA Astrophysics Data System (ADS)
Junaid, Ali Khan; Muhammad, Asif Zahoor Raja; Ijaz Mansoor, Qureshi
2011-02-01
We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
New variable separation solutions for the generalized nonlinear diffusion equations
NASA Astrophysics Data System (ADS)
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
Nonlinear stability of oscillatory pulses in the parametric nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Chang, Paul A. C.; Promislow, Keith
2007-03-01
We extend the renormalization group method, developed for the study of pulse interaction in damped wave equations, to the study of oscillatory motion of supercritical pulses in the parametrically forced nonlinear Schrödinger equation (PNLS). We construct a global manifold which asymptotically attracts the flow into an {\\cal O}(r^4) neighbourhood in the H1 norm, where r is the amplitude of the internal oscillations. The oscillatory and translational dynamics of the pulses are rigorously recovered as a finite-dimensional flow on the manifold. The normal form for the projected dynamics of the oscillatory pulse shows that it is created in a supercritical Poincaré-Hopf bifurcation.
A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations
Güner, Özkan; Cevikel, Adem C.
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972
Nonlinear periodic waves solutions of the nonlinear self-dual network equations
Laptev, Denis V. Bogdan, Mikhail M.
2014-04-15
The new classes of periodic solutions of nonlinear self-dual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.
Charged anisotropic matter with linear or nonlinear equation of state
Varela, Victor; Rahaman, Farook; Ray, Saibal; Chakraborty, Koushik; Kalam, Mehedi
2010-08-15
Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplifications achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or nonlinear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge (10{sup 19}C) and maximum electric field intensities are very large (10{sup 23}-10{sup 24} statvolt/cm) even in the case of zero net charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations with larger net charges and electric field intensities than those found in studies of charged isotropic fluids. Links of these results with charged strange quark stars as well as models of dark matter including massive charged particles are highlighted. The van der Waals equation of state leading to matter densities constrained by cubic polynomial equations is briefly considered. The fundamental question of stability is left open.
NASA Astrophysics Data System (ADS)
Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua
2014-11-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
Belmonte-Beitia, J.; Cuevas, J.
2011-03-15
In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schroedinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions
Fibich, G. . E-mail: fibich@math.tau.ac.il; Tsynkov, S. . E-mail: tsynkov@math.ncsu.edu
2005-11-20
In [J. Comput. Phys. 171 (2001) 632-677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632-677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.
NASA Astrophysics Data System (ADS)
Zecca, Antonio
2016-02-01
The Dirac equation with nonlinear terms induced by torsion is studied in Robertson-Walker (RW) space-time. An extension of a separation method of the equation, based on the Newman-Penrose formalism and previously applied to the nonlinear case, is considered. Accordingly the angular dependence of the Dirac spinor solution is separated, under a special assumption, in the general time-dependent RW metric. In the case of static RW metric the time dependence of the Dirac spinor factors out and one is left with a pair of two coupled nonlinear radial equations. The radial equations are disentangled by a suitable substitution of the spinor solution. The problem amounts then to the solution of a single second-order highly nonlinear differential equation. Some elementary considerations are done on the asymptotic behavior of the solution of the equation.
Numerical solution of the nonlinear Schrödinger equation with wave operator on unbounded domains.
Li, Hongwei; Wu, Xiaonan; Zhang, Jiwei
2014-09-01
In this paper, we generalize the unified approach proposed in Zhang et al. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method. PMID:25314566
On the dynamics of approximating schemes for dissipative nonlinear equations
NASA Technical Reports Server (NTRS)
Jones, Donald A.
1993-01-01
Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.
Symmetry analysis and exact solutions for nonlinear equations in mathematical physics
NASA Astrophysics Data System (ADS)
Fushchich, Vil'gel'm. I.; Shtelen', Vladimir M.; Serov, Nikolai I.
The book provides an overview of the current status of theoretical-algebraic methods in relation to linear and nonlinear multidimensional equations in mathematical and theoretical physics that are invariant with respect to the Poincare and Galilean groups and the wider Lie groups. Particular attention is given to the construction, in explicit form, of wide classes of accurate solutions to specific nonlinear partial differential equations, such as nonlinear wave equations for scalar, spinor, and vector fields, Young-Mills equations, and nonlinear quantum electrodynamic equations. A group-theory approach is used to analyze the classical three-body problem.
A new method for parameter estimation in nonlinear dynamical equations
NASA Astrophysics Data System (ADS)
Wang, Liu; He, Wen-Ping; Liao, Le-Jian; Wan, Shi-Quan; He, Tao
2015-01-01
Parameter estimation is an important scientific problem in various fields such as chaos control, chaos synchronization and other mathematical models. In this paper, a new method for parameter estimation in nonlinear dynamical equations is proposed based on evolutionary modelling (EM). This will be achieved by utilizing the following characteristics of EM which includes self-organizing, adaptive and self-learning features which are inspired by biological natural selection, and mutation and genetic inheritance. The performance of the new method is demonstrated by using various numerical tests on the classic chaos model—Lorenz equation (Lorenz 1963). The results indicate that the new method can be used for fast and effective parameter estimation irrespective of whether partial parameters or all parameters are unknown in the Lorenz equation. Moreover, the new method has a good convergence rate. Noises are inevitable in observational data. The influence of observational noises on the performance of the presented method has been investigated. The results indicate that the strong noises, such as signal noise ratio (SNR) of 10 dB, have a larger influence on parameter estimation than the relatively weak noises. However, it is found that the precision of the parameter estimation remains acceptable for the relatively weak noises, e.g. SNR is 20 or 30 dB. It indicates that the presented method also has some anti-noise performance.
Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence
Hahm, T. S.; Wang, Lu; Madsen, J.
2008-08-01
An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E Χ B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρ_{i}<< ρ_{θ¡} ~ L_{E} ~ L_{p} << R (here ρ_{i} is the thermal ion Larmor radius and ρ_{θ¡} = B/B_{θ}] ρ_{i}), as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. We take κ perpendicular to ρ_{i} ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ /Τ_{i} ~ δΒ / Β up to the second order. Extending the electrostatic theory in the presence of high E Χ B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.
Hypocoercivity of linear degenerately dissipative kinetic equations
NASA Astrophysics Data System (ADS)
Duan, Renjun
2011-08-01
In this paper we develop a general approach of studying the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator are considered when the spatial domain takes the whole space or torus and when there is a confining force or not. The key part of the developed approach is to construct some equivalent temporal energy functionals for obtaining time rates of the solution trending towards equilibrium in some Hilbert spaces. The result in the case of the linear Boltzmann equation with confining forces is new. The proof mainly makes use of the macro-micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic-parabolic system. At the end, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.
Bayesian Analysis of Structural Equation Models with Nonlinear Covariates and Latent Variables
ERIC Educational Resources Information Center
Song, Xin-Yuan; Lee, Sik-Yum
2006-01-01
In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the…
Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative
Li, Qi; Duan, Qiu-yuan; Zhang, Jian-bing
2014-01-01
Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions. PMID:25013858
Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
Shao, Sihong; Quintero, Niurka R; Mertens, Franz G; Cooper, Fred; Khare, Avinash; Saxena, Avadh
2014-09-01
We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t(c), it takes for the instability to set in, is an exponentially increasing function of ω and t(c) decreases monotonically with increasing κ. PMID:25314512
Code System for Solving Nonlinear Systems of Equations via the Gauss-Newton Method.
1981-08-31
Version 00 REGN solves nonlinear systems of numerical equations in difficult cases: high nonlinearity, poor initial approximations, a large number of unknowns, ill condition or degeneracy of a problem.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1995-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that we currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Karr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
NASA Astrophysics Data System (ADS)
Reyes, M. A.; Gutiérrez-Ruiz, D.; Mancas, S. C.; Rosu, H. C.
2016-01-01
We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations when p = 2.
Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio
2014-01-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530
A globalization procedure for solving nonlinear systems of equations
NASA Astrophysics Data System (ADS)
Shi, Yixun
1996-09-01
A new globalization procedure for solving a nonlinear system of equationsF(x)D0 is proposed based on the idea of combining Newton step and the steepest descent step WITHIN each iteration. Starting with an arbitrary initial point, the procedure converges either to a solution of the system or to a local minimizer off(x)D1/2F(x)TF(x). Each iteration is chosen to be as close to a Newton step as possible and could be the Newton step itself. Asymptotically the Newton step will be taken in each iteration and thus the convergence is quadratic. Numerical experiments yield positive results. Further generalizations of this procedure are also discussed in this paper.
Canonical equations of Hamilton for the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Liang, Guo; Guo, Qi; Ren, Zhanmei
2015-09-01
We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.
Hyperbolicity of the Nonlinear Models of Maxwell's Equations
NASA Astrophysics Data System (ADS)
Serre, Denis
. We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class Hs with s>1+d/2. The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex. Since the energy conservation law does not derive from the system of conservation laws itself (Faraday's and Ampère's laws), but also needs the compatibility relations divB=divD=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.
Controlled Nonlinear Stochastic Delay Equations: Part I: Modeling and Approximations
Kushner, Harold J.
2012-08-15
This two-part paper deals with 'foundational' issues that have not been previously considered in the modeling and numerical optimization of nonlinear stochastic delay systems. There are new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. There are two basic and interconnected themes for these models. The first, dealt with in this part, concerns the definition of admissible control. The classical definition of an admissible control as a nonanticipative relaxed control is inadequate for these models and needs to be extended. This is needed for the convergence proofs of numerical approximations for optimal controls as well as to have a well-defined model. It is shown that the new classes of admissible controls do not enlarge the range of the value functions, is closed (together with the associated paths) under weak convergence, and is approximatable by ordinary controls. The second theme, dealt with in Part II, concerns transportation equation representations, and their role in the development of numerical algorithms with much reduced memory and computational requirements.
Statistical mechanics of the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Lebowitz, Joel L.; Rose, Harvey A.; Speer, Eugene R.
1988-02-01
We investigate the statistical mechanics of a complex field ø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian H(φ ) = int_Ω {[1/2|nabla φ |^2 - (1/p) |φ |^p ] dx} is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, when Ω is the circle and the L 2 norm of the field (which is conserved by the dynamics) is bounded by N, the Gibbs measure υ obtained is absolutely continuous with respect to Wiener measure and normalizable if and only if p and N are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, as N and the temperature are varied.
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Tadmor, Eitan
1989-01-01
Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.
Nonlinear Dirac equation solitary waves in external fields.
Mertens, Franz G; Quintero, Niurka R; Cooper, Fred; Khare, Avinash; Saxena, Avadh
2012-10-01
We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalar-scalar self-interaction g2/κ+1(Ψ[over ¯]Ψ)κ+1 in the presence of various external electromagnetic fields. We find exact solutions for special external fields and we study the behavior of solitary-wave solutions to the NLDE in the presence of a wide variety of fields in a variational approximation depending on collective coordinates which allows the position, width, and phase of these waves to vary in time. We find that in this approximation the position q(t) of the center of the solitary wave obeys the usual behavior of a relativistic point particle in an external field. For time-independent external fields, we find that the energy of the solitary wave is conserved but not the momentum, which becomes a function of time. We postulate that, similarly to the nonlinear Schrödinger equation (NLSE), a sufficient dynamical condition for instability to arise is that dP(t)/dq[over ̇](t)<0. Here P(t) is the momentum of the solitary wave, and q[over ̇] is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials that we always have dP(t)/dq[over ̇](t)>0, so, when instabilities do occur, they are due to a different source. We investigate the accuracy of our variational approximation using numerical simulations of the NLDE and find that, when the forcing term is small and we are in a regime where the solitary wave is stable, that the behavior of the solutions of the collective coordinate equations agrees very well with the numerical simulations. We found that the time evolution of the collective coordinates of the solitary wave in our numerical simulations, namely the position of the average charge density and the momentum of the solitary wave, provide good indicators for when the solitary wave first becomes unstable. When these variables stop being smooth functions of time (t), then the solitary wave starts to distort
A path integral approach to the Langevin equation
NASA Astrophysics Data System (ADS)
Das, Ashok K.; Panda, Sudhakar; Santos, J. R. L.
2015-02-01
We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck-Ornstein model). The path integral description also leads to a simple derivation of the Fokker-Planck equation for the generalized Langevin equation.
Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance
NASA Astrophysics Data System (ADS)
Fujiwara, Kazumasa; Ozawa, Tohru
2016-08-01
A lifespan estimate and a condition of the initial data for finite time blowup for the nonlinear Schrödinger equation are presented from a view point of ordinary differential equation (ODE) mechanism.
NASA Astrophysics Data System (ADS)
Martens, W.; von Wagner, U.; Litak, G.
2013-09-01
Recent years have shown increasing interest of researchers in energy harvesting systems designed to generate electrical energy from ambient energy sources, such as mechanical excitations. In a lot of cases excitation patterns of such systems exhibit random rather than deterministic behaviour with broad-band frequency spectra. In this paper, we study the efficiency of vibration energy harvesting systems with stochastic ambient excitations by solving corresponding Fokker-Planck equations. In the system under consideration, mechanical energy is transformed by a piezoelectric transducer in the presence of mechanical potential functions which are governed by magnetic fields applied to the device. Depending on the magnet positions and orientations the vibrating piezo beam system is subject to characteristic potential functions, including single and double well shapes. Considering random excitation, the probability density function (pdf) of the state variables can be calculated by solving the corresponding Fokker-Planck equation. For this purpose, the pdf is expanded into orthogonal polynomials specially adapted to the problem and the residual is minimized by a Galerkin procedure. The power output has been estimated as a function of basic potential function parameters determining the characteristic pdf shape.
Carasso, Alfred S
2013-01-01
Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930’s, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes. PMID:26401430
Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems
NASA Astrophysics Data System (ADS)
Colli, Pierluigi; Fukao, Takeshi
2016-05-01
An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved.
Stochastic approach to the generalized Schrödinger equation: A method of eigenfunction expansion.
Tsuchida, Satoshi; Kuratsuji, Hiroshi
2015-05-01
Using a method of eigenfunction expansion, a stochastic equation is developed for the generalized Schrödinger equation with random fluctuations. The wave field ψ is expanded in terms of eigenfunctions: ψ=∑(n)a(n)(t)ϕ(n)(x), with ϕ(n) being the eigenfunction that satisfies the eigenvalue equation H(0)ϕ(n)=λ(n)ϕ(n), where H(0) is the reference "Hamiltonian" conventionally called the "unperturbed" Hamiltonian. The Langevin equation is derived for the expansion coefficient a(n)(t), and it is converted to the Fokker-Planck (FP) equation for a set {a(n)} under the assumption of Gaussian white noise for the fluctuation. This procedure is carried out by a functional integral, in which the functional Jacobian plays a crucial role in determining the form of the FP equation. The analyses are given for the FP equation by adopting several approximate schemes. PMID:26066158
Exact solutions to a class of nonlinear Schrödinger-type equations
NASA Astrophysics Data System (ADS)
Zhang, Jin-Liang; Wang, Ming-Liang
2006-12-01
A class of nonlinear Schrödinger-type equations, including the Rangwal--Rao equation, the Gerdjikov--Ivanov equation, the Chen--Lee--Lin equation and the Ablowitz--Ramani--Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).
Analytical Solution of the Space-Time Fractional Nonlinear Schrödinger Equation
NASA Astrophysics Data System (ADS)
Abdel-Salam, Emad A.-B.; Yousif, Eltayeb A.; El-Aasser, Mostafa A.
2016-02-01
The space-time fractional nonlinear Schrödinger equation is solved by mean of on the fractional Riccati expansion method. These solutions include generalized trigonometric and hyperbolic functions which could be useful for further understanding of mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.
Stochastic Landau-Lifshitz-Gilbert Equation with Delayed Feedback Field
NASA Astrophysics Data System (ADS)
Tutu, H.; Horita, T.
2008-08-01
A time-delayed feedback control to stabilize a swinging motion of magnetic moment in a single-domain magnetic system under AC field is studied. The system has a uniaxial anisotropy, and the AC field is parallel to this. Without control, it prefers the Ising state that is (anti)parallel to the anisotropy axis. The control stabilizes the oscillation across the equatorial plane perpendicular to the anisotropy axis (swinging motion). Employing a stochastic Landau-Lifshitz-Gilbert (LLG) equation, we study the effects of thermal fluctuation on the controlled state. Linear fluctuation, in which variance linearly depends on noise intensity, around the controlled state is analyzed in terms of correlation function and spectral density, and a criterion for the existence of such a linear relationship is obtained. Several technical improvements in the treatment of the stochastic LLG equation and the corresponding Fokker-Planck equation with stereographic coordinate system are also show n.
ISDEP: Integrator of stochastic differential equations for plasmas
NASA Astrophysics Data System (ADS)
Velasco, J. L.; Bustos, A.; Castejón, F.; Fernández, L. A.; Martin-Mayor, V.; Tarancón, A.
2012-09-01
In this paper we present a general description of the ISDEP code (Integrator of Stochastic Differential Equations for Plasmas) and a brief overview of its physical results and applications so far. ISDEP is a Monte Carlo code that calculates the distribution function of a minority population of ions in a magnetized plasma. It solves the ion equations of motion taking into account the complex 3D structure of fusion devices, the confining electromagnetic field and collisions with other plasma species. The Monte Carlo method used is based on the equivalence between the Fokker-Planck and Langevin equations. This allows ISDEP to run in distributed computing platforms without communication between nodes with almost linear scaling. This paper intends to be a general description and a reference paper in ISDEP.
Soliton Theory of Two-Dimensional Lattices: The Discrete Nonlinear Schroedinger Equation
Arevalo, Edward
2009-06-05
We theoretically investigate the motion of collective excitations in the two-dimensional nonlinear Schroedinger equation with cubic nonlinearity. The form of these excitations for a broad range of parameters is derived. Their evolution and interaction is numerically studied and the modulation instability is discussed. The case of saturable nonlinearity is revisited.
NASA Astrophysics Data System (ADS)
Polyanin, Andrei D.; Zhurov, Alexei I.
2014-03-01
We propose a new method for constructing exact solutions to nonlinear delay reaction-diffusion equations of the form ut=kuxx+F(u,w), where u=u(x,t),w=u(x,t-τ), and τ is the delay time. The method is based on searching for solutions in the form u=∑n=1Nξn(x)ηn(t), where the functions ξn(x) and ηn(t) are determined from additional functional constraints (which are difference or functional equations) and the original delay partial differential equation. All of the equations considered contain one or two arbitrary functions of a single argument. We describe a considerable number of new exact generalized separable solutions and a few more complex solutions representing a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The results are extended to a wide class of nonlinear partial differential-difference equations involving arbitrary linear differential operators of any order with respect to the independent variables x and t (in particular, this class includes the nonlinear delay Klein-Gordon equation) as well as to some partial functional differential equations with time-varying delay.
Habib, S.
1994-10-01
We consider a simple quantum system subjected to a classical random force. Under certain conditions it is shown that the noise-averaged Wigner function of the system follows an integro-differential stochastic Liouville equation. In the simple case of polynomial noise-couplings this equation reduces to a generalized Fokker-Planck form. With nonlinear noise injection new ``quantum diffusion`` terms rise that have no counterpart in the classical case. Two special examples that are not of a Fokker-Planck form are discussed: the first with a localized noise source and the other with a spatially modulated noise source.
Stochastic cooling: recent theoretical directions
Bisognano, J.
1983-03-01
A kinetic-equation derivation of the stochastic-cooling Fokker-Planck equation of correlation is introduced to describe both the Schottky spectrum and signal suppression. Generalizations to nonlinear gain and coupling between degrees of freedom are presented. Analysis of bunch beam cooling is included.
Motsa, S S; Magagula, V M; Sibanda, P
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252
Motsa, S. S.; Magagula, V. M.; Sibanda, P.
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252
A simple and direct method for generating travelling wave solutions for nonlinear equations
Bazeia, D. Das, Ashok; Silva, A.
2008-05-15
We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.
On the Stability of Self-Similar Solutions to Nonlinear Wave Equations
NASA Astrophysics Data System (ADS)
Costin, Ovidiu; Donninger, Roland; Glogić, Irfan; Huang, Min
2016-04-01
We consider an explicit self-similar solution to an energy-supercritical Yang-Mills equation and prove its mode stability. Based on earlier work by one of the authors, we obtain a fully rigorous proof of the nonlinear stability of the self-similar blowup profile. This is a large-data result for a supercritical wave equation. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations.
Exact finite difference schemes for the non-linear unidirectional wave equation
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1985-01-01
Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.
Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
NASA Astrophysics Data System (ADS)
Yaşar, Emrullah; San, Sait; Özkan, Yeşim Sağlam
2016-01-01
In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.
Damping models in the truncated derivative nonlinear Schroedinger equation
Sanchez-Arriaga, G.; Sanmartin, J. R.; Elaskar, S. A.
2007-08-15
Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schroedinger equation, has been analyzed; wave 1 is linearly unstable with growth rate {gamma}, and waves 2 and 3 are stable with damping {gamma}{sub 2} and {gamma}{sub 3}, respectively. The dependence of gross dynamical features on the damping model (as characterized by the relation between damping and wave-vector ratios, {gamma}{sub 2}/{gamma}{sub 3}, k{sub 2}/k{sub 3}), and the polarization of the waves, is discussed; two damping models, Landau ({gamma}{proportional_to}k) and resistive ({gamma}{proportional_to}k{sup 2}), are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist for {gamma}<2({gamma}{sub 2}+{gamma}{sub 3})/3, as against flow contraction just requiring {gamma}<{gamma}{sub 2}+{gamma}{sub 3}. In the case of right-hand (RH) polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if {gamma}<({gamma}{sub 2}+{gamma}{sub 3})/2.
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics
NASA Astrophysics Data System (ADS)
Mirzazadeh, Mohammad; Ekici, Mehmet; Sonmezoglu, Abdullah; Ortakaya, Sami; Eslami, Mostafa; Biswas, Anjan
2016-05-01
This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi's elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.
NASA Astrophysics Data System (ADS)
Mohammed, K. Elboree
2015-10-01
In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation and complex coupled KdV system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations (NLEEs) in mathematical physics.
A more general model equation of nonlinear Rayleigh waves and their quasilinear solutions
NASA Astrophysics Data System (ADS)
Zhang, Shuzeng; Li, Xiongbing; Jeong, Hyunjo
2016-03-01
A more general two-dimensional wave motion equation with consideration of attenuation and nonlinearity is proposed to describe propagating nonlinear Rayleigh waves of finite amplitude. Based on the quasilinear theory, the numerical solutions for the sound beams of fundamental and second harmonic waves are constructed with Green’s function method. Compared with solutions from the parabolic approximate equation, results from the general equation have more accuracy in both the near distance of the propagation direction and the far distance of the transverse direction, as quasiplane waves are used and non-paraxial Green’s functions are obtained. It is more effective to obtain the nonlinear Rayleigh sound beam distributions accurately with the proposed general equation and solutions. Brief consideration is given to the measurement of nonlinear parameter using nonlinear Rayleigh waves.
On a hierarchy of nonlinearly dispersive generalized Korteweg - de Vries evolution equations
Christov, Ivan C.
2015-08-20
We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. Two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves (“peakompactons”) are presented.
Is the Langevin phase equation an efficient model for oscillating neurons?
NASA Astrophysics Data System (ADS)
Ota, Keisuke; Tsunoda, Takamasa; Omori, Toshiaki; Watanabe, Shigeo; Miyakawa, Hiroyoshi; Okada, Masato; Aonishi, Toru
2009-12-01
The Langevin phase model is an important canonical model for capturing coherent oscillations of neural populations. However, little attention has been given to verifying its applicability. In this paper, we demonstrate that the Langevin phase equation is an efficient model for neural oscillators by using the machine learning method in two steps: (a) Learning of the Langevin phase model. We estimated the parameters of the Langevin phase equation, i.e., a phase response curve and the intensity of white noise from physiological data measured in the hippocampal CA1 pyramidal neurons. (b) Test of the estimated model. We verified whether a Fokker-Planck equation derived from the Langevin phase equation with the estimated parameters could capture the stochastic oscillatory behavior of the same neurons disturbed by periodic perturbations. The estimated model could predict the neural behavior, so we can say that the Langevin phase equation is an efficient model for oscillating neurons.
Integrable pair-transition-coupled nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.
Integrable pair-transition-coupled nonlinear Schrödinger equations.
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system. PMID:26382492
Global solutions to two nonlinear perturbed equations by renormalization group method
NASA Astrophysics Data System (ADS)
Kai, Yue
2016-02-01
In this paper, according to the theory of envelope, the renormalization group (RG) method is applied to obtain the global approximate solutions to perturbed Burger's equation and perturbed KdV equation. The results show that the RG method is simple and powerful for finding global approximate solutions to nonlinear perturbed partial differential equations arising in mathematical physics.
NASA Technical Reports Server (NTRS)
Simon, M. K.
1980-01-01
A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated.
Projection methods for solving nonlinear systems of equations
Brown, P.N. ); Saad, Y. . Ames Research Center)
1990-04-01
This paper describes several nonlinear projection methods based on Krylov subspaces and analyzes their convergence. The prototype of these methods is a technique that generalizes the conjugate direction method by minimizing the norm of the function F over some subspace. The emphasis of this paper is on nonlinear least squares problems which can also be handled by this general approach.
Lump solitons in a higher-order nonlinear equation in 2 +1 dimensions
NASA Astrophysics Data System (ADS)
Estévez, P. G.; Díaz, E.; Domínguez-Adame, F.; Cerveró, Jose M.; Diez, E.
2016-06-01
We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to 2 +1 dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived after applying the singular manifold method. We also present an iterative procedure to construct the solutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed.
Lump solitons in a higher-order nonlinear equation in 2+1 dimensions.
Estévez, P G; Díaz, E; Domínguez-Adame, F; Cerveró, Jose M; Diez, E
2016-06-01
We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to 2+1 dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived after applying the singular manifold method. We also present an iterative procedure to construct the solutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed. PMID:27415266
Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America
Symmetry analysis for a class of nonlinear dispersive equations
NASA Astrophysics Data System (ADS)
Charalambous, K.; Sophocleous, C.
2015-05-01
A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.
Slyusarchuk, V. E. E-mail: V.Ye.Slyusarchuk@NUWM.rv.ua
2014-06-01
The well-known theorems of Favard and Amerio on the existence of almost periodic solutions to linear and nonlinear almost periodic differential equations depend to a large extent on the H-classes and the requirement that the bounded solutions of these equations be separated. The present paper provides different conditions for the existence of almost periodic solutions. These conditions, which do not depend on the H-classes of the equations, are formulated in terms of a special functional on the set of bounded solutions of the equations under consideration. This functional is used, in particular, to test whether solutions are separated. Bibliography: 24 titles. (paper)
NASA Astrophysics Data System (ADS)
Rashidi, M. M.; Erfani, E.
2009-09-01
In this study, we present a numerical comparison between the differential transform method (DTM) and the homotopy analysis method (HAM) for solving Burgers' and nonlinear heat transfer problems. The first differential equation is the Burgers' equation serves as a useful model for many interesting problems in applied mathematics. The second one is the modeling equation of a straight fin with a temperature dependent thermal conductivity. In order to show the effectiveness of the DTM, the results obtained from the DTM is compared with available solutions obtained using the HAM [M.M. Rashidi, G. Domairry, S. Dinarvand, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708-717; G. Domairry, M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 489-499] and whit exact solutions. The method can easily be applied to many linear and nonlinear problems. It illustrates the validity and the great potential of the differential transform method in solving nonlinear partial differential equations. The obtained results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations and nonlinear ordinary differential equations that we are found to be in good agreement with the exact solutions.
Stationary states of extended nonlinear Schrödinger equation with a source
NASA Astrophysics Data System (ADS)
Borich, M. A.; Smagin, V. V.; Tankeev, A. P.
2007-02-01
Structure of nonlinear stationary states of the extended nonlinear Schrödinger equation (ENSE) with a source has been analyzed with allowance for both third-order and nonlinearity dispersion. A new class of particular solutions (solitary waves) of the ENSe has been obtained. The scenario of the destruction of these states under the effect of an external perturbation has been investigated analytically and numerically. The results obtained can be used to interpret experimental data on the weakly nonlinear dynamics of the magnetostatic envelope in heterophase ferromagnet-insulator-metal, metal-insulator-ferromagnet-insulator-metal, and other similar structures and upon the simulation of nonlinear processes in optical systems.
NASA Astrophysics Data System (ADS)
Triki, Houria; Biswas, Anjan; Milović, Daniela; Belić, Milivoj
2016-05-01
We consider a high-order nonlinear Schrödinger equation with competing cubic-quintic-septic nonlinearities, non-Kerr quintic nonlinearity, self-steepening, and self-frequency shift. The model describes the propagation of ultrashort (femtosecond) optical pulses in highly nonlinear optical fibers. A new ansatz is adopted to obtain nonlinear chirp associated with the propagating femtosecond soliton pulses. It is shown that the resultant elliptic equation of the problem is of high order, contains several new terms and is more general than the earlier reported results, thus providing a systematic way to find exact chirped soliton solutions of the septic model. Novel soliton solutions, including chirped bright, dark, kink and fractional-transform soliton solutions are obtained for special choices of parameters. Furthermore, we present the parameter domains in which these optical solitons exist. The nonlinear chirp associated with each of the solitonic solutions is also determined. It is shown that the chirping is proportional to the intensity of the wave and depends on higher-order nonlinearities. Of special interest is the soliton solution of the bright and dark type, determined for the general case when all coefficients in the equation have nonzero values. These results can be useful for possible chirped-soliton-based applications of highly nonlinear optical fiber systems.
Some exact solutions of a system of nonlinear Schroedinger equations in three-dimensional space
Moskalyuk, S.S.
1988-02-01
Interactions that break the symmetry of systems of nonrelativistic Schroedinger equations but preserve their symmetry with respect to one-parameter subgroups of the Schroedinger group are described. Ansatzes for invariant solutions and the corresponding systems of reduced equations in invariant variables for Galileo-invariant Schroedinger equations are found. Exact solutions for the system of nonlinear Schroedinger equations in three-dimensional space for the generalized Hubbard model are obtained.
Construction of rogue wave and lump solutions for nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Lü, Zhuosheng; Chen, Yinnan
2015-07-01
Based on symbolic computation and an ansatz, we present a constructive algorithm to seek rogue wave and lump solutions for nonlinear evolution equations. As illustrative examples, we consider the potential-YTSF equation and a variable coefficient KP equation, and obtain nonsingular rational solutions of the two equations. The solutions can be rogue wave or lump solutions under different parameter conditions. We also present graphic illustration of some special solutions which would help better understand the evolution of solution waves.
Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation
Avelar, A. T.; Cardoso, W. B.; Bazeia, D.
2010-11-15
In this paper we present analytical breather solutions of the three-dimensional nonlinear generalized Gross-Pitaevskii equation. We use an Ansatz to reduce the three-dimensional equation with space- and time-dependent coefficients into a one-dimensional equation with constant coefficients. The key point is to show that both the space- and time-dependent coefficients of the nonlinear equation can contribute to modulate the breather excitations. We briefly discuss the experimental feasibility of the results in Bose-Einstein condensates.
Evolution of higher order nonlinear equation for the dust ion-acoustic waves in nonextensive plasma
Yasmin, S.; Asaduzzaman, M.; Mamun, A. A.
2012-10-15
There are three different types of nonlinear equations, namely, Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and mixed modified K-dV (mixed mK-dV) equations, for the nonlinear propagation of the dust ion-acoustic (DIA) waves. The effects of electron nonextensivity on DIA solitary waves propagating in a dusty plasma (containing negatively charged stationary dust, inertial ions, and nonextensive q distributed electrons) are examined by solving these nonlinear equations. The basic features of mixed mK-dV (higher order nonlinear equation) solitons are found to exist beyond the K-dV limit. The properties of mK-dV solitons are compared with those of mixed mK-dV solitons. It is found that both positive and negative solitons are obtained depending on the q (nonextensive parameter).
Generalized mean-field or master equation for nonlinear cavities with transverse effects.
Dunlop, A M; Firth, W J; Heatley, D R; Wright, E M
1996-06-01
We present a general form of master equation for nonlinear-optical cavities that can be described by an ABCD matrix. It includes as special cases some previous models of spatiotemporal effects in lasers. PMID:19876153
Analytical-numerical solution of a nonlinear integrodifferential equation in econometrics
NASA Astrophysics Data System (ADS)
Kakhktsyan, V. M.; Khachatryan, A. Kh.
2013-07-01
A mixed problem for a nonlinear integrodifferential equation arising in econometrics is considered. An analytical-numerical method is proposed for solving the problem. Some numerical results are presented.
On the structure of nonlinear constitutive equations for fiber reinforced composites
NASA Technical Reports Server (NTRS)
Jansson, Stefan
1992-01-01
The structure of constitutive equations for nonlinear multiaxial behavior of transversely isotropic fiber reinforced metal matrix composites subject to proportional loading was investigated. Results from an experimental program were combined with numerical simulations of the composite behavior for complex stress to reveal the full structure of the equations. It was found that the nonlinear response can be described by a quadratic flow-potential, based on the polynomial stress invariants, together with a hardening rule that is dominated by two different hardening mechanisms.
Vázquez, J. L.
2010-01-01
The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy–Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. PMID:20823259
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks
Smadbeck, Patrick; Kaznessis, Yiannis N.
2014-01-01
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized. PMID:25215268
NASA Astrophysics Data System (ADS)
Wang, Zhi-Cheng; Bu, Zhen-Hui
2016-04-01
This paper is concerned with nonplanar traveling fronts in reaction-diffusion equations with combustion nonlinearity and degenerate Fisher-KPP nonlinearity. Our study contains two parts: in the first part we establish the existence of traveling fronts of pyramidal shape in R3, and in the second part we establish the existence and stability of V-shaped traveling fronts in R2.
Garbow, B.S.; Hillstrom, K.E.; More, J.J.
1980-07-01
MINPACK-1 is a package of Fortran subprograms for the numerical solution of systems of nonlinear equations and nonlinear least-squares problems. This report describes how to implement the package from the tape on which it is transmitted. 3 tables.
Infinite hierarchy of nonlinear Schrödinger equations and their solutions
NASA Astrophysics Data System (ADS)
Ankiewicz, A.; Kedziora, D. J.; Chowdury, A.; Bandelow, U.; Akhmediev, N.
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.
Infinite hierarchy of nonlinear Schrödinger equations and their solutions.
Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex. PMID:26871072
Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades
NASA Technical Reports Server (NTRS)
Hodges, D. H.; Dowell, E. H.
1974-01-01
The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.
Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations
NASA Astrophysics Data System (ADS)
Collier, N.; Knepley, M.
2015-12-01
The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).
Differential geometry techniques for sets of nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
NASA Astrophysics Data System (ADS)
Akbari, M. R.; Ganji, D. D.; Rostami, A. K.; Nimafar, M.
2015-03-01
In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by a simple and innovative approach, which has been called Akbari-Ganji's method (AGM). AGM is a very suitable computational process and is usable for solving various nonlinear differential equations. Moreover, using AGM which solving a set of algebraic equations, complicated nonlinear equations can easily be solved without any mathematical operations. Also, the damping ratio and energy lost per cycle for three cycles have been investigated. Furthermore, comparisons have been made between the obtained results by numerical method (Runk45) and AGM. Results showed the high accuracy of AGM. The results also showed that by increasing the amount of initial amplitude of vibration ( A), the value of damping ratio will be increased, and the energy lost per cycle decreases by increasing the number of cycle. It is concluded that AGM is a reliable and precise approach for solving differential equations. On the other hand, it is better to say that AGM is able to solve linear and nonlinear differential equations directly in most of the situations. This means that the final solution can be obtained without any dimensionless procedure. Therefore, AGM can be considered as a significant progress in nonlinear sciences.
NASA Astrophysics Data System (ADS)
Nakao, Mitsuhiro
We prove the existence of global decaying solutions to the exterior problem for the Klein-Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity. To derive the required estimates of solutions we employ a 'loan' method.
NASA Astrophysics Data System (ADS)
Zhang, Zai-yun; Li, Yun-xiang; Liu, Zhen-hai; Miao, Xiu-jin
2011-08-01
In this paper, the modified trigonometric function series method is employed to solve the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. Exact traveling wave solutions are obtained.
On the numerical treatment of nonlinear source terms in reaction-convection equations
NASA Technical Reports Server (NTRS)
Lafon, A.; Yee, H. C.
1992-01-01
The objectives of this paper are to investigate how various numerical treatments of the nonlinear source term in a model reaction-convection equation can affect the stability of steady-state numerical solutions and to show under what conditions the conventional linearized analysis breaks down. The underlying goal is to provide part of the basic building blocks toward the ultimate goal of constructing suitable numerical schemes for hypersonic reacting flows, combustions and certain turbulence models in compressible Navier-Stokes computations. It can be shown that nonlinear analysis uncovers much of the nonlinear phenomena which linearized analysis is not capable of predicting in a model reaction-convection equation.
Choas and instabilities in finite difference approximations to nonlinear differential equations
Cloutman, L. D., LLNL
1998-07-01
The numerical solution of time-dependent ordinary and partial differential equations by finite difference techniques is a common task in computational physics and engineering The rate equations for chemical kinetics in combustion modeling are an important example. They not only are nonlinear, but they tend to be stiff, which makes their solution a challenge for transient problems. We show that one must be very careful how such equations are solved In addition to the danger of large time-marching errors, there can be unphysical chaotic solutions that remain numerically stable for a range of time steps that depends on the particular finite difference method used We point out that the solutions of the finite difference equations converge to those of the differential equations only in the limit as the time step approaches zero for stable and consistent finite difference approximations The chaotic behavior observed for finite time steps in some nonlinear difference equations is unrelated to solutions of the differential equations, but is connected with the onset of numerical instabilities of the finite difference equations This behavior suggests that the use of the theory of chaos in nonlinear iterated maps may be useful in stability anlaysis of finite difference approximations to nonlinear differential equations, providing more stringent time step limits than the formal linear stability analysis that tests only for unbounded solutions This observation implies that apparently stable numerical solutions of nonlinear differential equations by finite difference techniques may in fact be contaminated (if not dominated) by nonphysical chaotic parasitic solutions that degrade the accuracy of the numerical solution We demonstrate this phenomenon with some solutions of the logistic equation and a simple two-dimensional computational fluid dynamics example
Exact Solutions for N-Coupled Nonlinear Schrödinger Equations With Variable Coefficients
NASA Astrophysics Data System (ADS)
Tang, Bo; Fan, Yingzhe; Wang, Jixiu; Chen, Shijun
2016-07-01
In this paper, based on similarity transformation and auxiliary equation method, we construct many exact solutions of N-coupled nonlinear Schrödinger equations with variable coefficients, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions. These solutions may give insight into many considerable physical processes.
Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave
NASA Technical Reports Server (NTRS)
Khazanov, G. V.; Krivorutsky, E. N.; Whitaker, Ann F. (Technical Monitor)
2001-01-01
Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account.
Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave
NASA Technical Reports Server (NTRS)
Khazanov, G. V.; Krivorutsky, E. N.; Six, N. Frank (Technical Monitor)
2002-01-01
Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account. The time dependent part of the ponderomotive force is discussed.
NASA Astrophysics Data System (ADS)
Wang, Jing; You, Jiangong
2016-07-01
We study the boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequencies. We proved that if the forcing is quasi-periodic in time with two frequencies which is not super-Liouvillean, then all solutions of the equation are bounded. The proof is based on action-angle variables and modified KAM theory.
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities
NASA Technical Reports Server (NTRS)
Beeler, Scott C.; Cox, David E. (Technical Monitor)
2004-01-01
The state-dependent Riccati equations (SDRE) is the basis of a technique for suboptimal feedback control of a nonlinear quadratic regulator (NQR) problem. It is an extension of the Riccati equation used for feedback control of linear problems, with the addition of nonlinearities in the state dynamics of the system resulting in a state-dependent gain matrix as the solution of the equation. In this paper several variations on the SDRE-based method will be considered for the feedback control problem with control nonlinearities. The control nonlinearities may result in complications in the numerical implementation of the control, which the different versions of the SDRE method must try to overcome. The control methods will be applied to three test problems and their resulting performance analyzed.
Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation
NASA Astrophysics Data System (ADS)
Mani Rajan, M. S.; Mahalingam, A.; Uthayakumar, A.
2014-07-01
We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.
Construction of the wave operator for non-linear dispersive equations
NASA Astrophysics Data System (ADS)
Tsuruta, Kai Erik
In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution ψlin to the dispersive equation with no non-linearity and initial data ψ +, there exists a unique solution ψ to the non-linear equation with initial data ψ0 such that ψ behaves as ψ lin as t → infinity. Then the wave operator is the map W+ that takes ψ + to ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the "semi-relativistic" Schrodinger equation as well as the Klein-Gordon-Schrodinger system on R2 . In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.
Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types
NASA Astrophysics Data System (ADS)
İsmail, Aslan
2016-01-01
Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous. Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations, the majority of the results deal with polynomial types. Limited research has been reported regarding such equations of rational type. In this paper we present an adaptation of the (G‧/G)-expansion method to solve nonlinear rational differential-difference equations. The procedure is demonstrated using two distinct equations. Our approach allows one to construct three types of exact traveling wave solutions (hyperbolic, trigonometric, and rational) by means of the simplified form of the auxiliary equation method with reduced parameters. Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.
A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms
NASA Astrophysics Data System (ADS)
Lee, Hyun Geun; Lee, June-Yub
2015-08-01
Allen-Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decompose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.
Brugarino, Tommaso; Sciacca, Michele
2010-09-15
In this paper, we investigate the integrability of an inhomogeneous nonlinear Schroedinger equation, which has several applications in many branches of physics, as in Bose-Einstein condensates and fiber optics. The main issue deals with Painleve property (PP) and Liouville integrability for a nonlinear Schroedinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose-Einstein condensates are proposed (including Bose-Einstein condensates in three-dimensional in cylindrical symmetry).
NASA Astrophysics Data System (ADS)
Chen, Lin-Jie; Ma, Chang-Feng
2010-01-01
This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form ut + αuux + βunux + γuxx + δuxxx + ζuxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.
Equations of nonlinear dynamics of elastic shells in cylindrical Eulerian coordinates
NASA Astrophysics Data System (ADS)
Zubov, L. M.
2016-05-01
The equations of dynamics of elastic shells subjected to large deformations are formulated. The Eulerian coordinates on a circular cylinder and time are accepted as independent variables, and one of the unknown functions is the distance from a point of the shell surface to the cylinder axis. The equations of dynamics of nonlinearly elastic shells in the Eulerian coordinates are convenient for exact formulation of the problem on the interaction of strongly deformable shells with moving fluids and gases. The equations obtained can be used for dynamic calculations of fluids and gases flowings in pipelines, blood vessels, hoses, and other nonlinearly deformable thin-walled tubular elements of constructions.
Describing function method applied to solution of nonlinear heat conduction equation
Nassersharif, B. )
1989-08-01
Describing functions have traditionally been used to obtain the solutions of systems of ordinary differential equations. The describing function concept has been extended to include the non-linear, distributed parameter solid heat conduction equation. A four-step solution algorithm is presented that may be applicable to many classes of nonlinear partial differential equations. As a specific application of the solution technique, the one-dimensional nonlinear transient heat conduction equation in an annular fuel pin is considered. A computer program was written to calculate one-dimensional transient heat conduction in annular cylindrical geometry. It is found that the quasi-linearization used in the describing function method is as accurate as and faster than true linearization methods.
Numerical solutions of Maxwell's equations for nonlinear-optical pulse propagation
NASA Astrophysics Data System (ADS)
Hile, Cheryl V.; Kath, William L.
1996-06-01
A model and numerical solutions of Maxwell's equations describing the propagation of short, solitonlike pulses in nonlinear dispersive optical media are presented. The model includes linear dispersion expressed in the time domain, a Kerr nonlinearity, and a coordinate system moving with the group velocity of the pulse. Numerical solutions of Maxwell's equations are presented for circularly polarized and linearly polarized electromagnetic fields. When the electromagnetic fields are assumed to be circularly polarized, numerical solutions are compared directly with solutions of the nonlinear Schrodinger (NLS) equation. These comparisons show good agreement and indicate that the NLS equation provides an excellent model for short-pulse propagation. When the electromagnetic fields are assumed to be linearly polarized, the propagation of daughter pulses, small-amplitude pulses at three times the frequency of the solitonlike pulse, are observed in the numerical solution. These daughter pulses are shown to be the direct result of third harmonics generated by the main, solitonlike, pulse.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions
Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.
2015-01-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256
Application of variational and Galerkin equations to linear and nonlinear finite element analysis
NASA Technical Reports Server (NTRS)
Yu, Y.-Y.
1974-01-01
The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.
Nonlinear Schroedinger equation and the Bogolyubov-Whitham method of averaging
Pavlov, M.V.
1987-12-01
An averaging is investigated for the nonlinear Schroedinger equation using the technique of finite-gap averaging. For the single-gap case, the results are given explicitly. Some characteristics of the original equation needed for applied calculations are averaged. Finally, recursion and functional formulas connecting the densities of the integrals of the motion of the Schroedinger equation, the fluxes, and the variational derivatives are given.
NASA Astrophysics Data System (ADS)
Yan, Zhenya; Bluman, George
2002-11-01
The special exact solutions of nonlinearly dispersive Boussinesq equations (called B( m, n) equations), utt- uxx- a( un) xx+ b( um) xxxx=0, is investigated by using four direct ansatze. As a result, abundant new compactons: solitons with the absence of infinite wings, solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions of these two equations are obtained. The variant is extended to include linear dispersion to support compactons and solitary patterns in the linearly dispersive Boussinesq equations with m=1. Moreover, another new compacton solution of the special case, B(2,2) equation, is also found.
NASA Technical Reports Server (NTRS)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Parashar, R.; Cushman, J.H.
2008-06-20
Microbial motility is often characterized by 'run and tumble' behavior which consists of bacteria making sequences of runs followed by tumbles (random changes in direction). As a superset of Brownian motion, Levy motion seems to describe such a motility pattern. The Eulerian (Fokker-Planck) equation describing these motions is similar to the classical advection-diffusion equation except that the order of highest derivative is fractional, {alpha} element of (0, 2]. The Lagrangian equation, driven by a Levy measure with drift, is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries. The Eulerian equation is used to non-dimensionalize parameters. The amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values. Salient features of first passage time are studied with respect to scaled parameters.
Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations
NASA Astrophysics Data System (ADS)
Sotoudeh, Zahra
2011-07-01
Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis. This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables. In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research. This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.
NASA Astrophysics Data System (ADS)
Cheng, Xing; Miao, Changxing; Zhao, Lifeng
2016-09-01
We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in H1 (Rd) and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in H1 (Rd) below the threshold for radial data when d ≤ 4.
Study of nonlinear waves described by the cubic Schroedinger equation
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
Polynomial elimination theory and non-linear stability analysis for the Euler equations
NASA Technical Reports Server (NTRS)
Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.
1986-01-01
Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.
Entropy Solutions to a Genuinely Nonlinear Ultraparabolic Kolmogorov-Type Equation
NASA Astrophysics Data System (ADS)
Sazhenkov, S. A.
2007-04-01
We consider a non-isotropic convection-diffusion-reaction equation of a very general form, in which the diffusion matrix is nonnegative and may change its rank depending on temporal and spatial variables, and convection and reaction terms may be discontinuous. This equation arises in astrophysics and plasma physics, in fluid dynamics, mathematical biology and financial mathematics. We assume that the equation a priori admits the maximum principle and is genuinely nonlinear, and we prove that there exists at least one entropy solution and that the genuinely nonlinear structure of the equation rules out fine oscillatory regimes in entropy solutions. The proofs rely on the method of kinetic equation and on theory of H-measures.
NASA Technical Reports Server (NTRS)
Nemeth, Michael P.
2010-01-01
A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.
Dromion interactions of (2+1)-dimensional nonlinear evolution equations
Ruan; Chen
2000-10-01
Starting from two line solitons, the solution of integrable (2+1)-dimensional mKdV system and KdV system in bilinear form yields a dromion solution or a "Solitoff" solution. Such a dromion solution is localized in all directions and the Solitoff solution decays exponentially in all directions except a preferred one for the physical field or a suitable potential. The interactions between two dromions and between the dromion and Solitoff are studied by the method of figure analysis for a (2+1)-dimensional modified KdV equation and a (2+1)-dimensional KdV type equation. Our analysis shows that the interactions between two dromions may be elastic or inelastic for different forms of solutions. PMID:11089133
Solutions of perturbed p-Laplacian equations with critical nonlinearity
NASA Astrophysics Data System (ADS)
Wang, Chunhua; Wang, Jiangtao
2013-01-01
In this paper, we study a perturbed p-Laplacian equation. Under some given conditions on V(x), we prove that the equation has at least one positive solution provided that ɛ le E; for any n^{*}in {N}, it has at least n* pairs solutions if ɛ le E_{n^{*}}; and suppose there exists an orthogonal involution T:{R}NrArr {R}N such that V(x), P(x), and K(x) are T-invariant, then it has at least one pair of solutions, which change sign exactly once provided that ɛ le E, where E and E_{n^{*}} are sufficiently small positive numbers. Moreover, these solutions uɛ → 0 in W^{1,p}({R}N) as ɛ → 0.
Dynamics of cubic–quintic nonlinear Schrödinger equation with different parameters
NASA Astrophysics Data System (ADS)
Wei, Hua; Xue-Shen, Liu; Shi-Xing, Liu
2016-05-01
We study the dynamics of the cubic–quintic nonlinear Schrödinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter. Project supported by the National Natural Science Foundation of China (Grant Nos. 11301350, 11472124, and 11271158) and the Doctor Start-up Fund in Liaoning Province, China (Grant No. 20141050).
NASA Astrophysics Data System (ADS)
Wang, Lei; Zhu, Yu-Jie; Wang, Zi-Zhe; Qi, Feng-Hua; Guo, Rui
2016-04-01
We present the semirational solution in terms of the determinant form for the derivative nonlinear Schrödinger equation. It describes the nonlinear combinations of breathers and rogue waves (RWs). We show here that the solution appears as a mixture of polynomials with exponential functions. The k-order semirational solution includes k - 1 types of nonlinear superpositions, i.e., the l-order RW and (k-l)-order breather for l = 1 , 2 , … , k - 1 . By adjusting the shift and spectral parameters, we display various patterns of the semirational solutions for describing the interactions among the RWs and breathers. We find that k-order RW can be derived from a l-order RW interacting with 1/2(k - l) (k + l + 1) neighboring elements of a (k - l)-order breather for l = 1 , 2 , … , k - 1 .
Initial Value Problem Solution of Nonlinear Shallow Water-Wave Equations
Kanoglu, Utku; Synolakis, Costas
2006-10-06
The initial value problem solution of the nonlinear shallow water-wave equations is developed under initial waveforms with and without velocity. We present a solution method based on a hodograph-type transformation to reduce the nonlinear shallow water-wave equations into a second-order linear partial differential equation and we solve its initial value problem. The proposed solution method overcomes earlier limitation of small waveheights when the initial velocity is nonzero, and the definition of the initial conditions in the physical and transform spaces is consistent. Our solution not only allows for evaluation of differences in predictions when specifying an exact initial velocity based on nonlinear theory and its linear approximation, which has been controversial in geophysical practice, but also helps clarify the differences in runup observed during the 2004 and 2005 Sumatran tsunamigenic earthquakes.
An ansatz for solving nonlinear partial differential equations in mathematical physics.
Akbar, M Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems. PMID:26783508
Coding of Nonlinear States for NLS-Type Equations with Periodic Potential
NASA Astrophysics Data System (ADS)
Alfimov, G. L.; Avramenko, A. I.
The problem of complete description of nonlinear states for NLS-type equations with periodic potential is considered. We show that in some cases all nonlinear states for equations of such kind can be coded by bi-infinite sequences of symbols of N-symbol alphabet (words). Sufficient conditions for one-to-one correspondence between the set of nonlinear states and the set of these bi-infinite words are given in the form convenient for numerical verification (Hypotheses 1-3). We report on numerical check of these hypotheses for the case of Gross-Pitaevskii equation with cosine potential and indicate regions in the space of governing parameters where this coding is possible.
Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Ma, Zheng-Yi; Ma, Song-Hua
2012-03-01
Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.
Crosta, M.; Fratalocchi, A.; Trillo, S.
2011-12-15
We characterize the full family of soliton solutions sitting over a background plane wave and ruled by the cubic-quintic nonlinear Schroedinger equation in the regime where a quintic focusing term represents a saturation of the cubic defocusing nonlinearity. We discuss the existence and properties of solitons in terms of catastrophe theory and fully characterize bistability and instabilities of the dark-antidark pairs, revealing mechanisms of decay of antidark solitons into dispersive shock waves.
NASA Technical Reports Server (NTRS)
Rosen, A.; Friedmann, P. P.
1978-01-01
A set of nonlinear equations of equilibrium for an elastic wind turbine or helicopter blades are presented. These equations are derived for the case of small strains and moderate rotations (slopes). The derivation includes several assumptions which are carefully stated. For the convenience of potential users the equations are developed with respect to two different systems of coordinates, the undeformed and the deformed coordinates of the blade. Furthermore, the loads acting on the blade are given in a general form so as to make them suitable for a variety of applications. The equations obtained in the study are compared with those obtained in previous studies.
NASA Astrophysics Data System (ADS)
Katzav, Eytan
2002-06-01
In this paper I discuss a generalization of the well-known Kardar-Parisi-Zhang (KPZ) equation that includes long-range interactions. This Non-local Kardar-Parisi-Zhang (NKPZ) equation has been suggested in the past to describe physical phenomena such as burning paper or deposition of colloids. I show that the steady state strong coupling solution for a subfamily of the NKPZ models can be solved exactly in one dimension, using the Fokker-Planck form of the equation, and yields a Gaussian distribution. This exact result does not agree with a previous result obtained by dynamic renormalization group (DRG) analysis. The reasons for this disagreement are not yet clear.
A class of nonlinear differential equations with fractional integrable impulses
NASA Astrophysics Data System (ADS)
Wang, JinRong; Zhang, Yuruo
2014-09-01
In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki's normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki-Ulam's type stability are introduced and generalized Ulam-Hyers-Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.
Modeling taper charge with a non-linear equation
NASA Technical Reports Server (NTRS)
Mcdermott, P. P.
1985-01-01
Work aimed at modeling the charge voltage and current characteristics of nickel-cadmium cells subject to taper charge is presented. Work reported at previous NASA Battery Workshops has shown that the voltage of cells subject to constant current charge and discharge can be modeled very accurately with the equation: voltage = A + (B/(C-X)) + De to the -Ex where A, B, D, and E are fit parameters and x is amp-hr of charge removed during discharge or returned during charge. In a constant current regime, x is also equivalent to time on charge or discharge.
NASA Astrophysics Data System (ADS)
Ayhan, Burcu; Özer, M. Naci; Bekir, Ahmet
2016-08-01
In this article, we applied the method of multiple scales for Korteweg-de Vries (KdV) type equations and we derived nonlinear Schrödinger (NLS) type equations. So we get a relation between KdV type equations and NLS type equations. In addition, exact solutions were found for KdV type equations. The ( G'} over G )-expansion methods and the ( {G'} over G, {1 over G}} )-expansion methods were proposed to establish new exact solutions for KdV type differential equations. We obtained periodic and hyperbolic function solutions for these equations. These methods are very effective for getting travelling wave solutions of nonlinear evolution equations (NEEs).
Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation
Mani Rajan, M.S.; Mahalingam, A.; Uthayakumar, A.
2014-07-15
We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management. -- Highlights: •We consider the nonlinear tunneling of soliton in birefringence fiber. •3-coupled NLS (CNLS) equation with variable coefficients is considered. •Two soliton solutions are obtained via Darboux transformation using constructed Lax pair. •Soliton tunneling through dispersion barrier and well are investigated. •Finally, cascade compression of soliton has been achieved.
NASA Astrophysics Data System (ADS)
Irving, A. D.; Dewson, T.
1997-02-01
A new method is described for extracting mixed linear-nonlinear coupled differential equations from multivariate discrete time series data. It is assumed in the present work that the solution of the coupled ordinary differential equations can be represented as a multivariate Volterra functional expansion. A tractable hierarchy of moment equations is generated by operating on a suitably truncated Volterra functional expansion. The hierarchy facilitates the calculation of the coefficients of the coupled differential equations. In order to demonstrate the method's ability to accurately estimate the coefficients of the governing differential equations, it is applied to data derived from the numerical solution of the Lorenz equations with additive noise. The method is then used to construct a dynamic global mid- and high-magnetic latitude ionospheric model where nonlinear phenomena such as period doubling and quenching occur. It is shown that the estimated inhomogeneous coupled second-order differential equation model for the ionospheric foF2 peak plasma density can accurately forecast the future behaviour of a set of ionosonde stations which encompass the earth. Finally, the method is used to forecast the future behaviour of a portfolio of Japanese common stock prices. The hierarchy method can be used to characterise the observed behaviour of a wide class of coupled linear and mixed linear-nonlinear phenomena.
Evaluation of Uncertainty in Runoff Analysis Incorporating Theory of Stochastic Process
NASA Astrophysics Data System (ADS)
Yoshimi, Kazuhiro; Wang, Chao-Wen; Yamada, Tadashi
2015-04-01
The aim of this paper is to provide a theoretical framework of uncertainty estimate on rainfall-runoff analysis based on theory of stochastic process. SDE (stochastic differential equation) based on this theory has been widely used in the field of mathematical finance due to predict stock price movement. Meanwhile, some researchers in the field of civil engineering have investigated by using this knowledge about SDE (stochastic differential equation) (e.g. Kurino et.al, 1999; Higashino and Kanda, 2001). However, there have been no studies about evaluation of uncertainty in runoff phenomenon based on comparisons between SDE (stochastic differential equation) and Fokker-Planck equation. The Fokker-Planck equation is a partial differential equation that describes the temporal variation of PDF (probability density function), and there is evidence to suggest that SDEs and Fokker-Planck equations are equivalent mathematically. In this paper, therefore, the uncertainty of discharge on the uncertainty of rainfall is explained theoretically and mathematically by introduction of theory of stochastic process. The lumped rainfall-runoff model is represented by SDE (stochastic differential equation) due to describe it as difference formula, because the temporal variation of rainfall is expressed by its average plus deviation, which is approximated by Gaussian distribution. This is attributed to the observed rainfall by rain-gauge station and radar rain-gauge system. As a result, this paper has shown that it is possible to evaluate the uncertainty of discharge by using the relationship between SDE (stochastic differential equation) and Fokker-Planck equation. Moreover, the results of this study show that the uncertainty of discharge increases as rainfall intensity rises and non-linearity about resistance grows strong. These results are clarified by PDFs (probability density function) that satisfy Fokker-Planck equation about discharge. It means the reasonable discharge can be
NASA Astrophysics Data System (ADS)
An, Yulian; Kim, Chan-Gyun; Shi, Junping
2016-02-01
A p-Laplacian nonlinear elliptic equation with positive and p-superlinear nonlinearity and Dirichlet boundary condition is considered. We first prove the existence of two positive solutions when the spatial domain is symmetric or strictly convex by using a priori estimates and topological degree theory. For the ball domain in RN with N ≥ 4 and the case that 1 < p < 2, we prove that the equation has exactly two positive solutions when a parameter is less than a critical value. Bifurcation theory and linearization techniques are used in the proof of the second result.
Combined solitons in generalized coupled mode equations of a nonlinear optical Bragg grating
Alatas, Husin
2007-08-15
We discuss the existence of combined dark and antidark soliton forms or combined solitons in the generalized coupled mode equations of a nonlinear optical Bragg grating. These solitons are not allowed in the conventional coupled mode equations with uniform nonlinearity and exist outside the linear grating band gap. Their related Hamiltonian phase portrait was briefly reported by de Sterke et al. [Phys. Rev. E 54, 1969 (1996)]. The explicit expressions for the corresponding solitons are presented, as well as their bifurcation process. We demonstrate the unstable propagation of perturbed combined solitons with zero velocity by means of direct numerical integration.
Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation
NASA Astrophysics Data System (ADS)
Xiong, Chi; Good, Michael R. R.; Guo, Yulong; Liu, Xiaopei; Huang, Kerson
2014-12-01
We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the nonrelativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.
Nonlinear Schrödinger equation in foundations: summary of 4 catches
NASA Astrophysics Data System (ADS)
Diósi, Lajos
2016-03-01
Fundamental modifications of the standard Schrödinger equation by additional nonlinear terms have been considered for various purposes over the recent decades. It came as a surprise when, inverting Abner Shimonyi's observation of “peaceful coexistence” between standard quantum mechanics and relativity, N. Gisin proved in 1990 that any (deterministic) nonlinear Schrödinger equation would allow for superluminal communication. This is by now the most spectacular and best known anomaly. We discuss further anomalies, simple but foundational, less spectacular but not less dramatic.