Multi-diffusive nonlinear Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Ribeiro, Mauricio S.; Casas, Gabriela A.; Nobre, Fernando D.
2017-02-01
Nonlinear Fokker-Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker-Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker-Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker-Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution.
Curl forces and the nonlinear Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Wedemann, R. S.; Plastino, A. R.; Tsallis, C.
2016-12-01
Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are q exponentials of an appropriate potential function, are determined. It is proved that when these stationary solutions exist, the nonlinear Fokker-Planck equations satisfy an H theorem in terms of a free-energy-like quantity involving the Sq entropy. A particular two-dimensional model admitting analytical, time-dependent q -Gaussian solutions is discussed in detail. This model describes a system of particles with short-range interactions, performing overdamped motion under drag effects due to a rotating resisting medium. It is related to models that have been recently applied to the study of type-II superconductors. The relevance of the present developments to the study of complex systems in physics, astronomy, and biology is discussed.
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.
Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution.
Sicuro, Gabriele; Rapčan, Peter; Tsallis, Constantino
2016-12-01
We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.
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.
Daly, Edoardo; Porporato, Amilcare
2004-11-01
Similarity solutions of the shallow-water equation with a generalized resistance term are studied for open channel flows when both inertial and gravity forces are negligible. The resulting model encompasses various particular cases that appear, in addition to mathematical hydraulics, in diverse physical phenomena, such as gravity currents, creeping flows of Newtonian and non-Newtonian fluids, thin films, and nonlinear Fokker-Planck equations. Solutions of both source-type and dam-break problems are analyzed. Closed-form solutions are discussed, when possible, along with a qualitative study of two phase-plane formulations based on two different variable transformations.
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.
How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?
Grima, Ramon; Thomas, Philipp; Straube, Arthur V
2011-08-28
The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.
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.
Fokker Planck equation with fractional coordinate derivatives
NASA Astrophysics Data System (ADS)
Tarasov, Vasily E.; Zaslavsky, George M.
2008-11-01
Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.
Invariants of Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Abe, Sumiyoshi
2017-02-01
A weak invariant of a stochastic system is defined in such a way that its expectation value with respect to the distribution function as a solution of the associated Fokker-Planck equation is constant in time. A general formula is given for time evolution of the fluctuations of the invariant. An application to the problem of share price in finance is illustrated. It is shown how this theory makes it possible to reduce the growth rate of the fluctuations.
NASA Astrophysics Data System (ADS)
Stahl, A.; Landreman, M.; Embréus, O.; Fülöp, T.
2017-03-01
Energetic electrons are of interest in many types of plasmas, however previous modeling of their properties has been restricted to the use of linear Fokker-Planck collision operators or non-relativistic formulations. Here, we describe a fully non-linear kinetic-equation solver, capable of handling large electric-field strengths (compared to the Dreicer field) and relativistic temperatures. This tool allows modeling of the momentum-space dynamics of the electrons in cases where strong departures from Maxwellian distributions may arise. As an example, we consider electron runaway in magnetic-confinement fusion plasmas and describe a transition to electron slide-away at field strengths significantly lower than previously predicted.
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.
Kheifets, S.
1982-07-01
For an electron storage ring the beam size evaluation including beam-beam interaction gives an example of such a problem. Another good example is finding the beam size for a nonlinear machine. The present work gives a way to solve some of these problems, at least in principle. The approach described here is an application of the well known Green's function method, which in this case is applied to the Fokker-Planck equation governing the distribution function in the phase space of particle motion. The new step made in this paper is to consider the particle motion in two degrees of freedom rather than in one dimension, a characteristic of all the previous work. This step seems to be necessary for an adequate description of the problem, at least for the class of problems which are considered below. This work consists of the formal solution of the Fokker-Planck equation in terms of its Green's function and describing the Green's function itself. The Green's function and the description of some of its properties can be found in the Appendices. I discuss the distribution function in the transverse phase space of a particle and it's Fokker-Planck equation for a simple case of a weak focusing machine. Part of this paper is devoted to describing the Green's function and solution of this equation. Then this technique is applied to a strong focusing machine and finally there is a discussion of applicability of this method, its limitations and relation to other methods. 13 refs.
Simplified Derivation of the Fokker-Planck Equation.
ERIC Educational Resources Information Center
Siegman, A. E.
1979-01-01
Presents an alternative derivation of the Fokker-Planck equation for the probability density of a random noise process, starting from the Langevin equation. The derivation makes use of the first two derivatives of the Dirac delta function. (Author/GA)
NASA Astrophysics Data System (ADS)
Chavanis, Pierre-Henri
2003-09-01
We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations can have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in “classes of equivalence” and provide an aesthetic connection between topics (vortices, stars, bacteria,…) which were previously disconnected.
Buet, C.; Cordier; Degond, P.; Lemou, M.
1997-05-15
We present fast numerical algorithms to solve the nonlinear Fokker-Planck-Landau equation in 3D velocity space. The discretization of the collision operator preserves the properties required by the physical nature of the Fokker-Planck-Landau equation, such as the conservation of mass, momentum, and energy, the decay of the entropy, and the fact that the steady states are Maxwellians. At the end of this paper, we give numerical results illustrating the efficiency of these fast algorithms in terms of accuracy and CPU time. 20 refs., 7 figs.
The Fokker-Planck equation for a bistable potential
NASA Astrophysics Data System (ADS)
Caldas, Denise; Chahine, Jorge; Filho, Elso Drigo
2014-10-01
The Fokker-Planck equation is studied through its relation to a Schrödinger-type equation. The advantage of this combination is that we can construct the probability distribution of the Fokker-Planck equation by using well-known solutions of the Schrödinger equation. By making use of such a combination, we present the solution of the Fokker-Planck equation for a bistable potential related to a double oscillator. Thus, we can observe the temporal evolution of the system describing its dynamic properties such as the time τ to overcome the barrier. By calculating the rates k=1/τ as a function of the inverse scaled temperature 1/D, where D is the diffusion coefficient, we compare the aspect of the curve k×1/D, with the ones obtained from other studies related to four different kinds of activated process. We notice that there are similarities in some ranges of the scaled temperatures, where the different processes follow the Arrhenius behavior. We propose that the type of bistable potential used in this study may be used, qualitatively, as a simple model, whose rates share common features with the rates of some single rate-limited thermally activated processes.
Chaotic universe dynamics using a Fokker-Planck equation
Coule, D.H.; Olynyk, K.O.
1987-07-01
A Fokker-Planck equation that accounts for fluctuations in field and its conjugate momentum is solved numerically for the case of a lambda phi/sup 4/ potential. Although the amount of inflation agrees closely with that expected classically, in certain cases (large initial fields or large dispersions),the ''slow rolling'' approximation appears invalid. In such cases inflation would stop prematurely before possibly restarting. 18 refs., 2 figs.
Derivative pricing with non-linear Fokker-Planck dynamics
NASA Astrophysics Data System (ADS)
Michael, Fredrick; Johnson, M. D.
2003-06-01
We examine how the Black-Scholes derivative pricing formula is modified when the underlying security obeys non-extensive statistics and Fokker-Planck dynamics. An unusual feature of such securities is that the volatility in the underlying Ito-Langevin equation depends implicitly on the actual market rate of return. This complicates most approaches to valuation. Here we show that progress is possible using variations of the Cox-Ross valuation technique.
Solving the Fokker-Planck kinetic equation on a lattice
NASA Astrophysics Data System (ADS)
Moroni, Daniele; Rotenberg, Benjamin; Hansen, Jean-Pierre; Succi, Sauro; Melchionna, Simone
2006-06-01
We propose a discrete lattice version of the Fokker-Planck kinetic equation in close analogy with the lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension D . A generalized Hermite-Gauss procedure is used to construct a discretized kinetic equation and a Chapman-Enskog expansion is applied to adapt the scheme so as to correctly reproduce the macroscopic continuum equations. The linear stability of the algorithm with respect to the finite time step Δt is characterized by the eigenvalues of the collision matrix. A heuristic second-order algorithm in Δt is applied to investigate the time evolution of the distribution function of simple model systems, and compared to known analytical solutions. Preliminary investigations of sedimenting Brownian particles subjected to an orthogonal centrifugal force illustrate the numerical efficiency of the Lattice-Fokker-Planck algorithm to simulate nontrivial situations. Interactions between Brownian particles may be accounted for by adding a standard Bhatnagar-Gross-Krook collision operator to the discretized Fokker-Planck kernel.
Computing generalized Langevin equations and generalized Fokker-Planck equations.
Darve, Eric; Solomon, Jose; Kia, Amirali
2009-07-07
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.
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.
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.
Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker-Planck and Fractional Equations
NASA Astrophysics Data System (ADS)
Boon, Jean Pierre; Lutsko, James F.
2017-03-01
The temporal Fokker-Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker-Planck equation for the first passage distribution function f_j(r,t) of a particle moving on a substrate with time delays τ _j. Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability P_j is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, P_j ∝ f_j^{ν - 1}, the generalized Fokker-Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, P_j ∝ τ _j^{-1-α } (with 0< α < 2), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.
Adjoint Fokker-Planck equation and runaway electron dynamics
Liu, Chang; Brennan, Dylan P.; Bhattacharjee, Amitava; Boozer, Allen H.
2016-01-15
The adjoint Fokker-Planck equation method is applied to study the runaway probability function and the expected slowing-down time for highly relativistic runaway electrons, including the loss of energy due to synchrotron radiation. In direct correspondence to Monte Carlo simulation methods, the runaway probability function has a smooth transition across the runaway separatrix, which can be attributed to effect of the pitch angle scattering term in the kinetic equation. However, for the same numerical accuracy, the adjoint method is more efficient than the Monte Carlo method. The expected slowing-down time gives a novel method to estimate the runaway current decay time in experiments. A new result from this work is that the decay rate of high energy electrons is very slow when E is close to the critical electric field. This effect contributes further to a hysteresis previously found in the runaway electron population.
Adjoint Fokker-Planck equation and runaway electron dynamics
NASA Astrophysics Data System (ADS)
Liu, Chang; Brennan, Dylan P.; Bhattacharjee, Amitava; Boozer, Allen H.
2016-01-01
The adjoint Fokker-Planck equation method is applied to study the runaway probability function and the expected slowing-down time for highly relativistic runaway electrons, including the loss of energy due to synchrotron radiation. In direct correspondence to Monte Carlo simulation methods, the runaway probability function has a smooth transition across the runaway separatrix, which can be attributed to effect of the pitch angle scattering term in the kinetic equation. However, for the same numerical accuracy, the adjoint method is more efficient than the Monte Carlo method. The expected slowing-down time gives a novel method to estimate the runaway current decay time in experiments. A new result from this work is that the decay rate of high energy electrons is very slow when E is close to the critical electric field. This effect contributes further to a hysteresis previously found in the runaway electron population.
NASA Astrophysics Data System (ADS)
Lasuik, J.; Fiege, J. D.; Shalchi, A.
2017-01-01
We solve the focused transport equation of cosmic rays numerically to investigate non-isotropic models of the pitch-angle scattering coefficient. In previous work, the Fokker-Planck equation was solved either analytically by using approximations, or by using a numerical approach together with simple models for the pitch-angle scattering coefficient. It is the purpose of the current article so compute particle distribution functions as well as the parallel diffusion coefficient by solving numerically the focused transport equation for a more realistic Fokker-Planck coefficient of pitch-angle scattering. Our analytical form for the scattering parameter is based on non-linear diffusion theory that takes into account realistic scattering at pitch-angles close to 90 ° . This general form contains the isotropic form as well as the quasi-linear limit as special cases. We show that the ratio of the diffusion coefficients with and without focusing sensitively depends on the ratio of the turbulent magnetic field and the mean field. The assumed form of the pitch-angle Fokker-Planck coefficient has an influence on the parallel diffusion coefficient. In all considered cases we found a reduction of the ratio of the diffusion coefficients if the ratio of magnetic fields is reduced.
A pseudospectral solution of a Fokker-Planck equation to model isomerization reactions
NASA Astrophysics Data System (ADS)
Shizgal, Bernie D.
2016-11-01
A Fokker-Planck equation is used to model a reactive system with two stable states. The barrier of the potential that separates the states is controlled with a parameter, ɛ, that alters the height of the barrier that separates the two states of the system. The rate of transitions between the two states, equivalently the rate of reaction, can be treated with a transition state theory as for a large class of chemical reactions. The Fokker-Planck equation is solved with a pseudospectral method based on nonclassical basis polynomials. The time dependent solution is expressed in terms of the eigenvalues and eigenfunctions of the linear Fokker-Planck operator. This eigenvalue problem can be written as the solution of a Schrödinger equation with a potential function defined by the drift and diffusion coefficients in the Fokker-Planck equation.
An efficient iterative method for solving the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
AL-Jawary, M. A.
In the present paper, the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) (2006) is used to solve the linear and nonlinear Fokker-Planck equations and some similar equations. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian decomposition method (ADM). It does not require to calculate Lagrange multiplier as in variational iteration method (VIM) and for solving a nonlinear case, the terms of the sequence become complex after several iterations. Thus, analytical evaluation of terms becomes very difficult or impossible in VIM. No needs to construct a homotopy and solve the corresponding algebraic equations as in homotopy perturbation method (HPM). In this work, the applications of the DJM for 1D, 2D, 3D linear and nonlinear Fokker-Planck equations are given and the results demonstrate that the presented method is very effective and reliable and does not require any restrictive assumptions for nonlinear terms and provide the analytic solutions. A symbolic manipulator Mathematica® 10.0 was used to evaluate terms in the iterative process.
Numerical Study on Fokker-Planck Equation of Bistable System Driven by Colored Noise
NASA Astrophysics Data System (ADS)
Lu, Zhiheng; Hu, Gang; L, Schoendorff; H, Risken
1992-06-01
A finite difference method is used to solve a Fokker-Planck equation of bistable system with Landau potential. The detailed dynamical relaxation process in the case of large correlation time is manifested via the phenomena including the saddle point appearance, the hole formation and distortion. The method is used to obtain the stationary solutions of Fokker-Planck equation of bistable system driven by rather weak noise.
Gaseous microflow modeling using the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Singh, S. K.; Thantanapally, Chakradhar; Ansumali, Santosh
2016-12-01
We present a comparative study of gaseous microflow systems using the recently introduced Fokker-Planck approach and other methods such as: direct simulation Monte Carlo, lattice Boltzmann, and variational solution of Boltzmann-BGK. We show that this Fokker-Plank approach performs efficiently at intermediate values of Knudsen number, a region where direct simulation Monte Carlo becomes expensive and lattice Boltzmann becomes inaccurate. We also investigate the effectiveness of a recently proposed Fokker-Planck model in simulations of heat transfer, as a function of relevant parameters such as the Prandtl, Knudsen numbers. Furthermore, we present simulation of shock wave as a function of Mach number in transonic regime. Our results suggest that the performance of the Fokker-Planck approach is superior to that of the other methods in transition regime for rarefied gas flow and transonic regime for shock wave.
NASA Astrophysics Data System (ADS)
Zhang, Chenglong; Gamba, Irene M.
2016-11-01
We propose a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with Poisson equation. Through time-splitting scheme, a Vlasov-Poisson (collisionless) problem and a homogeneous Landau (collisional) problem are obtained. These two subproblems can be treated separately. We use operator splitting where the transport dynamics for Runge-Kutta Discontinuous Galerkin (RK-DG) method and the collisional dynamics for homogeneous conservative spectral method are adopted respectively. Since two different numerical schemes are applied separately, we have designed a new conservation correction process such that, after projecting the conservative spectral solution onto the DG mesh, there is no loss of moment consvervation. Parallelization is readily implemented. To verify our solver, numerical experiments on linear and nonlinear Landau damping are provided.
Fokker Planck equations for globally coupled many-body systems with time delays
NASA Astrophysics Data System (ADS)
Frank, T. D.; Beek, P. J.
2005-10-01
A Fokker-Planck description for globally coupled many-body systems with time delays was developed by integrating previously derived Fokker-Planck equations for many-body systems and for time-delayed systems. By means of the Fokker-Planck description developed, we examined the dependence of the variability of many-body systems on attractive coupling forces and time delays. For a fundamental class of systems exemplified by a time-delayed Shimizu-Yamada model for muscular contractions, we established that the variability is an invertible one-to-one mapping of coupling forces and time delays and that coupling forces and time delays have opposite effects on system variability, allowing time delays to annihilate the impact of coupling forces. Furthermore, we showed how variability measures could be used to determine coupling parameters and time delays from experimental data.
Invariance principle and model reduction for the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Karlin, I. V.
2016-11-01
The principle of dynamic invariance is applied to obtain closed moment equations from the Fokker-Planck kinetic equation. The analysis is carried out to explicit formulae for computation of the lowest eigenvalue and of the corresponding eigenfunction for arbitrary potentials. This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.
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.
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.
Fokker-Planck quantum master equation for mixed quantum-semiclassical dynamics.
Ding, Jin-Jin; Wang, Yao; Zhang, Hou-Dao; Xu, Rui-Xue; Zheng, Xiao; Yan, YiJing
2017-01-14
We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.
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.
NASA Astrophysics Data System (ADS)
Sierakowski, Adam J.; Lukassen, Laura J.
2016-11-01
In the shear flow of non-Brownian particles, we describe the long-time diffusive processes stochastically using a Fokker-Planck equation. Previous work has indicated that a Fokker-Planck equation coupling the probability densities of position and velocity spaces may be appropriate for describing this phenomenon. The stochastic description, integrated over velocity space to obtain a reduced position-space Fokker-Planck equation, contains unknown space diffusion coefficients. In this work, we use the Physalis method for simulating disperse particle flows to verify the colored-noise velocity space model (an Ornstein-Uhlenbeck process) by comparing the simulated long-time diffusion rate with the diffusion rate proposed by the theory. We then use the simulated data to calculate the unknown space diffusion coefficients that appear in the reduced position-space Fokker-Planck equation and summarize the results. This study was partially supported by US NSF Grant CBET1335965.
Group of contact transformations: Symmetry classification of Fokker-Planck type equations
NASA Astrophysics Data System (ADS)
Rudra, P.
1999-11-01
Fokker-Planck type equations have been classified according to the groups of contact transformations to which they belong. It has been found that there are only five classes as in the case of groups of point transformations. We have also obtained the algebraic structures of the correspond-ing Lie algebras. However, there are isomorphies in their group properties. The corresponding basis sets of functionally independent invariants formed by the generators of these groups have also been obtained.
Stationary Fokker-Planck equation on noncompact manifolds and in unbounded domains
NASA Astrophysics Data System (ADS)
Noarov, A. I.
2016-12-01
We investigate the Fokker-Planck equation on an infinite cylindrical surface and in an infinite strip with reflecting boundary conditions, prove the existence of a positive (not necessarily integrable) solution, and derive various conditions on the vector field f that are sufficient for the existence of a solution that is the probability density. In particular, these conditions are satisfied for some vector fields f with integral trajectories going to infinity.
Shear-induced diffusion of non-Brownian suspensions using a colored noise Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Lukassen, Laura; Oberlack, Martin
2013-11-01
In the Literature, shear-induced diffusion resulting from hydrodynamic interactions between particles, is described as a long-time diffusion. In contrast to the well-known Brownian diffusion which is described by a white noise force, several authors report that the former type of diffusion exhibits the particularity of a much longer correlation time of velocities. Further, Fokker-Planck equations describing this process of shear-induced diffusion have mostly been derived in position space. We present a considerably extended framework of the shear-induced diffusion problem, which essentially relies on the Markov process assumption under the consideration of long correlation times. Applying the mathematical machinery of Markov processes and Fokker-Planck equations, we conclude that this process may only be properly modelled by a Fokker-Planck approach if written in both position and velocity space. With this complementation we observe, that the long correlation times enter as a colored noise velocity. As a result, the Fokker-Planck equation also needs to be extended and we derive the Fokker-Planck equation for the shear-induced diffusion problem following the definitions of a colored noise Fokker-Planck equation. Graduate School of Excellence Computational Engineering.
Applications of the Fokker-Planck equation in circuit quantum electrodynamics
NASA Astrophysics Data System (ADS)
Elliott, Matthew; Ginossar, Eran
2016-10-01
We study exact solutions of the steady-state behavior of several nonlinear open quantum systems which can be applied to the field of circuit quantum electrodynamics. Using Fokker-Planck equations in the generalized P representation, we investigate the analytical solutions of two fundamental models. First, we solve for the steady-state response of a linear cavity that is coupled to an approximate transmon qubit and use this solution to study both the weak and strong driving regimes, using analytical expressions for the moments of both cavity and transmon fields, along with the Husimi Q function for the transmon. Second, we revist exact solutions of a quantum Duffing oscillator, which is driven both coherently and parametrically while also experiencing decoherence by the loss of single photons and pairs of photons. We use this solution to discuss both stabilization of Schrödinger cat states and the generation of squeezed states in parametric amplifiers, in addition to studying the Q functions of the different phases of the quantum system. The field of superconducting circuits, with its strong nonlinearities and couplings, has provided access to parameter regimes in which returning to these exact quantum optics methods can provide valuable insights.
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.
Asymptotic solution of Fokker-Planck equation for plasma in Paul traps
NASA Astrophysics Data System (ADS)
Shah, Kushal
2010-05-01
An exact analytic solution of the Vlasov equation for the plasma distribution in a Paul trap is known to be a Maxwellian and thus, immune to collisions under the assumption of infinitely fast relaxation [K. Shah and H. S. Ramachandran, Phys. Plasmas 15, 062303 (2008)]. In this paper, it is shown that even for a more realistic situation of finite time relaxation, solutions of the Fokker-Planck equation lead to an equilibrium solution of the form of a Maxwellian with oscillatory temperature. This shows that the rf heating observed in Paul traps cannot be caused due to collisional effects alone.
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.
Fokker-Planck-Boltzmann equation for dissipative particle dynamics
NASA Astrophysics Data System (ADS)
Marsh, C. A.; Backx, G.; Ernst, M. H.
1997-05-01
The algorithm for Dissipative Particle Dynamics (DPD), as modified by Español and Warren, is used as a starting point for proving an H-theorem for the free energy and deriving hydrodynamic equations. Equilibrium and transport properties of the DPD fluid are explicitly calculated in terms of the system parameters for the continuous time version of the model.
Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation
NASA Astrophysics Data System (ADS)
Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.
2017-01-01
The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.
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
NASA Astrophysics Data System (ADS)
Shizgal, Bernie D.
2016-09-01
The Fokker-Planck equation with a constant diffusion coefficient and a particular polynomial drift coefficient can exhibit a bistable equilibrium distribution. Such model systems have been used to study chemical reactions, nucleation, climate, optical bistability and other phenomena. In this paper, we consider a particular choice for the drift coefficient of the form A(x) = x^5 - x^3 to exemplify the statistical behaviour of such systems. The transformation of the Fokker-Planck equation to a Schrödinger equation leads to a potential that belongs to the class of potentials in supersymmetric (SUSY) quantum mechanics. A pseudospectral method based on nonclassical polynomials is used to determine the spectrum of the Fokker-Planck operator and of the Schrödinger equation. The converged numerical eigenvalues are compared with WKB and SWKB approximations of the eigenvalues.
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.
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.
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.
Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces
NASA Astrophysics Data System (ADS)
Choi, Young-Pil
2016-07-01
In this paper, we are concerned with the global well-posedness and time-asymptotic decay of the Vlasov-Fokker-Planck equation with local alignment forces. The equation can be formally derived from an agent-based model for self-organized dynamics called the Motsch-Tadmor model with noises. We present the global existence and uniqueness of classical solutions to the equation around the global Maxwellian in the whole space. For the large-time behavior, we show the algebraic decay rate of solutions towards the equilibrium under suitable assumptions on the initial data. We also remark that the rate of convergence is exponential when the spatial domain is periodic. The main methods used in this paper are the classical energy estimates combined with hyperbolic-parabolic dissipation arguments.
Entropy production in irreversible systems described by a Fokker-Planck equation.
Tomé, Tânia; de Oliveira, Mário J
2010-08-01
We analyze the irreversibility and the entropy production in nonequilibrium interacting particle systems described by a Fokker-Planck equation by the use of a suitable master equation representation. The irreversible character is provided either by nonconservative forces or by the contact with heat baths at distinct temperatures. The expression for the entropy production is deduced from a general definition, which is related to the probability of a trajectory in phase space and its time reversal, that makes no reference a priori to the dissipated power. Our formalism is applied to calculate the heat conductance in a simple system consisting of two Brownian particles each one in contact to a heat reservoir. We show also the connection between the definition of entropy production rate and the Jarzynski equality.
An analytical solution of the Fokker-Planck equation in the phase-locked loop transient analysis
NASA Technical Reports Server (NTRS)
Zhang, Weijian
1987-01-01
A probabilistic approach is used to obtain an analytical solution to the Fokker-Planck equation used in the transient analysis of the phase-locked loop phase error process of the first-order phase-locked loop. The solution procedure, which is based on the Girsanov transformation, is described.
Fokker-Planck equation for Boltzmann-type and active particles: transfer probability approach.
Trigger, S A
2003-04-01
A Fokker-Planck equation with velocity-dependent coefficients is considered for various isotropic systems on the basis of probability transition (PT) approach. This method provides a self-consistent and universal description of friction and diffusion for Brownian particles. Renormalization of the friction coefficient is shown to occur for two-dimensional and three-dimensional cases, due to the tensorial character of diffusion. The specific forms of PT are calculated for Boltzmann-type and absorption-type collisions (the latter are typical in dusty plasmas and some other systems). The validity of the Einstein's relation for Boltzmann-type collisions is analyzed for the velocity-dependent friction and diffusion coefficients. For Boltzmann-type collisions in the region of very high grain velocity as well as it is always for non-Boltzmann collisions, such as, absorption collisions, the Einstein relation is violated, although some other relations (determined by the structure of PT) can exist. The generalized friction force is investigated in dusty plasmas in the framework of the PT approach. The relation among this force, the negative collecting friction force, and scattering and collecting drag forces is established. The concept of probability transition is used to describe motion of active particles in an ambient medium. On basis of the physical arguments, the PT for a simple model of the active particle is constructed and the coefficients of the relevant Fokker-Planck equation are found. The stationary solution of this equation is typical for the simplest self-organized molecular machines.
A cross-diffusion system derived from a Fokker-Planck equation with partial averaging
NASA Astrophysics Data System (ADS)
Jüngel, Ansgar; Zamponi, Nicola
2017-02-01
A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.
Studies of parallel algorithms for the solution of a Fokker-Planck equation
Deck, D.; Samba, G.
1995-11-01
The study of laser-created plasmas often requires the use of a kinetic model rather than a hydrodynamic one. This model change occurs, for example, in the hot spot formation in an ICF experiment or during the relaxation of colliding plasmas. When the gradients scalelengths or the size of a given system are not small compared to the characteristic mean-free-path, we have to deal with non-equilibrium situations, which can be described by the distribution functions of every species in the system. We present here a numerical method in plane or spherical 1-D geometry, for the solution of a Fokker-Planck equation that describes the evolution of stich functions in the phase space. The size and the time scale of kinetic simulations require the use of Massively Parallel Computers (MPP). We have adopted a message-passing strategy using Parallel Virtual Machine (PVM).
Fokker-Planck equation with linear and time dependent load forces
NASA Astrophysics Data System (ADS)
Sau Fa, Kwok
2016-11-01
The motion of a particle described by the Fokker-Planck equation with constant diffusion coefficient, linear force (-γ (t)x) and time dependent load force (β (t)) is investigated. The solution for the probability density function is obtained and it has the Gaussian form; it is described by the solution of the linear force with the translation of the position coordinate x. The constant load force preserves the stationary state of the harmonic potential system, however the time dependent load force may not preserve the stationary state of the harmonic potential system. Moreover, the n-moment and variance are also investigated. The solutions are obtained in a direct and pedagogical manner readily understandable by undergraduate and graduate students.
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.
NASA Astrophysics Data System (ADS)
Zhou, Yanjun; Yin, Cangtao
2016-12-01
The Fokker-Planck equation (FPE) of the unimolecular reaction with Tsallis distribution is established by means of approximation to the master equation. The memory effect, taken into transition probability, is relevant and important for lots of anomalous phenomena. The Taylor expansion for large volume is applied to derive the power-law FPE. The steady-state solution of FPE and microscopic dynamics Ito-Langevin equation of concentration variables are therefore obtained and discussed. Two unimolecular reactions are taken as examples and the concentration distributions with different power-law parameters are analyzed, which may imply strong memory effect of hopping process.
NASA Astrophysics Data System (ADS)
Yuste, S. B.; Abad, E.; Escudero, C.
2016-09-01
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time τ (t ) , which we define via the relation τ ˙=1 /a2 , where a (t ) is the expansion scale factor. If the medium expansion is driven by a power law [a (t ) ∝tγ with γ >0 ] , then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value γ =1 /2 . The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.
Yuste, S B; Abad, E; Escudero, C
2016-09-01
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time τ(t), which we define via the relation τ[over ̇]=1/a^{2}, where a(t) is the expansion scale factor. If the medium expansion is driven by a power law [a(t)∝t^{γ} with γ>0], then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value γ=1/2. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.
A Note on Dynamical Models on Random Graphs and Fokker-Planck Equations
NASA Astrophysics Data System (ADS)
Delattre, Sylvain; Giacomin, Giambattista; Luçon, Eric
2016-11-01
We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e., a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős-Rényi graphs with edge probability p_n, n is the number of vertices, such that lim _{n → ∞}p_n n= ∞ . The purpose of this note is twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the n=∞ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite, for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.
NASA Astrophysics Data System (ADS)
Allawala, Altan; Marston, J. B.
2016-11-01
We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.
Allawala, Altan; Marston, J B
2016-11-01
We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.
Exact solution of the Fokker-Planck equation for isotropic scattering
NASA Astrophysics Data System (ADS)
Malkov, M. A.
2017-01-01
The Fokker-Planck (FP) equation ∂tf +μ ∂xf =∂μ(1 -μ2) ∂μf is solved analytically. Foremost among its applications, this equation describes the propagation of energetic particles through a scattering medium (in x - direction, with μ being the x - projection of particle velocity). The solution is found in terms of an infinite series of mixed moments of particle distribution, ⟨μjxk⟩. The second moment ⟨x2⟩ (j =0 , k =2 ) was obtained by G. I. Taylor (1920) in his classical study of random walk: ⟨x2⟩ =⟨x2⟩0+t /3 +[exp (-2 t ) -1 ] /6 (where t is given in units of an average time between collisions). It characterizes a spatial dispersion of a particle cloud released at t =0 , with √{⟨x2⟩0} being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, ⟨x2⟩ -⟨x2⟩0≈t2/3 to a time-asymptotic, diffusive phase, ⟨x2⟩ -⟨x2⟩0≈t /3 . The present paper provides all the higher moments by a recurrence formula. The full set of moments is equivalent to the full solution of the FP equation, expressed in form of an infinite series in moments ⟨μjxk⟩. An explicit, easy-to-use approximation for a point source spreading of a pitch-angle averaged distribution f0(x ,t ) (starting from f0(x ,0 ) =δ (x ) , i.e., Green's function), is also presented and verified by a numerical integration of the FP equation.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Debosscher, A.
1998-01-01
A one-dimensional Fokker-Planck equation with nonmonotonic exponentially dependent drift and diffusion coefficients is defined by further generalizing a previously studied ``unifying stochastic Markov process.'' The equation, which has six essential parameters, defines and unifies a large class of interdisciplinary relevant stochastic processes, many of them being ``embedded'' as limiting cases. In addition to several known processes that previously have been solved independently, the equation also covers a wide ``interpolating'' variety of different, more general stochastic systems that are characterized by a more complex state dependence of the stochastic forces determining the process. The systems can be driven by additive and/or multiplicative noises. They can have saturating or nonsaturating characteristics and they can have unimodal or bimodal equilibrium distributions. Mathematically, the generalization considered parallels the extension from the Gauss hypergeometric to the Heun differential equation, by adding one more finite regular singularity and its associated confluence possibilities. A previously developed constructive solution method, based upon double integral transforms and contour integral representation, is extended for the actual equation by introducing ``factorizers'' and by using a few of their fundamental properties (compiled in Appendix A). In addition, the equivalent Schrödinger equation and the reflection symmetry principle prove to be important tools for analysis. Fully analytical results including normalization are obtained for the discrete part of the generally mixed spectrum. Only the eigenvalues have to be numerically determined as zeros of a spectral kernel. This kernel generally is unknown, but its zeros are accessible via appropriate, infinite continued fraction based search schemes. The basic role of ``congruence'' in this context is highlighted. For clarity, the simpler standard case corresponding to directly accessible zeros is
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→∞.
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.
Lukassen, Laura J; Oberlack, Martin
2014-05-01
In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive.
NASA Astrophysics Data System (ADS)
Lukassen, Laura J.; Oberlack, Martin
2014-05-01
In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive.
SU-E-T-22: A Deterministic Solver of the Boltzmann-Fokker-Planck Equation for Dose Calculation
Hong, X; Gao, H; Paganetti, H
2015-06-15
Purpose: The Boltzmann-Fokker-Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state-of-art numerical acceleration techniques for rapid and accurate dose calculation. Methods: Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pair production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM-SH) is both accurate in dealing with forward-peaking scattering via FEM, and efficient for multi-energy-group computation via SH. In addition, FEM-SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH-based multi-energy-group method. Results: The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM-SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4. Conclusion: A deterministic solver of the Boltzmann-Fokker-Planck equation is developed for dose calculation, and benchmarked against Geant4. Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang
NASA Astrophysics Data System (ADS)
Moggia, Elsa; Chiabrera, Alessandro; Bianco, Bruno
1997-11-01
The statistical properties of the solution of the Langevin-Lorentz equation are analyzed by means of the Fokker-Planck approach. The equation describes the dynamics of an ion that is attracted by a central field and is interacting with a time-varying magnetic field and with the thermal bath. If the endogenous force is assumed to be elastic, then a closed-form expression for the probability density of the process can be obtained, in the case of constant magnetic exposure and, for the time-varying case, at least asymptotically. In the general case, a numerical integration of the resulting set of differential equations with periodically time-varying coefficients has been implemented. A framework for studying the possible effects of low-frequency, low-intensity electromagnetic fields on biological systems has been developed on the basis of the equation. The model assumes that an exogenous electromagnetic field may affect the binding of a messenger attracted by the endogenous force field of its receptor protein. The results are applicable to the analysis of experiments, e.g., exposing a Petri dish, containing a biological sample, to a periodically time-varying magnetic field generated by a pair of Helmholtz coils, most widely used in the scientific literature. The proposed model provides a theoretical mean for evaluating the biological effectiveness of low-frequency, low-intensity electromagnetic exposure.
Fokker-Planck Equations: Uncertainty in Network Security Games and Information
2012-02-12
out [40].. 3. Numerical methods were developed for elastic waves propagation in a Kelvin- Voigt media. The objective is to simulate an elastic wave...behavior of nonlinear stochastic differential equations, especially the stochastic resonance effects based on the probability density function ... Computation North Carolina State University Box 8205 Raleigh, NC 27695-8212 Abstract We have significant accomplishments on uncertainty
NASA Astrophysics Data System (ADS)
Jiang, Peng
2017-02-01
We are concerned with the global well-posedness of the fluid-particle system which describes the evolutions of disperse two-phase flows. The system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the compressible magnetohydrodynamics equations modelling a dense phase (fluid) through the friction forcing. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution and decay rate of the solution are proved based on the classical energy estimates and Fourier multiplier technique, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between particle and fluid.
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.
NASA Astrophysics Data System (ADS)
Shotorban, Babak
2010-04-01
The dynamic least-squares kernel density (LSQKD) model [C. Pantano and B. Shotorban, Phys. Rev. E 76, 066705 (2007)] is used to solve the Fokker-Planck equations. In this model the probability density function (PDF) is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least-squares approximation of the PDF in the phase space. In this work basis functions are set to be Gaussian for which the mean, variance, and covariances are governed by a set of partial differential equations (PDEs) or ordinary differential equations (ODEs) depending on what phase-space variables are approximated by Gaussian functions. Three sample problems of univariate double-well potential, bivariate bistable neurodynamical system [G. Deco and D. Martí, Phys. Rev. E 75, 031913 (2007)], and bivariate Brownian particles in a nonuniform gas are studied. The LSQKD is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases. For the double-well potential problem it is observed that for low to moderate diffusivity the dynamic LSQKD well predicts the stationary PDF for which there is an exact solution. A similar observation is made for the bistable neurodynamical system. In both these problems least-squares approximation is made on all phase-space variables resulting in a set of ODEs with time as the independent variable for the Gaussian function parameters. In the problem of Brownian particles in a nonuniform gas, this approximation is made only for the particle velocity variable leading to a set of PDEs with time and particle position as independent variables. Solving these PDEs, a very good performance by LSQKD is observed for a wide range of diffusivities.
Solving the two-dimensional Fokker-Planck equation for strongly correlated neurons
NASA Astrophysics Data System (ADS)
Deniz, Taşkın; Rotter, Stefan
2017-01-01
Pairs of neurons in brain networks often share much of the input they receive from other neurons. Due to essential nonlinearities of the neuronal dynamics, the consequences for the correlation of the output spike trains are generally not well understood. Here we analyze the case of two leaky integrate-and-fire neurons using an approach which is nonperturbative with respect to the degree of input correlation. Our treatment covers both weakly and strongly correlated dynamics, generalizing previous results based on linear response theory.
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.
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.
Gyrokinetic Fokker-Planck collision operator.
Li, B; Ernst, D R
2011-05-13
The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both field-particle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects can be precomputed. The field-particle operator at each time step requires the evaluation of a single two-dimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than conserving model operators.
NASA Astrophysics Data System (ADS)
Rotenberg, B.; Dufrêche, J.-F.; Bagchi, B.; Giffaut, E.; Hansen, J.-P.; Turq, P.
2006-04-01
We show how a two-state diffusion-reaction description of the mobility of ions confined within compacted clays can be constructed from the microscopic dynamics of ions in an external field. The diffusion-reaction picture provides the usual interpretation of the reduced ionic mobility in clays, but the required partitioning coefficient Kd between trapped and mobile ions is generally an empirical parameter. We demonstrate that it is possible to obtain Kd from the microscopic dynamics of ions interacting with the clay surfaces by evaluating the ionic mobility using a novel lattice implementation of the Fokker-Planck equation. The resulting Kd allows a clear-cut characterization of the trapping sites on the clay surfaces and determines the adsorption/desorption rates. The results highlight the limitations of standard approximation schemes and pinpoint the crossover from jump to Brownian diffusion regimes.
NASA Astrophysics Data System (ADS)
Kumar, Sunil
2013-12-01
The main aim of the present work is to propose a new and simple algorithm to obtain a quick and accurate analytical solution of the time fractional Fokker-Plank equation which arises in various fields in natural science, including solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. This new and simple algorithm is an innovative adjustment in Laplace transform algorithm which makes the calculations much simpler and applicable to several practical problems in science and engineering. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore reduces the numerical computations to a great extent. Furthermore, several numerical examples are presented to illustrate the accuracy and the stability of the method.
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.
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.
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.
A Fokker-Planck description for Parrondo's games
NASA Astrophysics Data System (ADS)
Toral, Raul; Amengual, Pau; Mangioni, Sergio
2003-05-01
We discuss in detail two recently proposed relations between the Parrondo's games and the Fokker-Planck equation describing the flashing ratchet as the overdamped motion of a particle in a potential landscape. In both cases it is possible to relate exactly the probabilities of the games to the potential in which the overdamped particle moves. We will discuss under which conditions current-less potentials correspond to fair games and vie versa.
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.
A Fokker-Planck model of hard sphere gases based on H-theorem
NASA Astrophysics Data System (ADS)
Gorji, M. Hossein; Torillhon, Manuel
2016-11-01
It has been shown recently that the Fokker-Planck kinetic model can be employed as an approximation of the Boltzmann equation for rarefied gas flow simulations [4, 5, 10]. Similar to the direct simulation Monte-Carlo (DSMC), the Fokker-Planck solution algorithm is based on the particle Monte-Carlo representation of the distribution function. Yet opposed to DSMC, here the particles evolve along independent stochastic paths where no collisions need to be resolved. This leads to significant computational advantages over DSMC, considering small Knudsen numbers [10]. The original Fokker-Planck model (FP) for rarefied gas flow simulations was devised according to the Maxwell type pseudo-molecules [4, 5]. In this paper a consistent Fokker-Planck equation is derived based on the Boltzmann collision integrals and maximum entropy distribution. Therefore the resulting model fulfills the H-theorem and leads to correct relaxation of velocity moments up to heat fluxes consistent with hard sphere interactions. For assessment of the model, simulations are performed for Mach 5 flow around a vertical plate using both Fokker-Planck and DSMC simulations. Compared to the original FP model, significant improvements are achieved at high Mach flows.
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.
Ibuki, Kazuyasu; Ueno, Masakatsu
2006-04-07
The validity of a Fokker-Planck-Kramers equation (FPKE) treatment of the rate of diffusion-controlled reaction at short times [K. Ibuki and M. Ueno, J. Chem. Phys. 119, 7054 (2003)] is tested in a supercritical Lennard-Jones fluid over a wide density range by comparing it with the Langevin dynamics and molecular dynamics simulations and other theories. The density n range studied is 0.323n(c)< or =n< or =2.58n(c) and the temperature 1.52T(c), where n(c) and T(c) are the critical density and temperature, respectively. For the rate of bimolecular reactions, the transition between the collision-limited and diffusion-limited regimes is expected to take place in this density range. The simulations show that the rate constant decays with time extensively at high densities, and that the magnitude of decay decreases gradually with decreasing density. The decay profiles of the rate constants obtained by the simulations are reproduced reasonably well by the FPKE treatment in the whole density range studied if a continuous velocity distribution is used in solving the FPKE approximately. If a discontinuous velocity distribution is used instead of the continuous one, the FPKE treatment leads to a rate constant much larger than the simulation results at medium and low densities. The rate constants calculated from the Smoluchowski-Collins-Kimball (SCK) theory based on the diffusion equation are somewhat smaller than the simulation results in medium and low densities when the intrinsic rate constant is chosen to adjust the steady state rate constant in the low density limit to that derived by the kinetic collision theory. The discrepancy is relatively small, so that the SCK theory provides a useful guideline for a qualitative discussion of the density effect on the rate constant.
Parallelized Vlasov-Fokker-Planck solver for desktop personal computers
NASA Astrophysics Data System (ADS)
Schönfeldt, Patrik; Brosi, Miriam; Schwarz, Markus; Steinmann, Johannes L.; Müller, Anke-Susanne
2017-03-01
The numerical solution of the Vlasov-Fokker-Planck equation is a well established method to simulate the dynamics, including the self-interaction with its own wake field, of an electron bunch in a storage ring. In this paper we present Inovesa, a modularly extensible program that uses opencl to massively parallelize the computation. It allows a standard desktop PC to work with appropriate accuracy and yield reliable results within minutes. We provide numerical stability-studies over a wide parameter range and compare our numerical findings to known results. Simulation results for the case of coherent synchrotron radiation will be compared to measurements that probe the effects of the microbunching instability occurring in the short bunch operation at ANKA. It will be shown that the impedance model based on the shielding effect of two parallel plates can not only describe the instability threshold, but also the presence of multiple regimes that show differences in the emission of coherent synchrotron radiation.
Noise on resistive switching: a Fokker-Planck approach
NASA Astrophysics Data System (ADS)
Patterson, G. A.; Grosz, D. F.; Fierens, P. I.
2016-05-01
We study the effect of internal and external noise on the phenomenon of resistive switching. We consider a non-harmonic external driving signal and provide a theoretical framework to explain the observed behavior in terms of the related Fokker-Planck equations. It is found that internal noise causes an enhancement of the resistive contrast and that noise proves to be advantageous when considering short driving pulses. In the case of external noise, however, noise only has the effect of degrading the resistive contrast. Furthermore, we find a relationship between the noise amplitude and the driving signal pulsewidth that constrains the persistence of the resistive state. In particular, results suggest that strong and short driving pulses favor a longer persistence time, an observation that may find applications in the field of high-integration high-speed resistive memory devices.
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.
Orbit-averaged guiding-center Fokker-Planck operator for numerical applications
Decker, J.; Peysson, Y.; Duthoit, F.-X.; Brizard, A. J.
2010-11-15
A guiding-center Fokker-Planck operator is derived in a coordinate system that is well suited for the implementation in a numerical code. This differential operator is transformed such that it can commute with the orbit-averaging operation. Thus, in the low-collisionality approximation, a three-dimensional Fokker-Planck evolution equation for the orbit-averaged distribution function in a space of invariants is obtained. This transformation is applied to a collision operator with nonuniform isotropic field particles. Explicit neoclassical collisional transport diffusion and convection coefficients are derived, and analytical expressions are obtained in the thin orbit approximation. To illustrate this formalism and validate our results, the bootstrap current is analytically calculated in the Lorentz limit.
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.
Relaxation of terrace-width distributions: Physical information from Fokker Planck time
NASA Astrophysics Data System (ADS)
Hamouda, Ajmi BH.; Pimpinelli, Alberto; Einstein, T. L.
2008-12-01
Recently some of us have constructed a Fokker-Planck formalism to describe the equilibration of the terrace-width distribution of a vicinal surface from an arbitrary initial configuration. However, the meaning of the associated relaxation time, related to the strength of the random noise in the underlying Langevin equation, was rather unclear. Here we present a set of careful kinetic Monte Carlo simulations that demonstrate convincingly that the time constant shows activated behavior with a barrier that has a physically plausible dependence on the energies of the governing microscopic model. Remarkably, the rate-limiting step for relaxation in the far-from-equilibrium regime is the generation of kink-antikink pairs, involving the breaking of three lateral bonds on a cubic {0 0 1} surface, in contrast to the processes breaking two bonds that dominate equilibrium fluctuations. After an initial regime, the Fokker-Planck time at least semiquantitatively tracks the actual physical time.
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.
Fokker-Planck analysis of transverse collective instabilities in electron storage rings
NASA Astrophysics Data System (ADS)
Lindberg, Ryan R.
2016-12-01
We analyze single bunch transverse instabilities due to wakefields using a Fokker-Planck model. We first expand on the work of T. Suzuki, Part. Accel. 12, 237 (1982) to derive the theoretical model including chromaticity, both dipolar and quadrupolar transverse wakefields, and the effects of damping and diffusion due to the synchrotron radiation. We reduce the problem to a linear matrix equation, whose eigenvalues and eigenvectors determine the collective stability of the beam. We then show that various predictions of the theory agree quite well with results from particle tracking simulations, including the threshold current for transverse instability and the profile of the unstable mode. In particular, we find that predicting collective stability for high energy electron beams at moderate to large values of chromaticity requires the full Fokker-Planck analysis to properly account for the effects of damping and diffusion due to synchrotron radiation.
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.
Bayesian inference based on stationary Fokker-Planck sampling.
Berrones, Arturo
2010-06-01
A novel formalism for bayesian learning in the context of complex inference models is proposed. The method is based on the use of the stationary Fokker-Planck (SFP) approach to sample from the posterior density. Stationary Fokker-Planck sampling generalizes the Gibbs sampler algorithm for arbitrary and unknown conditional densities. By the SFP procedure, approximate analytical expressions for the conditionals and marginals of the posterior can be constructed. At each stage of SFP, the approximate conditionals are used to define a Gibbs sampling process, which is convergent to the full joint posterior. By the analytical marginals efficient learning methods in the context of artificial neural networks are outlined. Offline and incremental bayesian inference and maximum likelihood estimation from the posterior are performed in classification and regression examples. A comparison of SFP with other Monte Carlo strategies in the general problem of sampling from arbitrary densities is also presented. It is shown that SFP is able to jump large low-probability regions without the need of a careful tuning of any step-size parameter. In fact, the SFP method requires only a small set of meaningful parameters that can be selected following clear, problem-independent guidelines. The computation cost of SFP, measured in terms of loss function evaluations, grows linearly with the given model's dimension.
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.
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.
Evaluation of the Fokker-Planck probability by Asymptotic Taylor Expansion Method
NASA Astrophysics Data System (ADS)
Firat, Kenan; Ozer, Okan
2017-02-01
The one-dimensional Fokker-Planck equation is solved by the Asymptotic Taylor Expansion Method for the time-dependent probability density of a particle. Using an ansatz wave function, one obtains the series expansion of the solution for the Schrödinger and it allows one to find out the eigen functions and eigen energies of the states to the evaluation of the probability. The eigen energies of some certain kind of Bistable potentials are calculated for some certain potential parameters. The probability function is determined and graphed for potential parameters. The numerical results are compared with existing literature, and a conclusion about the advantages and disadvantages on the method is given.
Importance sampling variance reduction for the Fokker-Planck rarefied gas particle method
NASA Astrophysics Data System (ADS)
Collyer, B. S.; Connaughton, C.; Lockerby, D. A.
2016-11-01
The Fokker-Planck approximation to the Boltzmann equation, solved numerically by stochastic particle schemes, is used to provide estimates for rarefied gas flows. This paper presents a variance reduction technique for a stochastic particle method that is able to greatly reduce the uncertainty of the estimated flow fields when the characteristic speed of the flow is small in comparison to the thermal velocity of the gas. The method relies on importance sampling, requiring minimal changes to the basic stochastic particle scheme. We test the importance sampling scheme on a homogeneous relaxation, planar Couette flow and a lid-driven-cavity flow, and find that our method is able to greatly reduce the noise of estimated quantities. Significantly, we find that as the characteristic speed of the flow decreases, the variance of the noisy estimators becomes independent of the characteristic speed.
NASA Astrophysics Data System (ADS)
Drozdov, Alexander N.; Talkner, Peter
1998-08-01
Fokker-Planck processes with a singular diffusion matrix are quite frequently met in Physics and Chemistry. For a long time the resulting noninvertability of the diffusion matrix has been looked as a serious obstacle for treating these Fokker-Planck equations by various powerful numerical methods of quantum and statistical mechanics. In this paper, a path-integral method is presented that takes advantage of the singularity of the diffusion matrix and allows one to solve such problems in a simple and economic way. The basic idea is to split the Fokker-Planck equation into one of a linear system and an anharmonic correction and then to employ a symmetric decomposition of the short time propagator, which is exact up to a high order in the time step. Just because of the singularity of the diffusion matrix, the factors of the resulting product formula consist of well behaved propagators. In this way one obtains a highly accurate propagation scheme, which is simultaneously fast, stable, and computationally simple. Because it allows much larger time steps, it is more efficient than the standard propagation scheme based on the Trotter splitting formula. The proposed method is tested for Brownian motion in different types of potentials. For a harmonic potential we compare to the known analytic results. For a symmetric double well potential we determine the transition rates between the two wells for different friction strengths and compare them with the crossover theories of Mel'nikov and Meshkov and Pollak, Grabert, and Hänggi. Using a properly defined energy loss of the deterministic particle dynamics, we obtain excellent agreement. The methodology is outlined for a large class of processes defined by generalized Langevin equations and processes driven by colored noise.
Kinetic equation for nonlinear resonant wave-particle interaction
NASA Astrophysics Data System (ADS)
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2016-09-01
We investigate the nonlinear resonant wave-particle interactions including the effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relationship between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations.
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.
Vlasov-Fokker-Planck modeling of magnetized plasma
Thomas, Alexander
2016-08-01
Understanding the magnetic fields that can develop in high-power-laser interactions with solid-density plasma is important because such fields significantly modify both the magnitude and direction of electron heat fluxes. The dynamics of such fields evidently have consequences for inertial fusion energy applications, as the coupling of the laser beams with the walls or pellet and the development of temperature inhomogeneities are critical to the uniformity of the implosion and potentially the success of, for example, the National Ignition Facility. To study these effects, we used the code Impacta, a two-dimensional, fully implicit, Vlasov-Fokker-Planck code with self-consistent magnetic fields and a hydrodynamic ion model, designed for nanosecond time-scale laser-plasma interactions. Heat-flux effects in Ohm’s law under non-local conditions was investigated; physics that is not well captured by standard numerical models but is nevertheless important in fusion-related scenarios. Under such conditions there are numerous interesting physical effects, such as collisional magnetic instabilities, amplification of magnetic fields, re-emergence of non-locality through magnetic convection, and reconnection of magnetic field lines and redistribution of thermal energy. In this project highlights included the first full scale kinetic simulations of a magnetized hohlraum [Joglekar 2016] and the discovery of a new magnetic reconnection mechanism [Joglekar 2014] as well as a completed PhD thesis and the production of a new code for Inertial Fusion research.
NASA Astrophysics Data System (ADS)
Tessarotto, M.; Ellero, M.; Sarmah, D.; Nicolini, P.
2008-12-01
Extending the statistical approach proposed in a parallel paper [1], purpose of this work is to propose a stochastic inverse kinetic theory for small-scale hydrodynamic turbulence based on the introduction of a suitable local phase-space probability density function (pdf). In particular, we pose the problem of the construction of Fokker-Planck kinetic models of hydrodynamic turbulence. The approach here adopted is based on the so-called IKT approach (inverse kinetic theory), developed by Tessarotto et al. (2004-2008) which permits an exact phase-space description of incompressible fluids based on the adoption of a local pdf. We intend to show that for prescribed models of stochasticity the present approach permits to determine uniquely the time evolution of the stochastic fluid fields. The stochastic-averaged local pdf is shown to obey a kinetic equation which, although generally non-Markovian, locally in velocity-space can be approximated by means of a suitable Fokker-planck kinetic equation. As a side result, the same pdf is proven to have generally a non-Gaussian behavior.
An Implicit Energy-Conservative 2D Fokker-Planck Algorithm. II. Jacobian-Free Newton-Krylov Solver
NASA Astrophysics Data System (ADS)
Chacón, L.; Barnes, D. C.; Knoll, D. A.; Miley, G. H.
2000-01-01
Energy-conservative implicit integration schemes for the Fokker-Planck transport equation in multidimensional geometries require inverting a dense, non-symmetric matrix (Jacobian), which is very expensive to store and solve using standard solvers. However, these limitations can be overcome with Newton-Krylov iterative techniques, since they can be implemented Jacobian-free (the Jacobian matrix from Newton's algorithm is never formed nor stored to proceed with the iteration), and their convergence can be accelerated by preconditioning the original problem. In this document, the efficient numerical implementation of an implicit energy-conservative scheme for multidimensional Fokker-Planck problems using multigrid-preconditioned Krylov methods is discussed. Results show that multigrid preconditioning is very effective in speeding convergence and decreasing CPU requirements, particularly in fine meshes. The solver is demonstrated on grids up to 128×128 points in a 2D cylindrical velocity space (vr, vp) with implicit time steps of the order of the collisional time scale of the problem, τ. The method preserves particles exactly, and energy conservation is improved over alternative approaches, particularly in coarse meshes. Typical errors in the total energy over a time period of 10τ remain below a percent.
NASA Astrophysics Data System (ADS)
Jaeger, E. F.; Berry, L. A.; Ahern, S. D.; Barrett, R. F.; Batchelor, D. B.; Carter, M. D.; D'Azevedo, E. F.; Moore, R. D.; Harvey, R. W.; Myra, J. R.; D'Ippolito, D. A.; Dumont, R. J.; Phillips, C. K.; Okuda, H.; Smithe, D. N.; Bonoli, P. T.; Wright, J. C.; Choi, M.
2006-05-01
Magnetically confined plasmas can contain significant concentrations of nonthermal plasma particles arising from fusion reactions, neutral beam injection, and wave-driven diffusion in velocity space. Initial studies in one-dimensional and experimental results show that nonthermal energetic ions can significantly affect wave propagation and heating in the ion cyclotron range of frequencies. In addition, these ions can absorb power at high harmonics of the cyclotron frequency where conventional two-dimensional global-wave models are not valid. In this work, the all-orders global-wave solver AORSA [E. F. Jaeger et al., Phys. Rev. Lett. 90, 195001 (2003)] is generalized to treat non-Maxwellian velocity distributions. Quasilinear diffusion coefficients are derived directly from the wave fields and used to calculate energetic ion velocity distributions with the CQL3D Fokker-Planck code [R. W. Harvey and M. G. McCoy, Proceedings of the IAEA Technical Committee Meeting on Simulation and Modeling of Thermonuclear Plasmas, Montreal, Canada, 1992 (USDOC NTIS Document No. DE93002962)]. For comparison, the quasilinear coefficients can be calculated numerically by integrating the Lorentz force equations along particle orbits. Self-consistency between the wave electric field and resonant ion distribution function is achieved by iterating between the global-wave and Fokker-Planck solutions.
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.
Nonparametric estimates of drift and diffusion profiles via Fokker-Planck algebra.
Lund, Steven P; Hubbard, Joseph B; Halter, Michael
2014-11-06
Diffusion processes superimposed upon deterministic motion play a key role in understanding and controlling the transport of matter, energy, momentum, and even information in physics, chemistry, material science, biology, and communications technology. Given functions defining these random and deterministic components, the Fokker-Planck (FP) equation is often used to model these diffusive systems. Many methods exist for estimating the drift and diffusion profiles from one or more identifiable diffusive trajectories; however, when many identical entities diffuse simultaneously, it may not be possible to identify individual trajectories. Here we present a method capable of simultaneously providing nonparametric estimates for both drift and diffusion profiles from evolving density profiles, requiring only the validity of Langevin/FP dynamics. This algebraic FP manipulation provides a flexible and robust framework for estimating stationary drift and diffusion coefficient profiles, is not based on fluctuation theory or solved diffusion equations, and may facilitate predictions for many experimental systems. We illustrate this approach on experimental data obtained from a model lipid bilayer system exhibiting free diffusion and electric field induced drift. The wide range over which this approach provides accurate estimates for drift and diffusion profiles is demonstrated through simulation.
Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation
2011-05-31
material science. We have com- puted the electronic structure of 2D quantum dot system, and compared the efficiency with the benchmark software OCTOPUS . For...one self-consistent iteration step with 512 electrons, OCTOPUS costs 1091 sec, and selected inversion costs 9.76 sec. The algorithm exhibits
Adaptive particle-cell algorithm for Fokker-Planck based rarefied gas flow simulations
NASA Astrophysics Data System (ADS)
Pfeiffer, M.; Gorji, M. H.
2017-04-01
Recently, the Fokker-Planck (FP) kinetic model has been devised on the basis of the Boltzmann equation (Jenny et al., 2010; Gorji et al., 2011). Particle Monte-Carlo schemes are then introduced for simulations of rarefied gas flows based on the FP kinetics. Here the particles follow independent stochastic paths and thus a spatio-temporal resolution coarser than the collisional scales becomes possible. In contrast to the direct simulation Monte-Carlo (DSMC), the computational cost is independent of the Knudsen number resulting in efficient simulations at moderate/low Knudsen flows. In order to further exploit the efficiency of the FP method, the required particle-cell resolutions should be found, and a cell refinement strategy has to be developed accordingly. In this study, an adaptive particle-cell scheme applicable to a general unstructured mesh is derived for the FP model. Virtual sub cells are introduced for the adaptive mesh refinement. Moreover a sub cell-merging algorithm is provided to honor the minimum required number of particles per cell. For assessments, the 70 degree blunted cone reentry flow (Allgre et al., 1997) is studied. Excellent agreement between the introduced adaptive FP method and DSMC is achieved.
A rapid fast ion Fokker-Planck solver for integrated modelling of tokamaks
NASA Astrophysics Data System (ADS)
Schneider, M.; Eriksson, L.-G.; Johnson, T.; Futtersack, R.; Artaud, J. F.; Dumont, R.; Wolle, B.; Contributors, ITM-TF
2015-01-01
The RISK (rapid ion solver for tokamaks) code for simulating the evolution of the distribution function of neutral beam injected ions (NBI) in tokamak plasmas is described. The code has been especially developed for use in integrated modelling frameworks. Within this context, a code needs to be modular, machine independent and fast. RISK fulfils all these conditions. The RISK code solves the bounce averaged Fokker-Planck equation for the species of the injected ions by expanding the distribution function in the eigenfunctions of the collisional pitch angle scattering operator. The velocity dependent coefficient functions are calculated with a finite element solver. Finite orbit width effects are handled by an ad hoc broadening algorithm of the NBI ionization source. In order to assess the validity of the approximations employed in RISK, a comparison with a full orbit following Monte Carlo code is presented. RISK is integrated into the CRONOS transport suite of codes (Artaud et al 2010 Nucl. Fusion 50 043001) and the European integrated modelling (EU-IM) framework (Falchetto et al 2014 Nucl. Fusion 54 043018). The RISK implementation in this platform is discussed and exemplified to show the strength of running simulation codes in a modular and machine independent environment for simulation of fusion plasmas.
2012-12-14
surprise since the I-POD uses different time -scales, as well as the adjoint information to get a higher fidelity ROM. 5. Application of I-POD to Filtering of...Turbulent and Magnetohydrodynamic Flows . Boston, MA: Systems and Control: Foundations and Applications , Birkhauser, 2007. [31] H. T. Banks and K...been one of the most successful applications of control theoretic techniques in the industry [4]. The MPC techniques solve a sequence of finite horizon
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.
On Non-Linear Sensitivity of Marine Biological Models to Parameter Variations
2007-01-01
M.B., 2002. Understanding uncertain enviromental systems. In: Grasman, J., van Straten, G. (Eds.), Predictability and Nonlinear Modelling in Natural...Sciences and Economics . Kluwer, Dordrecht, pp. 294–311. Chu, P.C., Ivanov, L.M., Fan, C.W., 2002a. Backward Fokker-Planck equation for determining
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.
Harvey, R.W.; Chan, V.S.
1996-12-31
Runaway of electrons to high energy during plasma disruptions occurs due to large induced toroidal electric fields which tend to maintain the toroidal plasma current, in accord with Lenz law. This has been observed in many tokamaks. Within the closed flux surfaces, the bounce-averaged CQL3D Fokker-Planck code is well suited to obtain the resulting electron distributions, nonthermal contributions to electrical conductivity, and runaway rates. The time-dependent 2D in momentum-space (p{sub {parallel}} and p{sub {perpendicular}}) distributions axe calculated on a radial array of noncircular flux surfaces, including bounce-averaging of the Fokker-Planck equation to account for toroidal trapping effects. In the steady state, the resulting distributions represent a balance between applied toroidal electric field, relativistic Coulomb collisions, and synchrotron radiation. The code can be run in a mode where the electrons are sourced at low velocity and run off the high velocity edge of the computational mesh, giving runaway rates at steady state. At small minor radius, the results closely match previous results reported by Kulsrud et al. It is found that the runaway rate has a strong dependence on inverse aspect ratio e, decreasing by a factor {approx} 5 as e increases from 0.0 to 0.3. The code can also be run with a radial diffusion and pinching term, simulating radial transport with plasma pinching to maintain a given density profile. Results show a transport reduction of runaways in the plasma center, and an enhancement towards the edge due to the electrons from the plasma center. Avalanching of runaways due to a knock-on electron source is being included.
A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields
NASA Astrophysics Data System (ADS)
Brizard, Alain J.
2004-11-01
A new formulation for collisional kinetic theory is presented based on the use of Lie-transform methods to eliminate fast orbital time scales from a general bilinear collision operator. As an application of this new formalism, a general guiding-center bilinear Fokker-Planck collision operator is derived following the elimination of the fast gyromotion time scale of a charged particle moving in a nonuniform magnetic field. It is expected that classical transport processes in a strongly magnetized nonuniform plasma can, thus, be described in terms of this reduced guiding-center Fokker-Planck kinetic theory. The poster introduces the reduced-collision formalism only while its applications are left to future work.
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.
Forecasting with the Fokker-Planck model: Bayesian setting of parameter
NASA Astrophysics Data System (ADS)
Montagnon, Chris
2017-04-01
Using a closed solution to a Fokker-Planck model of a time series, a probability distribution for the next point in the time series is developed. This probability distribution has one free parameter. Various Bayesian approaches to setting this parameter are tested by forecasting some real world time series. Results show a more than 25 % reduction in the ' 95 % point' of the probability distribution (the safety stock required in these real world situations), versus the conventional ARMA approach, without a significant increase in actuals exceeding this level.
3D Fokker-Planck modeling of axisymmetric collisional losses of fusion products in TFTR
Goloborod`ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A.; Zweben, S.J.
1995-10-01
Results of a 3D (in constants of motion space) Fokker-Planck simulation of collisional losses of fusion products in axisymmetric DT and DD discharges on TFTR are presented. The distributions of escaped ions over poloidal angle, pitch angle, and their energy spectra are obtained. Axisymmetric collisional losses of fusion products are found to be less than (2--5)%. The distribution of confined fusion products is shown to be strongly anisotropic and nonuniform in the radial coordinate mainly for slowed-down fusion products with small longitudinal energy. Comparison of these results of modeling and experimental data is done.
A fully-neoclassical finite-orbit-width version of the CQL3D Fokker-Planck code
NASA Astrophysics Data System (ADS)
Petrov, Yu V.; Harvey, R. W.
2016-11-01
The time-dependent bounce-averaged CQL3D flux-conservative finite-difference Fokker-Planck equation (FPE) solver has been upgraded to include finite-orbit-width (FOW) capabilities which are necessary for an accurate description of neoclassical transport, losses to the walls, and transfer of particles, momentum, and heat to the scrape-off layer. The FOW modifications are implemented in the formulation of the neutral beam source, collision operator, RF quasilinear diffusion operator, and in synthetic particle diagnostics. The collisional neoclassical radial transport appears naturally in the FOW version due to the orbit-averaging of local collision coefficients coupled with transformation coefficients from local (R, Z) coordinates along each guiding-center orbit to the corresponding midplane computational coordinates, where the FPE is solved. In a similar way, the local quasilinear RF diffusion terms give rise to additional radial transport of orbits. We note that the neoclassical results are obtained for ‘full’ orbits, not dependent on a common small orbit-width approximation. Results of validation tests for the FOW version are also presented.
Fokker-Planck diffusive law: its interpretation in the context of plasma transport modeling
NASA Astrophysics Data System (ADS)
Sanchez, Raul; Carreras, Ben A.; van Milligen, Boudewijn Ph.
2006-10-01
It was recently proposed that, when building phenomenological transport models for particle transport in tokamaks, use of the Fokker-Planck diffusive law might be preferable to Fick's law to express particle fluxes [1]. In particular, it might offer a possible explanation for the excessive pinch velocites observed in some experimental situations with respect to the values expected from the forces and asymmetries existent in the system. In spite of the fact that Fokker-Planck's law was first proposed many years ago, it produces a series of counterintuitive results that at first sight seem in contradiction with the second law of thermodynamics. In this contribution we will review the basic concepts behind its formulation and show that, through the use of simple examples relevant to plasma physics, the second law of thermodynamics is not violated in any manner if properly used. The benefits of its use within the modelling of transport in tokamaks will also be clarified.REFERENCES: [1] R. Sanchez et al, Phys. Plasmas 12, 056105 (2005); B.Ph. van Milligen et al, Plasma Phys.Contr.Fusion 47, B743 (2005)
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 guiding-center Fokker-Planck collision operator for nonuniform magnetic fields
NASA Astrophysics Data System (ADS)
Brizard, Alain J.
2004-09-01
A formulation for collisional kinetic theory is presented based on the use of Lie-transform methods to eliminate fast orbital time scales from a general bilinear collision operator. As an application of this formalism, a general guiding-center bilinear Fokker-Planck (FP) collision operator is derived following the elimination of the fast gyromotion time scale of a charged particle moving in a nonuniform magnetic field. It is expected that classical transport processes in a strongly magnetized nonuniform plasma can, thus, be described in terms of this reduced guiding-center FP kinetic theory. The present paper introduces the reduced-collision formalism only, while its applications are left to future work.
Nonlinear Ginzburg-Landau-type approach to quantum dissipation.
López, José L
2004-02-01
We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
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.
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 description of single nucleosome repositioning by dimeric chromatin remodelers
NASA Astrophysics Data System (ADS)
Vandecan, Yves; Blossey, Ralf
2013-07-01
Recent experiments have demonstrated that the ATP-utilizing chromatin assembly and remodeling factor (ACF) is a dimeric, processive motor complex which can move a nucleosome more efficiently towards longer flanking DNA than towards shorter flanking DNA strands, thereby centering an initially ill-positioned nucleosome on DNA substrates. We give a Fokker-Planck description for the repositioning process driven by transitions between internal chemical states of the remodelers. In the chemical states of ATP hydrolysis during which the repositioning takes place a power stroke is considered. The slope of the effective driving potential is directly related to ATP hydrolysis and leads to the unidirectional motion of the nucleosome-remodeler complex along the DNA strand. The Einstein force relation allows us to deduce the ATP-concentration dependence of the diffusion constant of the nucleosome-remodeler complex. We have employed our model to study the efficiency of positioning of nucleosomes as a function of the ATP sampling rate between the two motors which shows that the synchronization between the motors is crucial for the remodeling mechanism to work.
Fokker-Planck model for collisional loss of fast ions in tokamaks
NASA Astrophysics Data System (ADS)
Yavorskij, V.; Goloborod'ko, V.; Schoepf, K.
2016-11-01
Modelling of the collisional loss of fast ions from tokamak plasmas is important from the point of view of the impact of fusion alphas and neutral beam injection ions on plasma facing components as well as for the development of diagnostics of fast ion losses [1-3]. This paper develops a Fokker-Planck (FP) method for the assessment of distributions of collisional loss of fast ions as depending on the coordinates of the first wall surface and on the velocities of lost ions. It is shown that the complete 4D drift FP approach for description of fast ions in axisymmetric tokamak plasmas can be reduced to a 2D FP problem for lost ions with a boundary condition delivered by the solution of a 3D boundary value problem for confined ions. Based on this newly developed FP approach the poloidal distribution of neoclassical loss, depending on pitch-angle and energy, of fast ions from tokamak plasma may be examined as well as the contribution of this loss to the signal detected by the scintillator probe may be evaluated. It is pointed out that the loss distributions obtained with the novel FP treatment may serve as an alternative approach with respect to Monte-Carlo models [4, 5] commonly used for simulating fast ion loss from toroidal plasmas.
Bounce-averaged Fokker-Planck Simulation of Runaway Avalanche from Secondary Knock-on Production
NASA Astrophysics Data System (ADS)
Chiu, S. C.; Chan, V. S.; Harvey, R. W.; Rosenbluth, M. N.
1996-11-01
It has been pointed out that secondary production of runaway electrons by knock-on collisions with very energetic confined electrons can significantly change the runaway rate,(M.N. Rosenbluth, Bull. Amer. Phys. Soc. 40), 1804 (1995).^,(N.T. Besedin, I.M. Pankratov, Nucl. Fusion 26), 807 (1986).^,(R. Jaspers, K.H. Finden, G. Mank et al.), Nucl. Fusion 33, 1775 (1993). and is potentially a serious problem in reactors. Previous calculations of the effect have only partially included important effects such as toroidal trapping, synchrotron radiation, and bremsstrahlung. Furthermore, in a normal constant current operation, the increase of the density of runaway electrons causes a decrease of the ohmic field and all these effects can balance to a steady-state. The purpose of the present paper is to present results on bounce-averaged Fokker-Planck simulations of knock-on avalanching runaways including these effects. Initially, an energetic seed component is inserted to initiate knock-on avalanching. Results on the dependence of the steady-state runaway current on Z_eff, density, and radial location will be presented.
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.
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.
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)
Küchlin, Stephan; Jenny, Patrick
2017-01-01
A major challenge for the conventional Direct Simulation Monte Carlo (DSMC) technique lies in the fact that its computational cost becomes prohibitive in the near continuum regime, where the Knudsen number (Kn)-characterizing the degree of rarefaction-becomes small. In contrast, the Fokker-Planck (FP) based particle Monte Carlo scheme allows for computationally efficient simulations of rarefied gas flows in the low and intermediate Kn regime. The Fokker-Planck collision operator-instead of performing binary collisions employed by the DSMC method-integrates continuous stochastic processes for the phase space evolution in time. This allows for time step and grid cell sizes larger than the respective collisional scales required by DSMC. Dynamically switching between the FP and the DSMC collision operators in each computational cell is the basis of the combined FP-DSMC method, which has been proven successful in simulating flows covering the whole Kn range. Until recently, this algorithm had only been applied to two-dimensional test cases. In this contribution, we present the first general purpose implementation of the combined FP-DSMC method. Utilizing both shared- and distributed-memory parallelization, this implementation provides the capability for simulations involving many particles and complex geometries by exploiting state of the art computer cluster technologies.
FOKKER-PLANCK MODELS FOR M15 WITHOUT A CENTRAL BLACK HOLE: THE ROLE OF THE MASS FUNCTION
Murphy, Brian W.; Cohn, Haldan N.; Lugger, Phyllis M.
2011-05-10
We have developed a set of dynamically evolving Fokker-Planck models for the collapsed-core globular star cluster M15, which directly address the issue of whether a central black hole is required to fit Hubble Space Telescope (HST) observations of the stellar spatial distribution and kinematics. As in our previous work reported by Dull et al., we find that a central black hole is not needed. Using local mass-function data from HST studies, we have also inferred the global initial stellar mass function. As a consequence of extreme mass segregation, the local mass functions differ from the global mass function at every location. In addition to reproducing the observed mass functions, the models also provide good fits to the star-count and velocity-dispersion profiles, and to the millisecond pulsar accelerations. We address concerns about the large neutron star populations adopted in our previous Fokker-Planck models for M15. We find that good model fits can be obtained with as few as 1600 neutron stars; this corresponds to a retention fraction of 5% of the initial population for our best-fit initial mass function. The models contain a substantial population of massive white dwarfs, that range in mass up to 1.2M{sub sun} . The combined contribution by the massive white dwarfs and neutron stars provides the gravitational potential needed to reproduce HST measurements of the central velocity-dispersion profile.
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
Fractional Fokker-Planck Equations and Artificial Neural Networks for Stochastic Control of Tokamak
NASA Astrophysics Data System (ADS)
Rastovic, Danilo
2008-09-01
The general form of description of Kolmogorov-Arnold-Moser (KAM) theorem in controlled plasma fusion, is obtained via the theory of artificial fuzzy neural networks. Without of the global maximum entropy principle, the complexity function is used for the Monte Carlo simulations.
Kwok, Sau Fa
2012-08-15
A Langevin equation with multiplicative white noise and its corresponding Fokker-Planck equation are considered in this work. From the Fokker-Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: Black-Right-Pointing-Pointer Fokker-Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. Black-Right-Pointing-Pointer Transformation of diffusion processes into the Wiener process in different prescriptions is provided. Black-Right-Pointing-Pointer The prescription parameter is associated with the growth rate for a Gompertz-type model.
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.
Transport equation for plasmas in a stationary-homogeneous turbulence
Wang, Shaojie
2016-02-15
For a plasma in a stationary homogeneous turbulence, the Fokker-Planck equation is derived from the nonlinear Vlasov equation by introducing the entropy principle. The ensemble average in evaluating the kinetic diffusion tensor, whose symmetry has been proved, can be computed in a straightforward way when the fluctuating particle trajectories are provided. As an application, it has been shown that a mean parallel electric filed can drive a particle flux through the Stokes-Einstein relation, independent of the details of the fluctuations.
Stability and Bifurcation of a Class of Stochastic Closed Orbit Equations
NASA Astrophysics Data System (ADS)
Luo, Chaoliang; Guo, Shangjiang
In this paper, by using Lyapunov functions and exponents, Feller's scale functions, and the Fokker-Planck equations, we investigate the stability and bifurcation of stochastic closed orbit equations with singular diffusion coefficients.
NASA Astrophysics Data System (ADS)
Boghosian, Bruce M.; Devitt-Lee, Adrian; Johnson, Merek; Li, Jie; Marcq, Jeremy A.; Wang, Hongyan
2017-06-01
The ;Yard-Sale Model; of asset exchange is known to result in complete inequality-all of the wealth in the hands of a single agent. It is also known that, when this model is modified by introducing a simple model of redistribution based on the Ornstein-Uhlenbeck process, it admits a steady state exhibiting some features similar to the celebrated Pareto Law of wealth distribution. In the present work, we analyze the form of this steady-state distribution in much greater detail, using a combination of analytic and numerical techniques. We find that, while Pareto's Law is approximately valid for low redistribution, it gives way to something more similar to Gibrat's Law when redistribution is higher. Additionally, we prove in this work that, while this Pareto or Gibrat behavior may persist over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Also in this work, we introduce a bias in favor of the wealthier agent-what we call Wealth-Attained Advantage (WAA)-and show that this leads to the phenomenon of ;wealth condensation; when the bias exceeds a certain critical value. In the wealth-condensed state, a finite fraction of the total wealth of the population ;condenses; to the wealthiest agent. We examine this phenomenon in some detail, and derive the corresponding modification to the Fokker-Planck equation. We observe a second-order phase transition to a state of coexistence between an oligarch and a distribution of non-oligarchs. Finally, by studying the asymptotic behavior of the distribution in some detail, we show that the onset of wealth condensation has an abrupt reciprocal effect on the character of the non-oligarchical part of the distribution. Specifically, we show that the above-mentioned gaussian decay at extremely large wealth is valid both above and below criticality, but degenerates to exponential decay precisely at criticality.
Variational Derivation of Dissipative Equations
NASA Astrophysics Data System (ADS)
Sogo, Kiyoshi
2017-03-01
A new variational principle is formulated to derive various dissipative equations. Model equations considered are the damping equation, Bloch equation, diffusion equation, Fokker-Planck equation, Kramers equation and Smoluchowski equation. Each equation and its time reversal equation are simultaneously obtained in our variational principle.
NASA Astrophysics Data System (ADS)
Tableman, Adam; Tzoufras, Michail; Fonseca, Ricardo; Mori, W. B.
2016-10-01
We present physics results and general updates for two plasma kinetic simulation codes developed under the UCLA PICKSE initiative. We also discuss the issues around making these codes open source such that they can be used (and contributed too) by a large audience. The first code discussed is Oshun - a Vlasov-Fokker-Planck (VFP) code. Recent simulations with the VFP code OSHUN will be presented for all of the aforementioned problems. The algorithmic improvements that have facilitated these studies will be also be discussed. The second code discussed is the PIC code Osiris. Osiris is a widely respected code used in hundreds of papers. Osiris was first developed for laser-plasma interactions but has grown into a robust framework covering most areas of plasma research. One defining feature of Osiris is that it is highly optimized for a variety of hardware configurations and scales linearly over 1 million + CPU nodes. We will discuss the recently released version 4.0 written in modern, fully-object oriented FORTRAN. Funding provided by Grants NSF ACI 1339893 and DOE DE NA 0001833.
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.
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.
Perturbed nonlinear differential equations
NASA Technical Reports Server (NTRS)
Proctor, T. G.
1974-01-01
For perturbed nonlinear systems, a norm, other than the supremum norm, is introduced on some spaces of continuous functions. This makes possible the study of new types of behavior. A study is presented on a perturbed nonlinear differential equation defined on a half line, and the existence of a family of solutions with special boundedness properties is established. The ideas developed are applied to the study of integral manifolds, and examples are given.
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.
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
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.
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.
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.
Perturbed nonlinear differential equations
NASA Technical Reports Server (NTRS)
Proctor, T. G.
1972-01-01
The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t,y) + F(t,y). The unperturbed system x = f(t,x) has a dichotomy in which some solutions exist and are well behaved as t increases to infinity, and some solution exists and are well behaved as t decreases to minus infinity. A similar study is made for a perturbed nonlinear differential equation defined on a half line, R+, and the existence of a family of solutions with special boundedness properties is established. The ideas are applied to integral manifolds.
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.
Adaptive approach for nonlinear sensitivity analysis of reaction kinetics.
Horenko, Illia; Lorenz, Sönke; Schütte, Christof; Huisinga, Wilhelm
2005-07-15
We present a unified approach for linear and nonlinear sensitivity analysis for models of reaction kinetics that are stated in terms of systems of ordinary differential equations (ODEs). The approach is based on the reformulation of the ODE problem as a density transport problem described by a Fokker-Planck equation. The resulting multidimensional partial differential equation is herein solved by extending the TRAIL algorithm originally introduced by Horenko and Weiser in the context of molecular dynamics (J. Comp. Chem. 2003, 24, 1921) and discussed it in comparison with Monte Carlo techniques. The extended TRAIL approach is fully adaptive and easily allows to study the influence of nonlinear dynamical effects. We illustrate the scheme in application to an enzyme-substrate model problem for sensitivity analysis w.r.t. to initial concentrations and parameter values.
Effects of introducing nonlinear components for a random excited hybrid energy harvester
NASA Astrophysics Data System (ADS)
Zhou, Xiaoya; Gao, Shiqiao; Liu, Haipeng; Guan, Yanwei
2017-01-01
This work is mainly devoted to discussing the effects of introducing nonlinear components for a hybrid energy harvester under random excitation. For two different types of nonlinear hybrid energy harvesters subjected to random excitation, the analytical solutions of the mean output power, voltage and current are derived from Fokker-Planck (FP) equations. Monte Carlo simulation exhibits qualitative agreement with FP theory, showing that load values and excitation’s spectral density have an effect on the total mean output power, piezoelectric (PE) power and electromagnetic power. Nonlinear components affect output characteristics only when the PE capacitance of the hybrid energy harvester is non-negligible. Besides, it is also demonstrated that for this type of nonlinear hybrid energy harvesters under random excitation, introducing nonlinear components can improve output performances effectively.
Probabilistic approach to nonlinear wave-particle resonant interaction
NASA Astrophysics Data System (ADS)
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2017-02-01
In this paper we provide a theoretical model describing the evolution of the charged-particle distribution function in a system with nonlinear wave-particle interactions. Considering a system with strong electrostatic waves propagating in an inhomogeneous magnetic field, we demonstrate that individual particle motion can be characterized by the probability of trapping into the resonance with the wave and by the efficiency of scattering at resonance. These characteristics, being derived for a particular plasma system, can be used to construct a kinetic equation (or generalized Fokker-Planck equation) modeling the long-term evolution of the particle distribution. In this equation, effects of charged-particle trapping and transport in phase space are simulated with a nonlocal operator. We demonstrate that solutions of the derived kinetic equations agree with results of test-particle tracing. The applicability of the proposed approach for the description of space and laboratory plasma systems is also discussed.
Resonant-test-field model of fluctuating nonlinear waves
NASA Astrophysics Data System (ADS)
West, Bruce J.
1982-03-01
A Hamiltonian system of nonlinear dispersive waves is used as a basis for generalizing the test-wave model to a set of resonantly interacting waves. The resonant test field (RTF) is shown to obey a nonlinear generalized Langevin equation in general. In the Markov limit a Fokker-Planck equation is obtained and the exact steady-state solution is determined. An algebraic expression for the power spectral density is obtained in terms of the number of resonantly interacting waves (n) in the RTF, the interaction strength (Vk), and the dimensionality of the wave field (d). For gravity waves on the ocean surface a k-4 spectrum is obtained, and for capillary waves a k-8 spectrum, both of which are in essential agreement with data.
Quantum fluctuation of nonlinear degenerate optical parametric amplification
NASA Astrophysics Data System (ADS)
Zhao, C. Y.; Tan, W. H.
An analytical solution of the Fokker Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).
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.
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.
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.
NASA Astrophysics Data System (ADS)
Kiesewetter, Simon; Drummond, Peter D.
2017-03-01
A variance reduction method for stochastic integration of Fokker-Planck equations is derived. This unifies the cumulant hierarchy and stochastic equation approaches to obtaining moments, giving a performance superior to either. We show that the brute force method of reducing sampling error by just using more trajectories in a sampled stochastic equation is not the best approach. The alternative of using a hierarchy of moment equations is also not optimal, as it may converge to erroneous answers. Instead, through Bayesian conditioning of the stochastic noise on the requirement that moment equations are satisfied, we obtain improved results with reduced sampling errors for a given number of stochastic trajectories. The method used here converges faster in time-step than Ito-Euler algorithms. This parallel optimized sampling (POS) algorithm is illustrated by several examples, including a bistable nonlinear oscillator case where moment hierarchies fail to converge.
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.
Nonlinear equations of 'variable type'
NASA Astrophysics Data System (ADS)
Larkin, N. A.; Novikov, V. A.; Ianenko, N. N.
In this monograph, new scientific results related to the theory of equations of 'variable type' are presented. Equations of 'variable type' are equations for which the original type is not preserved within the entire domain of coefficient definition. This part of the theory of differential equations with partial derivatives has been developed intensively in connection with the requirements of mechanics. The relations between equations of the considered type and the problems of mathematical physics are explored, taking into account quasi-linear equations, and models of mathematical physics which lead to equations of 'variable type'. Such models are related to transonic flows, problems involving a separation of the boundary layer, gasdynamics and the van der Waals equation, shock wave phenomena, and a combustion model with a turbulent diffusion flame. Attention is also given to nonlinear parabolic equations, and nonlinear partial differential equations of the third order.
NASA Astrophysics Data System (ADS)
Azoulay, M.; George, M. A.; Burger, A.; Collins, W. E.; Silberman, E.
A two-dimensional bounce averaged Fokker-Planck code is used to study the fusion yield and the wave absorption by residual hydrogen ions in higher harmonic ICRF heating of D (120 keV) and 3He (80 keV) beams in the JT-60U tokamak. Both for the fourth harmonic resonance of 3He (ω = 4ωc3He(0), which is accompanied by the third harmonic resonance of hydrogen (ω = 3ωcH) at the low field side, and for the third harmonic resonance of 3He (ω = 4ωcD(0) = 3ωc3He(0)) = 2ωcH(0)), a few per cent of hydrogen ions are found to absorb a large fraction of the ICRF power and to degrade the fusion output power. In the latter case, D beam acceleration due to the fourth harmonic resonance in the 3He(D) regime can enhance the fusion yield more effectively. A discussion is given of the effect of D beam acceleration due to the fifth harmonic resonance (ω = 5ωcD) at the high field side in the case of ω = 4ωc3He(0) and of the optimization of the fusion yield in the case of lower electron density and higher electron temperature
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
NASA Astrophysics Data System (ADS)
Maas, Jan; Matthes, Daniel
2016-07-01
We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d≥slant 1 . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.
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.
NASA Technical Reports Server (NTRS)
Ljepojevic, N. N.; Macneice, P.
1988-01-01
The high-velocity tail of the electron distribution has been calculated by solving the high-velocity form of the Landau equation for a thermal structure representative of a flaring coronal loop. These calculations show an enhancement of the tail population above Maxwellian for electrons moving down the temperature gradient. The results obtained are used to test the reliability of the BGK approximation. The comparison shows that the BGK technique can estimate contributions to the heat flux from the high-energy tail to within an order of magnitude.
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
Modeling water table fluctuations by means of a stochastic differential equation
NASA Astrophysics Data System (ADS)
Bierkens, Marc F. P.
1998-10-01
The combined system of soil-water and shallow groundwater is modeled with simple mass balance equations assuming equilibrium soil moisture conditions. This results in an ordinary but nonlinear differential equation of water table depth at a single location. If errors in model inputs, errors due to model assumptions and parameter uncertainty are lumped and modeled as a wide band noise process, a stochastic differential equation (SDE) results. A solution for the stationary probability density function is given through use of the Fokker-Planck equation. For the nonstationary case, where the model inputs are given as daily time series, sample functions of water table depth, soil saturation, and drainage discharge can be simulated by numerically solving the SDE. These sample functions can be used for designing drainage systems and to perform risk analyses. The parameters and noise statistics of the SDE are calibrated on time series of water table depths by embedding the SDE in a Kaiman filter algorithm and using the filter innovations in a filter-type maximum likelihood criterion. The stochastic model is calibrated and validated at two locations: a peat soil with a very shallow water table and a loamy sand soil with a moderately shallow water table. It is shown in both cases that sample functions simulated with the SDE are able to reproduce a wide range of statistics of water table depth. Despite its unrealistic assumption of constant inputs, the stationary solution derived from the Fokker-Planck equation gives good results for the peat soil, most likely because the characteristic response time of the water table is very small.
Global Weak Solutions for Kolmogorov-Vicsek Type Equations with Orientational Interactions
NASA Astrophysics Data System (ADS)
Gamba, Irene M.; Kang, Moon-Jin
2016-10-01
We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov-Vicsek models that can be considered as non-local, non-linear, Fokker-Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin-Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339-343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193-1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov-Vicsek models is the unit sphere. Our analysis for L p estimates and compactness properties take advantage of the orientational interaction property, meaning that the velocity space is a compact manifold.
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.
Solutions of the cylindrical nonlinear Maxwell equations.
Xiong, Hao; Si, Liu-Gang; Ding, Chunling; Lü, Xin-You; Yang, Xiaoxue; Wu, Ying
2012-01-01
Cylindrical nonlinear optics is a burgeoning research area which describes cylindrical electromagnetic wave propagation in nonlinear media. Finding new exact solutions for different types of nonlinearity and inhomogeneity to describe cylindrical electromagnetic wave propagation is of great interest and meaningful for theory and application. This paper gives exact solutions for the cylindrical nonlinear Maxwell equations and presents an interesting connection between the exact solutions for different cylindrical nonlinear Maxwell equations. We also provide some examples and discussion to show the application of the results we obtained. Our results provide the basis for solving complex systems of nonlinearity and inhomogeneity with simple systems.
Systems of Nonlinear Hyperbolic Partial Differential Equations
1997-12-01
McKinney) Travelling wave solutions of the modified Korteweg - deVries -Burgers Equation . J. Differential Equations , 116 (1995), 448-467. 4. (with D.G...SUBTITLE Systems of Nonlinear Hyperbolic Partial Differential Equations 6. AUTHOR’S) Michael Shearer PERFORMING ORGANIZATION NAMES(S) AND...DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This project concerns properties of wave propagation in partial differential equations that are nonlinear
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.
Nonlinear Poisson equation for heterogeneous media.
Hu, Langhua; Wei, Guo-Wei
2012-08-22
The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.
Nonlinear Poisson Equation for Heterogeneous Media
Hu, Langhua; Wei, Guo-Wei
2012-01-01
The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects. PMID:22947937
Nonlinear gyrokinetic equations for tokamak microturbulence
Hahm, T.S.
1988-05-01
A nonlinear electrostatic gyrokinetic Vlasov equation, as well as Poisson equation, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.
Probabilistic characteristics of noisy Van der Pol type oscillator with nonlinear damping
NASA Astrophysics Data System (ADS)
Dubkov, A. A.; Litovsky, I. A.
2016-05-01
The exact Fokker-Planck equation for the joint probability distribution of amplitude and phase of a Van der Pol oscillator perturbed by both additive and multiplicative noise sources with arbitrary nonlinear damping is first derived by the method of functional splitting of averages. We truncate this equation in the usual manner using the smallness of the damping parameter and obtain a general expression for the stationary probability density function of oscillation amplitude, which is valid for any nonlinearity in the feedback loop of the oscillator. We analyze the dependence of this stationary solution on system parameters and intensities of noise sources for two different situations: (i) Van der Pol generator with soft and hard excitation regimes; (ii) Van der Pol oscillator with negative nonlinear damping. As shown, in the first case the probability distribution of amplitude demonstrates one characteristic maximum, which indicates the presence of a stable limit cycle in the system. The non-monotonic dependence of stationary probability density function on oscillation frequency is also detected. In the second case we examine separately three situations: linear oscillator with two noise sources, nonlinear oscillator with additive noise and nonlinear oscillator with frequency fluctuations. For the last two situations, noise-induced transitions in the system under consideration are found.
Nonlinear Behavior of Magnetic Fluid in Brownian Relaxation
Yoshida, Takashi; Ogawa, Koutaro; Bhuiya, Anwarul K.; Enpuku, Keiji
2010-12-02
This study investigated the nonlinear behavior of magnetic fluids under high excitation fields due to nonlinear Brownian relaxation. As a direct indication of nonlinear behavior, we characterized the higher harmonics of the magnetization signal generated by the magnetic fluid. The amplitudes of the fundamental to the ninth harmonic of the magnetization signal were measured as a function of the external field. The experimental results were compared with numerical simulations based on the Fokker-Planck equation, which describes nonlinear Brownian relaxation. To allow a quantitative comparison, we estimated the size distribution and size dependence of the magnetic moment in the sample. In the present magnetic fluid, composed of agglomerates of Fe{sub 3}O{sub 4} particles, the magnetic moment was estimated to be roughly proportional to the diameter of the particles, in contrast to the case of single-domain particles. When the size distribution and the size dependence of the magnetic moment were taken into account, there was good quantitative agreement between the experiment and simulation.
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.
Identification for a Nonlinear Periodic Wave Equation
Morosanu, C.; Trenchea, C.
2001-07-01
This work is concerned with an approximation process for the identification of nonlinearities in the nonlinear periodic wave equation. It is based on the least-squares approach and on a splitting method. A numerical algorithm of gradient type and the numerical implementation are given.
Polynomial solutions of nonlinear integral equations
NASA Astrophysics Data System (ADS)
Dominici, Diego
2009-05-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
Iterative performance of various formulations of the SPN equations
NASA Astrophysics Data System (ADS)
Zhang, Yunhuang; Ragusa, Jean C.; Morel, Jim E.
2013-11-01
In this paper, the Standard, Composite, and Canonical forms of the Simplified PN (SPN) equations are reviewed and their corresponding iterative properties are compared. The Gauss-Seidel (FLIP), Explicit, and preconditioned Source Iteration iterative schemes have been analyzed for both isotropic and highly anisotropic (Fokker-Planck) scattering. The iterative performance of the various SPN forms is assessed using Fourier analysis, corroborated with numerical experiments.
Lax integrable nonlinear partial difference equations
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Nagavigneshwari, G.
2015-03-01
A systematic investigation to derive nonlinear lattice equations governed by partial difference equations admitting specific Lax representation is presented. Further whether or not the identified lattice equations possess other characteristics of integrability namely Consistency Around the Cube (CAC) property and linearizability through a global transformation is analyzed. Also it is presented that how to derive higher order ordinary difference equations or mappings from the obtained lattice equations through periodic reduction and investigated whether they are measure preserving or linearizable and admit sufficient number of integrals leading to their integrability.
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.
Some remarks on quantum physics, stochastic processes, and nonlinear filtering theory
NASA Astrophysics Data System (ADS)
Balaji, Bhashyam
2016-05-01
The mathematical similarities between quantum mechanics and stochastic processes has been studied in the literature. Some of the major results are reviewed, such as the relationship between the Fokker-Planck equation and the Schrödinger equation. Also reviewed are more recent results that show the mathematical similarities between quantum many particle systems and concepts in other areas of applied science, such as stochastic Petri nets. Some connections to filtering theory are discussed.
On Coupled Rate Equations with Quadratic Nonlinearities
Montroll, Elliott W.
1972-01-01
Rate equations with quadratic nonlinearities appear in many fields, such as chemical kinetics, population dynamics, transport theory, hydrodynamics, etc. Such equations, which may arise from basic principles or which may be phenomenological, are generally solved by linearization and application of perturbation theory. Here, a somewhat different strategy is emphasized. Alternative nonlinear models that can be solved exactly and whose solutions have the qualitative character expected from the original equations are first searched for. Then, the original equations are treated as perturbations of those of the solvable model. Hence, the function of the perturbation theory is to improve numerical accuracy of solutions, rather than to furnish the basic qualitative behavior of the solutions of the equations. PMID:16592013
Efficient numerical methods for nonlinear Schrodinger equations
NASA Astrophysics Data System (ADS)
Liang, Xiao
The nonlinear Schrodinger equations are widely used to model a number of important physical phenomena, including solitary wave propagations in optical fibers, deep water turbulence, laser beam transmissions, and the Bose-Einstein condensation, just to mention a few. In the field of optics and photonics, the systems of nonlinear Schrodinger equations can be used to model multi-component solitons and the interaction of self-focusing laser beams. In three spatial dimensions, the nonlinear Schrodinger equation is known as the Gross-Pitaevskii equation, which models the soliton in a low-cost graded-index fiber. Recently, research on nonlinear space fractional Schrodinger equations, which capture the self-similarity in the fractional environment, has become prevalent. Our study includes the systems of multi-dimensional nonlinear space fractional Schrodinger equations. To solve the systems of multi-dimensional nonlinear Schrodinger equations efficiently, several novel numerical methods are presented. The central difference and quartic spline approximation based exponential time differencing Crank-Nicolson method is introduced for solving systems of one- and two-dimensional nonlinear Schrodinger equations. A local extrapolation is employed to achieve fourth-order accuracy in time. The numerical examples include the transmission of a self-focusing laser beam. The local discontinuous Galerkin methods combined with the fourth-order exponential time differencing Runge-Kutta time discretization are studied for solving the systems of nonlinear Schrodinger equations with hyperbolic terms, which are critical in modeling optical solitons in the birefringent fibers. The local discontinuous Galerkin method is able to achieve any order of accuracy in space, thanks to the usage of piecewise polynomial spaces. The exponential time differencing methods are employed to deal with the coupled nonlinearities for the reason that there is no need to solve nonlinear systems at every time step
Prolongation structures of nonlinear evolution equations
NASA Technical Reports Server (NTRS)
Wahlquist, H. D.; Estabrook, F. B.
1975-01-01
A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.
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.
Exact solutions for nonlinear foam drainage equation
NASA Astrophysics Data System (ADS)
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2017-02-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Exact solutions for nonlinear foam drainage equation
NASA Astrophysics Data System (ADS)
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2016-09-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G) -expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Zhu, H T
2012-06-01
This paper studies the stationary probability density function (PDF) of the response of multi-degree-of-freedom nonlinear systems under external independent Poisson white noises. The PDF is governed by the high-dimensional generalized Fokker-Planck-Kolmogorov (FPK) equation. The state-space-split (3S) method is adopted to reduce the high-dimensional generalized FPK equation to a low-dimensional equation. Subsequently, the exponential-polynomial closure (EPC) method is further used to solve the reduced FPK equation for the PDF solution. Two illustrative examples are presented to examine the accuracy of the 3S-EPC solution procedure. One example involves a two-degree-of-freedom coupled nonlinear system. The other example is concerned with a ten-degree-of-freedom system with cubic terms in displacement. A Monte Carlo simulation is also performed for simulating the PDF solution of the response. The comparison with the simulated result shows that the 3S-EPC solution procedure can provide satisfactory PDF solutions. The good agreement is also observed in the tail regions of the PDF solutions.
A combination method for solving nonlinear equations
NASA Astrophysics Data System (ADS)
Silalahi, B. P.; Laila, R.; Sitanggang, I. S.
2017-01-01
This paper discusses methods for finding solutions of nonlinear equations: the Newton method, the Halley method and the combination of the Newton method, the Newton inverse method and the Halley method. Computational results in terms of the accuracy, the number of iterations and the running time for solving some given problems are presented.
The Stochastic Nonlinear Damped Wave Equation
Barbu, V. Da Prato, G.
2002-12-19
We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in R{sup n} of the Klein-Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied.
Gaudreault, Mathieu; Drolet, François; Viñals, Jorge
2010-11-01
Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average
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.
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.
Explicit integration of Friedmann's equation with nonlinear equations of state
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong E-mail: gwg1@damtp.cam.ac.uk
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.
Numerical solutions of nonlinear wave equations
Kouri, D.J.; Zhang, D.S.; Wei, G.W.; Konshak, T.; Hoffman, D.K.
1999-01-01
Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equation are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds. Earlier local, grid-based numerical studies have encountered difficulties, including numerically induced chaos for such initial conditions. The present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple, explicit method for the time evolution. The numerical solutions are chaos-free for the same conditions employed in previous work that encountered chaos. Moreover, stable results that are free of homoclinic-orbit crossing are obtained even when initial conditions are within 10{sup {minus}7} of the phase space separatrix value {pi}. It also is found that the present approach yields extremely accurate solutions for the Korteweg{endash}de Vries and nonlinear Schr{umlt o}dinger equations. Our results support Ablowitz and co-workers{close_quote} conjecture that ensuring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations. {copyright} {ital 1999} {ital The American Physical Society}
Invariant metrics, contractions and nonlinear matrix equations
NASA Astrophysics Data System (ADS)
Lee, Hosoo; Lim, Yongdo
2008-04-01
In this paper we consider the semigroup generated by the self-maps on the open convex cone of positive definite matrices of translations, congruence transformations and matrix inversion that includes symplectic Hamiltonians and show that every member of the semigroup contracts any invariant metric distance inherited from a symmetric gauge function. This extends the results of Bougerol for the Riemannian metric and of Liverani-Wojtkowski for the Thompson part metric. A uniform upper bound of the Lipschitz contraction constant for a member of the semigroup is given in terms of the minimum eigenvalues of its determining matrices. We apply this result to a variety of nonlinear equations including Stein and Riccati equations for uniqueness and existence of positive definite solutions and find a new convergence analysis of iterative algorithms for the positive definite solution depending only on the least contraction coefficient for the invariant metric from the spectral norm.
Galerkin Methods for Nonlinear Elliptic Equations.
NASA Astrophysics Data System (ADS)
Murdoch, Thomas
Available from UMI in association with The British Library. Requires signed TDF. This thesis exploits in the nonlinear situation the optimal approximation property of the finite element method for linear, elliptic problems. Of particular interest are the steady state semiconductor equations in one and two dimensions. Instead of discretising the differential equations by the finite element method and solving the nonlinear algebraic equations by Newton's method, a Newton linearisation of the continuous problem is preferred and a sequence of linear problems solved until some convergence criterion is achieved. For nonlinear Poisson equations, this approach reduces to solving a sequence of linear, elliptic, self -adjoint problems, their approximation by the finite element being optimal in a suitably defined energy norm. Consequently, there is the potential to recover a smoother representation of the underlying solution at each step of the Newton iteration. When this approach is applied to the continuity equations for semiconductor devices, a sequence of linear problems of the form -_{nabla }(anabla u - bu) = f must be solved. The Galerkin method in its crude form does not adequately represent the true solution: however, generalising the framework to permit Petrov-Galerkin approximations remedies the situation. For one dimensional problems, the work of Barrett and Morton allows an optimal test space to be chosen at each step of the Newton iteration so that the resulting approximation is near optimal in a norm closely related to the standard L^2 norm. More detailed information about the underlying solution can then be obtained by recovering a solution of an appropriate form. For two-dimensional problems, since the optimal test functions are difficult to find in practice, an upwinding method due to Heinrich et.al. is used at each step of the Newton iteration. Also, a framework is presented in which various upwind methods may be compared. The thesis also addresses the
Strongly Nonlinear Integral Equations of Hammerstein Type
Browder, Felix E.
1975-01-01
This paper studies the solution of the nonlinear Hammerstein equation u(x) + ʃ k(x,y)f[y,u(y)]μ(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) ≥ 0, that K maps L1 into L1loc and is compact from L1 [unk] L∞ into L1loc, that f(y,s) has the same sign as s for ǀsǀ ≥ R, and that for each constant r > 0, ǀf(y,s)ǀ ≤ gr(y) for ǀsǀ ≤ r where g is bounded and summable. The proof is obtained by combining a priori bounds, a truncation procedure, and a convergence argument using the Dunford-Pettis theorem. PMID:16578727
Notes on the Modified Nonlinear Schrodinger Equation
NASA Astrophysics Data System (ADS)
Pizzo, N. E.; Melville, W. K.
2011-12-01
In this study, we present the derivation of a modified Nonlinear Schrodinger equation (MNLSE) based on variational calculus. Using weakly nonlinear theory we derive an averaged Lagrangian, which in turn yields a slightly modified version of the MNLSE that conserves wave action. We also explore ramifications of the MNLSE with respect to the coupling between mean currents and non-uniform radiation stresses. We present this in the context of breaking waves and the free long waves they generate (Kristian Dysthe, personal communication). It has been noted in laboratory experiments (Meza et al, 1999) that breaking waves transfer some energy to modes far below the peak frequency of the spectrum. The transfer mechanism is widely believed to be the result of nonlinear four wave resonant interactions; however, the coupling between breaking-induced non-uniform radiation stresses and long wave radiation suggests a potential alternative explanation. Through direct numerical simulations, along with the theory, we test the feasibility of this mechanism by comparing it to data from wave tank experiments (Drazen et al., 2008).
Nonlinear Dynamics of the Leggett Equation
NASA Astrophysics Data System (ADS)
Ragan, Robert J.
1995-01-01
We study the nonlinear dynamics of spin-polarized Fermi liquids. Our starting point is the equation of motion for the magnetization derived by Leggett and Rice, which accounts for spin-rotation effects in the limit of small polarization. We also include later modifications to the theory by Meyerovich, and Jeon and Mullin, which account for polarization dependences of the transport coefficients. In the analysis of NMR experiments the methods of current research can be summarized as follows: (a) to linearize the Leggett equation by considering small amplitude oscillations (small tip angles), (b) to use perturbation theory to account for small spin-rotation effects, (c) to exploit the simple helical solution which describes spin-echo experiments. In this thesis, we report progress in several directions: (1) We extend the linear theory to describe bounded spin diffusion with spin-rotation and finite-polarization effects. The analysis is valid for arbitrary tip angles and arbitrary degree of nonlinearity. (2) We show that because of the spin-rotation effect, the helical solution exhibits a Castiang instability for large tip angles. In the limit of small damping, we use the inverse scattering theory developed by Levy to display the full nonlinear evolution of the instabilities. (3) We use perturbation theory to show that anisotropy in the spin diffusion coefficients gives rise to multiple spin echoes, even in the absence of spin -rotation effects. This description applies to experiments on ^3He-^4He solutions at ^3He concentrations of 3-5%. This experiment provides a unique means of verifying the theory of Jeon and Mullin. We also report some exact results in the theory of anisotropic spin diffusion.
Nonlinear scalar field equations involving the fractional Laplacian
NASA Astrophysics Data System (ADS)
Byeon, Jaeyoung; Kwon, Ohsang; Seok, Jinmyoung
2017-04-01
In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.
Some new solutions of nonlinear evolution equations with variable coefficients
NASA Astrophysics Data System (ADS)
Virdi, Jasvinder Singh
2016-05-01
We construct the traveling wave solutions of nonlinear evolution equations (NLEEs) with variable coefficients arising in physics. Some interesting nonlinear evolution equations are investigated by traveling wave solutions which are expressed by the hyperbolic functions, the trigonometric functions and rational functions. The applied method will be used in further works to establish more entirely new solutions for other kinds of such nonlinear evolution equations with variable coefficients arising in physics.
Nonlinear Parabolic Equations Involving Measures as Initial Conditions.
1981-09-01
CHART N N N Afl4Uf’t 1N II Il MRC Technical Summary Report # 2277 0 NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS I Haim Brezis ...NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS Haim Brezis and Avner Friedman Technical Summary Report #2277 September 1981...with NRC, and not with the authors of this report. * s ’a * ’ 4| NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS Haim Brezis
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.
Bifurcation and stability for a nonlinear parabolic partial differential equation
NASA Technical Reports Server (NTRS)
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
Exact solutions to nonlinear delay differential equations of hyperbolic type
NASA Astrophysics Data System (ADS)
Vyazmin, Andrei V.; Sorokin, Vsevolod G.
2017-01-01
We consider nonlinear delay differential equations of hyperbolic type, including equations with varying transfer coefficients and varying delays. The equations contain one or two arbitrary functions of a single argument. We present new classes of exact generalized and functional separable solutions. All the solutions involve free parameters and can be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for similar and more complex nonlinear differential-difference equations.
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.
Nonlinear Schrödinger equation with complex supersymmetric potentials
NASA Astrophysics Data System (ADS)
Nath, D.; Roy, P.
2017-03-01
Using the concept of supersymmetry we obtain exact analytical solutions of nonlinear Schrödinger equation with a number of complex supersymmetric potentials and power law nonlinearity. Linear stability of these solutions for self-focusing as well as de-focusing nonlinearity has also been examined.
Exploring the nonlinear cloud and rain equation.
Koren, Ilan; Tziperman, Eli; Feingold, Graham
2017-01-01
Marine stratocumulus cloud decks are regarded as the reflectors of the climate system, returning back to space a significant part of the income solar radiation, thus cooling the atmosphere. Such clouds can exist in two stable modes, open and closed cells, for a wide range of environmental conditions. This emergent behavior of the system, and its sensitivity to aerosol and environmental properties, is captured by a set of nonlinear equations. Here, using linear stability analysis, we express the transition from steady to a limit-cycle state analytically, showing how it depends on the model parameters. We show that the control of the droplet concentration (N), the environmental carrying-capacity (H0), and the cloud recovery parameter (τ) can be linked by a single nondimensional parameter (μ=N/(ατH0)), suggesting that for deeper clouds the transition from open (oscillating) to closed (stable fixed point) cells will occur for higher droplet concentration (i.e., higher aerosol loading). The analytical calculations of the possible states, and how they are affected by changes in aerosol and the environmental variables, provide an enhanced understanding of the complex interactions of clouds and rain.
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.
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.
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
The effect of nonlinearity on unstable zones of Mathieu equation
NASA Astrophysics Data System (ADS)
Saryazdi, M. Gh
2017-03-01
Mathieu equation is a well-known ordinary differential equation in which the excitation term appears as the non-constant coefficient. The mathematical modelling of many dynamic systems leads to Mathieu equation. The determination of the locus of unstable zone is important for the control of dynamic systems. In this paper, the stable and unstable regions of Mathieu equation are determined for three cases of linear and nonlinear equations using the homotopy perturbation method. The effect of nonlinearity is examined in the unstable zone. The results show that the transition curves of linear Mathieu equation depend on the frequency of the excitation term. However, for nonlinear equations, the curves depend also on initial conditions. In addition, increasing the amplitude of response leads to an increase in the unstable zone.
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.
A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects.
Xu, Zhengfu; Bao, Gang
2010-11-01
A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.
Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations
NASA Astrophysics Data System (ADS)
Ma, Wen-Xiu; Zhou, Yuan; Dougherty, Rachael
2016-08-01
Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on f are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on u, generated from taking a transformation of dependent variables u = 2(ln f)x.
Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities.
Khare, Avinash; Rasmussen, Kim Ø; Salerno, Mario; Samuelsen, Mogens R; Saxena, Avadh
2006-07-01
A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.
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.
Variable-coefficient extended mapping method for nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Zhang, Sheng; Xia, Tiecheng
2008-03-01
In this Letter, a variable-coefficient extended mapping method is proposed to seek new and more general exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients and ( 2+1)-dimensional Nizhnik-Novikov-Veselov equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the proposed method provides a very effective and powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.
Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation.
1980-06-01
Equation (1) may also be considered as an ordinary differential equation on a Banach space. This is the setting I prefer, as it usually seems much more... NONLINEAR WAVE EQUATION ~0 by gc~ Paul Massatt Lefschetz Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, Rhode...Interim -) DAMPED NONLINEAR WAVE EQUATION . 6. PERFORMING 0G. RMRT UMBER 7. AUTHOR(a) S. CONTRACT OR GRANT NUMBER(O) PAUL!MASSATT 47 -Xo AFdSR-76-3,992 / 9
Comparative study of homotopy continuation methods for nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
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.
Integrable nonlocal nonlinear Schrödinger equation.
Ablowitz, Mark J; Musslimani, Ziad H
2013-02-08
A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed. A method to find pure soliton solutions is given. An explicit breathing one soliton solution is found. Key properties are discussed and contrasted with the classical nonlinear Schrödinger equation.
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.
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.
Nonlinear acoustic wave equations with fractional loss operators.
Prieur, Fabrice; Holm, Sverre
2011-09-01
Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.
The nonlinear modified equation approach to analyzing finite difference schemes
NASA Technical Reports Server (NTRS)
Klopfer, G. H.; Mcrae, D. S.
1981-01-01
The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.
Slunyaev, A; Pelinovsky, E; Sergeeva, A; Chabchoub, A; Hoffmann, N; Onorato, M; Akhmediev, N
2013-07-01
The rogue wave solutions (rational multibreathers) of the nonlinear Schrödinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation, also known as the Dysthe equation. This numerical modeling allowed us to directly compare simulations with recent results of laboratory measurements in Chabchoub et al. [Phys. Rev. E 86, 056601 (2012)]. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near-breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.
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.)
GHM method for obtaining rationalsolutions of nonlinear differential equations.
Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo
2015-01-01
In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.
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 partial differential equations: Integrability, geometry and related topics
NASA Astrophysics Data System (ADS)
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
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
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
2013-12-28
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.
Model Predictive Control for Nonlinear Parabolic Partial Differential Equations
NASA Astrophysics Data System (ADS)
Hashimoto, Tomoaki; Yoshioka, Yusuke; Ohtsuka, Toshiyuki
In this study, the optimal control problem of nonlinear parabolic partial differential equations (PDEs) is investigated. Optimal control of nonlinear PDEs is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Model predictive control with a fast numerical solution method has been well established to solve the optimal control problem of nonlinear systems described by ordinary differential equations. In this study, we develop a design method of the model predictive control for nonlinear systems described by parabolic PDEs. Our approach is a direct infinite dimensional extension of the model predictive control method for finite-dimensional systems. The objective of this paper is to develop an efficient algorithm for numerically solving the model predictive control problem of nonlinear parabolic PDEs. The effectiveness of the proposed method is verified by numerical simulations.
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.
Stochastic differential equation model to Prendiville processes
NASA Astrophysics Data System (ADS)
Granita, Bahar, Arifah
2015-10-01
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.
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.
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.
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.
Nonlinear equations of dynamics for spinning paraboloidal antennas
NASA Technical Reports Server (NTRS)
Utku, S.; Shoemaker, W. L.; Salama, M.
1983-01-01
The nonlinear strain-displacement and velocity-displacement relations of spinning imperfect rotational paraboloidal thin shell antennas are derived for nonaxisymmetrical deformations. Using these relations with the admissible trial functions in the principle functional of dynamics, the nonlinear equations of stress inducing motion are expressed in the form of a set of quasi-linear ordinary differential equations of the undetermined functions by means of the Rayleigh-Ritz procedure. These equations include all nonlinear terms up to and including the third degree. Explicit expressions are given for the coefficient matrices appearing in these equations. Both translational and rotational off-sets of the axis of revolution (and also the apex point of the paraboloid) with respect to the spin axis are considered. Although the material of the antenna is assumed linearly elastic, it can be anisotropic.
A general non-linear multilevel structural equation mixture model
Kelava, Augustin; Brandt, Holger
2014-01-01
In the past 2 decades latent variable modeling has become a standard tool in the social sciences. In the same time period, traditional linear structural equation models have been extended to include non-linear interaction and quadratic effects (e.g., Klein and Moosbrugger, 2000), and multilevel modeling (Rabe-Hesketh et al., 2004). We present a general non-linear multilevel structural equation mixture model (GNM-SEMM) that combines recent semiparametric non-linear structural equation models (Kelava and Nagengast, 2012; Kelava et al., 2014) with multilevel structural equation mixture models (Muthén and Asparouhov, 2009) for clustered and non-normally distributed data. The proposed approach allows for semiparametric relationships at the within and at the between levels. We present examples from the educational science to illustrate different submodels from the general framework. PMID:25101022
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
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 γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(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.
Nonlinear waves described by the generalized Swift-Hohenberg equation
NASA Astrophysics Data System (ADS)
Ryabov, P. N.; Kudryashov, N. A.
2017-01-01
We study the wave processes described by the generalized Swift-Hohenberg equation. We show that the traveling wave reduction of this equation does not pass the Kovalevskaya test. Some solitary wave solutions and kink solutions of the generalized Swift-Hohenberg equation are found. We use the pseudo-spectral algorithm to perform the numerical simulation of the wave processes described by the mixed boundary value problem for the generalized Swift-Hohenberg equation. This algorithm was tested on the obtained solutions. Some features of the nonlinear waves evolution described by the generalized Swift-Hohenberg equation are studied.
Approximating a nonlinear advanced-delayed equation from acoustics
NASA Astrophysics Data System (ADS)
Teodoro, M. Filomena
2016-10-01
We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.
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.
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.
The Jeffcott equations in nonlinear rotordynamics
NASA Technical Reports Server (NTRS)
Zalik, R. A.
1989-01-01
The solutions of the Jeffcott equations describing the behavior of a rotating shaft are investigated analytically, with a focus on the case where deadband is taken into account. Bounds on the solutions are obtained from those for the linearized equations, and the onset of destructive vibrations is predicted by analyzing the Fourier transforms of the solutions; good agreement with numerical solutions and power-spectrum density plots is demonstrated. It is suggested that the present analytical approach could be applied to determine cryogenic-pump stability margins in flight and hot-fire ground testing of launch vehicles such as the Space Shuttle.
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…
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…
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.
Exact solutions of the derivative nonlinear Schrödinger equation for a nonlinear transmission line.
Kengne, E; Liu, W M
2006-02-01
We consider the derivative nonlinear Schrödinger equation with constant potential as a model for wave propagation on a discrete nonlinear transmission line. This equation can be derived in the small amplitude and long wavelength limit using the standard reductive perturbation method and complex expansion. We construct some exact soliton and elliptic solutions of the mentioned equation by perturbation of its Stokes wave solutions. We find that for some values of the coefficients of the equation and for some parameters of solutions, the graphical representations show some kinds of symmetries such as mirror symmetry and rotational symmetry.
Transport equations for subdiffusion with nonlinear particle interaction.
Straka, P; Fedotov, S
2015-02-07
We show how the nonlinear interaction effects 'volume filling' and 'adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.
A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation
NASA Astrophysics Data System (ADS)
Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.
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.
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.
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-08
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.
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.
Relativistic Langevin equation for runaway electrons
NASA Astrophysics Data System (ADS)
Mier, J. A.; Martin-Solis, J. R.; Sanchez, R.
2016-10-01
The Langevin approach to the kinetics of a collisional plasma is developed for relativistic electrons such as runaway electrons in tokamak plasmas. In this work, we consider Coulomb collisions between very fast, relativistic electrons and a relatively cool, thermal background plasma. The model is developed using the stochastic equivalence of the Fokker-Planck and Langevin equations. The resulting Langevin model equation for relativistic electrons is an stochastic differential equation, amenable to numerical simulations by means of Monte-Carlo type codes. Results of the simulations will be presented and compared with the non-relativistic Langevin equation for RE electrons used in the past. Supported by MINECO (Spain), Projects ENE2012-31753, ENE2015-66444-R.
Nonlinear Landau damping, and nonlinear envelope equation, for a driven plasma wave
NASA Astrophysics Data System (ADS)
Benisti, Didier; Morice, Olivier; Gremillet, Laurent; Strozzi, David
2009-11-01
A nonlinear envelope equation for a laser-driven electron plasma wave (EPW) is derived in a 3-D geometry, starting from first principles. This equation accounts the nonlinear variations of the EPW Landau damping rate, frequency, and group velocity, as well as for the nonlinear variations of the coupling of the EPW to the electromagnetic waves. All these quantities are moreover shown to be nonlocal because of nonlocal variations of the electron distribution function. Each piece of our model is carefully tested against Vlasov simulations of stimulated Raman scattering (SRS), and very good agreement is found between the numerical and theoretical results. Our envelope equations for both, the electrostatic and electromagnetic waves, are solved numerically, and comparisons with Vlasov simulations regarding the growth of SRS are provided. Finally, from our theory we can straightforwardly deduce a nonlinear gain factor which provides an alternate, simpler and faster method to quantify the SRS reflectivity. First results using this method will be shown.
Case-Deletion Diagnostics for Nonlinear Structural Equation Models
ERIC Educational Resources Information Center
Lee, Sik-Yum; Lu, Bin
2003-01-01
In this article, a case-deletion procedure is proposed to detect influential observations in a nonlinear structural equation model. The key idea is to develop the diagnostic measures based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. An one-step pseudo approximation is proposed to reduce the…
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.
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.
Embedded eigenvalues and the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Asad, R.; Simpson, G.
2011-03-01
A common challenge in proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola and Simpson [Nonlinearity 52, 389 (2011)], 10.1088/0951-7715/24/2/003, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrödinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and end point resonances. The proof is computer assisted as it depends on the signs of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://hdl.handle.net/1807/26121.
Long-time relaxation processes in the nonlinear Schroedinger equation
Ovchinnikov, Yu. N.; Sigal, I. M.
2011-03-15
The nonlinear Schroedinger equation, known in low-temperature physics as the Gross-Pitaevskii equation, has a large family of excitations of different kinds. They include sound excitations, vortices, and solitons. The dynamics of vortices strictly depends on the separation between them. For large separations, some kind of adiabatic approximation can be used. We consider the case where an adiabatic approximation can be used (large separation between vortices) and the opposite case of a decay of the initial state, which is close to the double vortex solution. In the last problem, no adiabatic parameter exists (the interaction is strong). Nevertheless, a small numerical parameter arises in the problem of the decay rate, connected with an existence of a large centrifugal potential, which leads to a small value of the increment. The properties of the nonlinear wave equation are briefly considered in the Appendix A.
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.
1/f noise from nonlinear stochastic differential equations
NASA Astrophysics Data System (ADS)
Ruseckas, J.; Kaulakys, B.
2010-03-01
We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fβ noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fβ noise, and provides further insights into the origin of 1/fβ noise.
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
Ramaswamy, Rajesh; Sbalzarini, Ivo F; González-Segredo, Nélido
2011-01-28
Stochastic effects from correlated noise non-trivially modulate the kinetics of non-linear chemical reaction networks. This is especially important in systems where reactions are confined to small volumes and reactants are delivered in bursts. We characterise how the two noise sources confinement and burst modulate the relaxation kinetics of a non-linear reaction network around a non-equilibrium steady state. We find that the lifetimes of species change with burst input and confinement. Confinement increases the lifetimes of all species that are involved in any non-linear reaction as a reactant. Burst monotonically increases or decreases lifetimes. Competition between burst-induced and confinement-induced modulation may hence lead to a non-monotonic modulation. We quantify lifetime as the integral of the time autocorrelation function (ACF) of concentration fluctuations around a non-equilibrium steady state of the reaction network. Furthermore, we look at the first and second derivatives of the ACF, each of which is affected in opposite ways by burst and confinement. This allows discriminating between these two noise sources. We analytically derive the ACF from the linear Fokker-Planck approximation of the chemical master equation in order to establish a baseline for the burst-induced modulation at low confinement. Effects of higher confinement are then studied using a partial-propensity stochastic simulation algorithm. The results presented here may help understand the mechanisms that deviate stochastic kinetics from its deterministic counterpart. In addition, they may be instrumental when using fluorescence-lifetime imaging microscopy (FLIM) or fluorescence-correlation spectroscopy (FCS) to measure confinement and burst in systems with known reaction rates, or, alternatively, to correct for the effects of confinement and burst when experimentally measuring reaction rates.
Kedziora, D J; Ankiewicz, A; Chowdury, A; Akhmediev, N
2015-10-01
We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.
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.
Physical dynamics of quasi-particles in nonlinear wave equations
NASA Astrophysics Data System (ADS)
Christov, Ivan; Christov, C. I.
2008-02-01
By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field.
On the Amplitude Equations for Weakly Nonlinear Surface Waves
NASA Astrophysics Data System (ADS)
Benzoni-Gavage, Sylvie; Coulombel, Jean-François
2012-09-01
Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185-202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3-4):1463-1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3-4):303-320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220-2240, 2011). In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.
Numerical Solution of a Nonlinear Integro-Differential Equation
NASA Astrophysics Data System (ADS)
Buša, Ján; Hnatič, Michal; Honkonen, Juha; Lučivjanský, Tomáš
2016-02-01
A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a(t) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a(t) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.
Connecting orbits for nonlinear differential equations at resonance
NASA Astrophysics Data System (ADS)
Kokocki, Piotr
We study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the Conley index of associated semiflow with respect to appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of the well-known in literature Landesman-Lazer and strong resonance conditions. Obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.
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.
Inverse Problem of Variational Calculus for Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Ali, Sk. Golam; Talukdar, B.; Das, U.
2007-06-01
We couple a nonlinear evolution equation with an associated one and derive the action principle. This allows us to write the Lagrangian density of the system in terms of the original field variables rather than Casimir potentials. We find that the corresponding Hamiltonian density provides a natural basis to recast the pair of equations in the canonical form. Amongst the case studies presented the KdV and modified KdV pairs exhibit bi-Hamiltonian structure and allow one to realize the associated fields in physical terms.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Williams, L.R.; Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Singular Solutions of Fully Nonlinear Elliptic Equations and Applications
NASA Astrophysics Data System (ADS)
Armstrong, Scott N.; Sirakov, Boyan; Smart, Charles K.
2012-08-01
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of {R^n} , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.
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
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)].
The exotic conformal Galilei algebra and nonlinear partial differential equations
NASA Astrophysics Data System (ADS)
Cherniha, Roman; Henkel, Malte
2010-09-01
The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.
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.
Chaoticons described by nonlocal nonlinear Schrödinger equation.
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-30
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).
Chaoticons described by nonlocal nonlinear Schrödinger equation
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-01
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions). PMID:28134268
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.
Stabilisation of second-order nonlinear equations with variable delay
NASA Astrophysics Data System (ADS)
Berezansky, Leonid; Braverman, Elena; Idels, Lev
2015-08-01
For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.
Lu, N.; Zhao, F.; Bergou, J.
1989-05-15
We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit.
Dispersion relation equation preserving FDTD method for nonlinear cubic Schrödinger equation
NASA Astrophysics Data System (ADS)
Sheu, Tony W. H.; Le Lin
2015-10-01
In this study we aim to solve the cubic nonlinear Schrödinger (CNLS) equation by the method of fractional steps. Over a time step from tn to tn+1, the linear part of the Schrödinger equation is solved firstly through four time integration steps. In this part of the simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term. The second-order spatial derivative term in the linear Schrödinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of the simulation, the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment. The proposed semi-discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution. Several problems have been solved to show that application of this new finite difference scheme for the calculation of one- and two-dimensional Schrödinger equations can deemed conserve Hamiltonian quantities and preserve dispersion relation equation (DRE).
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
NASA Astrophysics Data System (ADS)
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Xie, Xi-Yang; Tian, Bo Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-15
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.
Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation.
Bustamante, Miguel D; Nazarenko, Sergey
2015-11-01
We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines,H=κ(2)/8π∫(|s-s'|>ξ(*))(ds·ds')/|s-s'|,with cutoff length ξ(*)≈0.3416293/√(ρ(0)), where ρ(0) is the background condensate density far from the vortex lines and κ is the quantum of circulation.
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.
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.
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.
Coupled equations of electromagnetic waves in nonlinear metamaterial waveguides.
Azari, Mina; Hatami, Mohsen; Meygoli, Vahid; Yousefi, Elham
2016-11-01
Over the past decades, scientists have presented ways to manipulate the macroscopic properties of a material at levels unachieved before, and called them metamaterials. This research can be considered an important step forward in electromagnetics and optics. In this study, higher-order nonlinear coupled equations in a special kind of metamaterial waveguides (a planar waveguide with metamaterial core) will be derived from both electric and magnetic components of the transverse electric mode of electromagnetic pulse propagation. On the other hand, achieving the refractive index in this research is worthwhile. It is also shown that the coupled equations are not symmetric with respect to the electric and magnetic fields, unlike these kinds of equations in fiber optics and dielectric waveguides. Simulations on the propagation of a fundamental soliton pulse in a nonlinear metamaterial waveguide near the resonance frequency (a little lower than the magnetic resonant frequency) are performed to study its behavior. These pulses are recommended to practice in optical communications in controlled switching by external voltage, even in low power.
Approximate analytic solutions to coupled nonlinear Dirac equations
NASA Astrophysics Data System (ADS)
Khare, Avinash; Cooper, Fred; Saxena, Avadh
2017-03-01
We consider the coupled nonlinear Dirac equations (NLDEs) in 1 + 1 dimensions with scalar-scalar self-interactions g12 / 2 (ψ bar ψ) 2 + g22/2 (ϕ bar ϕ) 2 + g32 (ψ bar ψ) (ϕ bar ϕ) as well as vector-vector interactions of the form g1/22 (ψ bar γμ ψ) (ψ bar γμ ψ) + g22/2 (ϕ bar γμ ϕ) (ϕ bar γμ ϕ) + g32 (ψ bar γμ ψ) (ϕ bar γμ ϕ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ =e - iω1 t {R1 cos θ ,R1 sin θ }, ϕ =e - iω2 t {R2 cos η ,R2 sin η }, and assuming that θ (x) , η (x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri (x) which are valid for small values of g32 / g22 and g32 / g12. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ± ∞.
Approximate analytic solutions to coupled nonlinear Dirac equations
Khare, Avinash; Cooper, Fred; Saxena, Avadh
2017-01-30
Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g12/2(more » $$\\bar{ψ}$$ψ)2 + g22/2($$\\bar{Φ}$$Φ)2 + g23($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g12/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g22/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g23($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e–iω1tR1cosθ,R1sinθΦ=e–iω2tR2cosη,R2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g32/g22 and g32/g12. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less
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.
Continuous symmetries of certain nonlinear partial difference equations and their reductions
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Nagavigneshwari, G.
2014-09-01
In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
NASA Astrophysics Data System (ADS)
Khare, Avinash; Saxena, Avadh
2014-03-01
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, λϕ4, 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 dn2(x, m), it also admits solutions in terms of dn^2(x,m) ± sqrt{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.
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
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.
Some existence results on nonlinear fractional differential equations.
Baleanu, Dumitru; Rezapour, Shahram; Mohammadi, Hakimeh
2013-05-13
In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η
Fourth order wave equations with nonlinear strain and source terms
NASA Astrophysics Data System (ADS)
Liu, Yacheng; Xu, Runzhang
2007-07-01
In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u0)[greater-or-equal, slanted]0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.
New Analytical Solution for Nonlinear Shallow Water-Wave Equations
NASA Astrophysics Data System (ADS)
Aydin, Baran; Kânoğlu, Utku
2017-03-01
We solve the nonlinear shallow water-wave equations over a linearly sloping beach as an initial-boundary value problem under general initial conditions, i.e., an initial wave profile with and without initial velocity. The methodology presented here is extremely simple and allows a solution in terms of eigenfunction expansion, avoiding integral transform techniques, which sometimes result in singular integrals. We estimate parameters, such as the temporal variations of the shoreline position and the depth-averaged velocity, compare with existing solutions, and observe perfect agreement with substantially less computational effort.
Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
NASA Astrophysics Data System (ADS)
Enkov, Svetoslav; Georgieva, Atanaska; Nikolla, Renato
2016-12-01
In this work we investigate nonlinear Hammerstein fuzzy functional integral equation. Our aim is to provide an efficient iterative method of successive approximations by optimal quadrature formula for classes of fuzzy number-valued functions of Lipschitz type to approximate the solution. We prove the convergence of the method by Banach's fixed point theorem and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, illustrative numerical experiment demonstrate the accuracy and the convergence of the proposed method.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
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.
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
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
NASA Astrophysics Data System (ADS)
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
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.
Self-focusing and modulational analysis for nonlinear Schroedinger equations
Weinsten, M.I.
1982-01-01
For the initial-value problem (IVP) for the nonlinear Schroedinger equation, a sufficient condition for the existence of a unique global solution of the IVP is found. The condition is derived by solving a variational problem to obtain the best constant for a classical interpolation estimate of Nirenberg and Gagliardo. A systematic analysis of the singular structure is presented here for the first time. Methods apply to the general critical case. Linear modulational stability of the ground state relative to small perturbations in NLS and/or the initial data is established in the subcritical case. A sufficient condition for the existence of a unique global solution of a generalized Korteweg-de Vries equation is obtained in terms of the solitary (traveling) wave solution.
Ergodicity for Nonlinear Stochastic Equations in Variational Formulation
Barbu, Viorel Da Prato, Giuseppe
2006-03-15
This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods.The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P{sub t}. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure {nu}. Moreover, we solve the Kolmogorov equation and prove the so-called 'identite du carre du champs'. This will be used to study the Sobolev space W{sup 1,2}(H,{nu}) and to obtain information on the domain of the infinitesimal generator of P{sub t}.
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.
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Approximate symmetry and solutions of the nonlinear Klein-Gordon equation with a small parameter
NASA Astrophysics Data System (ADS)
Rahimian, Mohammad; Toomanian, Megerdich; Nadjafikhah, Mehdi
In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein-Gordon equation with a small parameter. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method.
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.
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.
Nonlinear Envelope Equation and Nonlinear Landau Damping Rate for a Driven Electron Plasma Wave
NASA Astrophysics Data System (ADS)
Bénisti, Didier; Morice, Olivier; Gremillet, Laurent; Strozzi, David J.
2011-10-01
In this article, we provide a theoretical description and calculate the nonlinear frequency shift, group velocity, and collionless damping rate, ν, of a driven electron plasma wave (EPW). All these quantities, whose physical content will be discussed, are identified as terms of an envelope equation allowing one to predict how efficiently an EPW may be externally driven. This envelope equation is derived directly from Gauss' law and from the investigation of the nonlinear electron motion, provided that the time and space rates of variation of the EPW amplitude, ?, are small compared to the plasma frequency or the inverse of the Debye length. ν arises within the EPW envelope equation as a more complicated operator than a plain damping rate and may only be viewed as such because [?]? remains nearly constant before abruptly dropping to zero. We provide a practical analytic formula for ν and show, without resorting to complex contour deformation, that in the limit ?0, ν is nothing but the Landau damping rate. We then term ν the "nonlinear Landau damping rate" of the driven plasma wave. As for the nonlinear frequency shift of the driven EPW, it is also derived theoretically and found to assume values significantly different from previously published ones, which were obtained by assuming that the wave was freely propagating. Moreover, we find no limitation in ?, ? being the plasma wavenumber and ? the Debye length, for a solution to the dispertion relation to exist, and want to stress here the importance of specifying how an EPW is generated to discuss its properties. Our theoretical predictions are in excellent agreement with results inferred from Vlasov simulations of stimulated Raman scattering (SRS), and an application of our theory to the study of SRS is presented.
NASA Astrophysics Data System (ADS)
Canoglu, Ahmet; Güldogan, Bahri; Salihoglu, Selâmi
We obtain new integrable coupled nonlinear partial differential equations by assuming the soliton connection having values in orthogonal-symplectic Lie superalgebras [B(m, n), C(n), D(m, n)]. These equations are coupled Nonlinear Schrödinger equations on various super symmetric spaces.
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
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.
2014-09-30
nonlinear Schrodinger equation. It is well known that dark solitons are exact solutions of such equation. In the present paper it has been shown that gray...in numerical computations of Nonlinear Schrodinger equation, and in the optical fibers experiments. In particular it has been shown that the
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.
An almost symmetric Strang splitting scheme for nonlinear evolution equations.
Einkemmer, Lukas; Ostermann, Alexander
2014-07-01
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.
Implementation of nonreflecting boundary conditions for the nonlinear Euler equations
NASA Astrophysics Data System (ADS)
Atassi, Oliver V.; Galán, José M.
2008-01-01
Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier-Bessel modes. This leads to an 'exact' nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier-Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.
Fokker-Planck Modelling of PISCES Linear Divertor Simulator
NASA Astrophysics Data System (ADS)
Batishchev, O. V.; Krasheninnikov, S. I.; Schmitz, L.
1996-11-01
The gas target operating regime in the PISCES [1] linear divertor simulator is characterized by a relatively high plasma density, 2.5 × 10^19 m-3, and low temperature, 8 eV, in the middle section of an ≈ 1 m long plasma column. Near the target, the plasma temperature and density as measured by Langmuir probes drop to 2 eV and 3.5 × 10^18 m-3, respectively, as a result of electron energy loss due to dissociation, ionization, and radiation. Such a sharp gradient in the plasma parameters can enhance non-local effects. To study these, we performed kinetic simulations of the relaxation of the electron energy distribution function on the experimentally measured background plasma using the adaptive finite-volumes code ALLA [2]. We discuss the effects of the observed incompletely equilibrated electron distribution function on key plasma parameter measurements and plasma - neutral particle interactions. cm [1] L.Schmitz et al., Physics of Plasmas 2 (1995) 3081. cm [2] A.A.Batishcheva et al., Physics of Plasmas 3 (1996) 1634. cm ^*Under U.S. DoE Contracts No.DE-FG02-91-ER-54109 at MIT, DE-FG02-88-ER-53263 at Lodestar, and DE-FG03-95ER54301 at UCSD.
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 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.
Nonlinear dirac and diffusion equations in 1+1 dimensions from stochastic considerations
Maharana
2000-08-01
We generalize the method of obtaining fundamental linear partial differential equations such as the diffusion and Schrodinger equation, the Dirac, and the telegrapher's equation from a simple stochastic consideration to arrive at a certain nonlinear form of these equations. A group classification through a one-parameter group of transformations for two of these equations is also carried out.
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
Currents driven by electron cyclotron waves
Karney, C.F.F.; Fisch, N.J.
1981-07-01
Certain aspects of the generation of steady-state currents by electron cyclotron waves are explored. A numerical solution of the Fokker-Planck equation is used to verify the theory of Fisch and Boozer and to extend their results into the nonlinear regime. Relativistic effects on the current generated are discussed. Applications to steady-state tokamak reactors are considered.
Estimation of Delays and Other Parameters in Nonlinear Functional Differential Equations.
1981-12-01
FSTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS by K. T. Banks and P. L. Daniel December 1981 LCDS Report #82...ESTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS H. T. Banks and P. L. Daniel ABSTRACT We discuss a spline...based approximation scheme for nonlinear nonautonomous delay differential equations . Convergence results (using dissipative type estimates on the
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.
Nonequilibrium discrete nonlinear Schrödinger equation.
Iubini, Stefano; Lepri, Stefano; Politi, Antonio
2012-07-01
We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e., transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.
Rogue waves of a (3 + 1) -dimensional nonlinear evolution equation
NASA Astrophysics Data System (ADS)
Shi, Yu-bin; Zhang, Yi
2017-03-01
General high-order rogue waves of a (3 + 1) -dimensional Nonlinear Evolution Equation ((3+1)-d NEE) are obtained by the Hirota bilinear method, which are given in terms of determinants, whose matrix elements possess plain algebraic expressions. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. Two subclass of nonfundamental rogue waves are analyzed in details. By proper means of the regulations of free parameters, the dynamics of multi-rogue waves and high-order rogue waves have been illustrated in (x,t) plane and (y,z) plane by three dimensional figures.
Self-similar solutions for a nonlinear radiation diffusion equation
Garnier, Josselin; Malinie, Guy; Saillard, Yves; Cherfils-Clerouin, Catherine
2006-09-15
This paper considers the hydrodynamic equations with nonlinear conduction when the internal energy and the opacity have power-law dependences in the density and in the temperature. This system models the situation in which a dense solid is brought into contact with a thermal bath. It supports self-similar solutions that depend on the surface temperature. The self-similar solution can exhibit a shock wave followed by an ablation front if the surface temperature does not increase too fast in time, but it can exhibit a heat front followed by an isothermal shock otherwise. These flows are carefully studied in order to clarify the role of the initial solid density in the energy absorption and the ablation process. Comparisons with numerical simulations show excellent agreement.
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.
Discrete nonlinear Schrödinger equation with defects.
Trombettoni, A; Smerzi, A; Bishop, A R
2003-01-01
We investigate the dynamical properties of the one-dimensional discrete nonlinear Schrödinger equation (DNLS) with periodic boundary conditions and with an arbitrary distribution of on-site defects. We study the propagation of a traveling plane wave with momentum k: the dynamics in Fourier space mainly involves two localized states with momenta +/-k (corresponding to a transmitted and a reflected wave). Within a two-mode ansatz in Fourier space, the dynamics of the system maps on a nonrigid pendulum Hamiltonian. The several analytically predicted (and numerically confirmed) regimes include states with a vanishing time average of the rotational states (implying complete reflections and refocusing of the incident wave), oscillations around fixed points (corresponding to quasi-stationary states), and, above a critical value of the nonlinearity, self-trapped states (with the wave traveling almost undisturbed through the impurity). We generalize this treatment to the case of several traveling waves and time-dependent defects. The validity of the two-mode ansatz and the continuum limit of the DNLS are discussed.
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.
Chaos control in the nonlinear Schrödinger equation with Kerr law nonlinearity
NASA Astrophysics Data System (ADS)
Yin, Jiu-Li; Zhao, Liu-Wei; Tian, Li-Xin
2014-02-01
The nonlinear Schrödinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.
Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrödinger Equations
NASA Astrophysics Data System (ADS)
Asad-uz-zaman, M.; Chachou Samet, H.; Khawaja, U. Al
2016-08-01
We derive the Lax pairs and integrability conditions of the nonlinear Schrödinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrödinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrödinger equation.
Solving evolutionary-type differential equations and physical problems using the operator method
NASA Astrophysics Data System (ADS)
Zhukovsky, K. V.
2017-01-01
We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker-Planck type, Black-Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.
Nonlinear Schrödinger equation with spatiotemporal perturbations.
Mertens, Franz G; Quintero, Niurka R; Bishop, A R
2010-01-01
We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation (NLSE) with the following perturbations: nonparametric spatiotemporal driving of the form f(x,t)=a exp[iK(t)x], damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations (ODEs) for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force f(x). The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of f(x). In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum P(t) and the soliton velocity V(t): This is a parameter representation of a curve P(V) which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.
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.
ON NONLINEAR EQUATIONS OF THE FORM F(x,\\, u,\\, Du,\\, \\Delta u) = 0
NASA Astrophysics Data System (ADS)
Soltanov, K. N.
1995-02-01
The Dirichlet problem for equations of the form F(x,\\, u,\\, Du,\\, \\Delta u) = 0 and also the initial boundary value problem for a parabolic equation with a nonlinearity are studied.Bibliography: 11 titles.
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
NASA Astrophysics Data System (ADS)
Parand, K.; Shahini, M.; Dehghan, Mehdi
2009-12-01
Lane-Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane-Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss-Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.
Analytic study of solutions for the Born-Infeld equation in nonlinear electrodynamics
NASA Astrophysics Data System (ADS)
Gao, Hui; Xu, Tianzhou; Fan, Tianyou; Wang, Gangwei
2017-03-01
The Born-Infeld equation is an important nonlinear partial differential equation in theoretical and mathematical physics. The Lie group method is used for simplifying the nonlinear partial differential equation, which is partly solved, in which there are some difficulties; to overcome the difficulties, we develop a power series method, and find the solutions in analytic form. In the mean time, a wave propagation (traveling wave) method is developed for solving the equation, and analytic solutions are also constructed.
Higher-order nonlinear Schrodinger equations for simulations of surface wavetrains
NASA Astrophysics Data System (ADS)
Slunyaev, Alexey
2016-04-01
Numerous recent results of numerical and laboratory simulations of waves on the water surface claim that solutions of the weakly nonlinear theory for weakly modulated waves in many cases allow a smooth generalization to the conditions of strong nonlinearity and dispersion, even when the 'envelope' is difficult to determine. The conditionally 'strongly nonlinear' high-order asymptotic equations still imply the smallness of the parameter employed in the asymptotic series. Thus at some (unknown a priori) level of nonlinearity and / or dispersion the asymptotic theory breaks down; then the higher-order corrections become useless and may even make the description worse. In this paper we use the higher-order nonlinear Schrodinger (NLS) equation, derived in [1] (the fifth-order NLS equation, or next-order beyond the classic Dysthe equation [2]), for simulations of modulated deep-water wave trains, which attain very large steepness (below or beyond the breaking limit) due to the Benjamin - Feir instability. The results are compared with fully nonlinear simulations of the potential Euler equations as well as with the weakly nonlinear theories represented by the nonlinear Schrodinger equation and the classic Dysthe equation with full linear dispersion [2]. We show that the next-order Dysthe equation can significantly improve the description of strongly nonlinear wave dynamics compared with the lower-order asymptotic models. [1] A.V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water. JETP 101, 926-941 (2005). [2] K. Trulsen, K.B. Dysthe, A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281-289 (1996).
New Traveling Wave Solutions for a Class of Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Bai, Cheng-Jie; Zhao, Hong; Xu, Heng-Ying; Zhang, Xia
The deformation mapping method is extended to solve a class of nonlinear evolution equations (NLEEs). Many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, and Jacobian elliptic function solutions, are obtained by a simple algebraic transformation relation between the solutions of the NLEEs and those of the cubic nonlinear Klein-Gordon (NKG) equation.
Soliton theory of two-dimensional lattices: the discrete nonlinear schrödinger equation.
Arévalo, Edward
2009-06-05
We theoretically investigate the motion of collective excitations in the two-dimensional nonlinear Schrödinger 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.
Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation.
Wang, L H; Porsezian, K; He, J S
2013-05-01
In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.
Bursting processes in plasmas and relevant nonlinear model equations
Basu, B.; Coppi, B.
1995-01-01
Important intrinsic plasma instabilities manifest themselves in the form of periodic bursts of fluctuations rather than as a state of stationary fluctuations, which a conventional application of quasilinear theory would lead to expect. A set of coupled nonlinear equations for the time evolution of the fluctuation amplitude and of the driving factor of the relevant instability is shown to have the features necessary to reproduce the variety of bursts that are observed experimentally. These are the periodicity, the duration, and the shape of the bursts, special consideration being given to the excitation of modes by high-energy particle populations in thermalized plasmas and to a model for the transition from a bursting state to one of stationary fluctuations. A model is introduced that is relevant to the case where the spatial dependence of the mode amplitude is important. The application of the given analysis to the bursty wave emissions observed in space is discussed. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.
Nonlinear grid error effects on numerical solution of partial differential equations
NASA Technical Reports Server (NTRS)
Dey, S. K.
1980-01-01
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
A new solution procedure for a nonlinear infinite beam equation of motion
NASA Astrophysics Data System (ADS)
Jang, T. S.
2016-10-01
Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.
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.
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
Konotop, V.V.; Pacciani, P.
2005-06-24
It is proven that periodically varying and sign definite nonlinearity in a general case does not prevent collapse in two-dimensional and three-dimensional nonlinear Schroedinger equations: at any oscillation frequency of the nonlinearity blowing up solutions exist. Contrary to the results known for a sign-alternating nonlinearity, an increase of the frequency of oscillations accelerates collapse. The effect is discussed from the viewpoint of scaling arguments. For the three-dimensional case a sufficient condition for the existence of collapse is rigorously established. The results are discussed in the context of the mean field theory of Bose-Einstein condensates with time-dependent scattering length.
NASA Astrophysics Data System (ADS)
Yan, Zhenya
2003-04-01
In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.
Analytical solutions with the improved (G’/G)-expansion method for nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Bekir, Ahmet; Akbulut, Arzu
2016-10-01
To seek the exact solutions of nonlinear partial differential equations (NPDEs), the improved (G'/G)-expansion method is proposed in the present work. With the aid of symbolic computation, this effective method is applied to construct exact solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and (3+1)- dimensional Kudryashov-Sinelshchikov equation. As a result, new types of exact solutions are obtained.
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 Schrödinger equation: generalized Darboux transformation and rogue wave solutions.
Guo, Boling; Ling, Liming; Liu, Q P
2012-02-01
In this paper, we construct a generalized Darboux transformation for the nonlinear Schrödinger equation. The associated N-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the Nth-order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.
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.
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.
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)
Zhang, B.; Billings, S. A.
2015-08-01
Although a vast number of techniques for the identification of nonlinear discrete-time systems have been introduced, the identification of continuous-time nonlinear systems is still extremely difficult. In this paper, the Nonlinear Difference Equation with Moving Average noise (NDEMA) model which is a general representation of nonlinear systems and contains, as special cases, both continuous-time and discrete-time models, is first proposed. Then based on this new representation, a systematic framework for the identification of nonlinear continuous-time models is developed. The new approach can not only detect the model structure and estimate the model parameters, but also work for noisy nonlinear systems. Both simulation and experimental examples are provided to illustrate how the new approach can be applied in practice.
Perfectly matched layer absorbing boundary condition for nonlinear two-fluid plasma equations
NASA Astrophysics Data System (ADS)
Sun, X. F.; Jiang, Z. H.; Hu, X. W.; Zhuang, G.; Jiang, J. F.; Guo, W. X.
2015-04-01
Numerical instability occurs when coupled Maxwell equations and nonlinear two-fluid plasma equations are solved using finite difference method through parallel algorithm. Thus, a perfectly matched layer (PML) boundary condition is set to avoid the instability caused by velocity and density gradients between vacuum and plasma. A splitting method is used to first decompose governing equations to time-dependent nonlinear and linear equations. Then, a proper complex variable is used for the spatial derivative terms of the time-dependent nonlinear equation. Finally, with several auxiliary function equations, the governing equations of the absorbing boundary condition are derived by rewriting the frequency domain PML in the original physical space and time coordinates. Numerical examples in one- and two-dimensional domains show that the PML boundary condition is valid and effective. PML stability depends on the absorbing coefficient and thickness of absorbing layers.
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.
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.
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.
Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave
NASA Astrophysics Data System (ADS)
Khachatryan, A. Kh.; Khachatryan, Kh. A.
2016-11-01
We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L 1[-r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.
Nonexistence of small, odd breathers for a class of nonlinear wave equations
NASA Astrophysics Data System (ADS)
Kowalczyk, Michał; Martel, Yvan; Muñoz, Claudio
2016-11-01
In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and φ ^4 and φ ^6 models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9-74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one.
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
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)
Ndzana, Fabien; Mohamadou, Alidou; Kofané, Timoléon C
2008-12-01
We study wave propagation in a nonlinear transmission line with dissipative elements. We show analytically that the telegraphers' equations of the electrical transmission line can be modeled by a pair of continuous coupled complex Ginzburg-Landau equations, coupled by purely nonlinear terms. Based on this system, we investigated both analytically and numerically the modulational instability (MI). We produce characteristics of the MI in the form of typical dependence of the instability growth rate on the wavenumbers and system parameters. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. We find that the initial modulated plane wave disintegrates into waves train. An apparently turbulent state takes place in the system during the propagation.
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.
The (G'/G)-expansion method for the nonlinear time fractional differential equations
NASA Astrophysics Data System (ADS)
Unsal, Omer; Guner, Ozkan; Bekir, Ahmet; Cevikel, Adem C.
2017-01-01
In this paper, we obtain exact solutions of two time fractional differential equations using Jumarie's modified Riemann-Liouville derivative which is encountered in mathematical physics and applied mathematics; namely (3 + 1)-dimensional time fractional KdV-ZK equation and time fractional ADR equation by using fractional complex transform and (G/'G )-expansion method. It is shown that the considered transform and method are very useful in solving nonlinear fractional differential equations.
NASA Astrophysics Data System (ADS)
Gupta, A. K.; Ray, S. Saha
2014-09-01
In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
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.
NASA Astrophysics Data System (ADS)
Chen, Yong; Yan, Zhenya
2017-01-01
The effect of derivative nonlinearity and parity-time-symmetric (PT -symmetric) potentials on the wave propagation dynamics is explored in the derivative nonlinear Schrödinger equation, where the physically interesting Scarf-II and harmonic-Hermite-Gaussian potentials are chosen. We study numerically the regions of unbroken and broken linear PT -symmetric phases and find some stable bright solitons of this model in a wide range of potential parameters even though the corresponding linear PT -symmetric phases are broken. The semielastic interactions between particular bright solitons and exotic incident waves are illustrated such that we find that particular nonlinear modes almost keep their shapes after interactions even if the exotic incident waves have evidently been changed. Moreover, we exert the adiabatic switching on PT -symmetric potential parameters such that a stable nonlinear mode with the unbroken linear PT -symmetric phase can be excited to another stable nonlinear mode belonging to the broken linear PT -symmetric phase.
Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M
2011-01-01
We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.
Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2015-02-01
We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.
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.
Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations
NASA Astrophysics Data System (ADS)
Byun, Sun-Sig; Lee, Mikyoung; Palagachev, Dian K.
2016-03-01
We prove global regularity in weighted Lebesgue spaces for the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equations. As a consequence, regularity in Morrey spaces of the Hessian is derived as well.
NASA Astrophysics Data System (ADS)
Yang, Zhijian; Liu, Zhiming
2017-03-01
The paper investigates the well-posedness and the longtime dynamics of the quasilinear wave equations with structural damping and supercritical nonlinearities: {{u}tt}- Δ u+{{≤ft(- Δ \\right)}α}{{u}t}-\
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).
Analytic solutions for time-dependent Schrödinger equations with linear of nonlinear Hamiltonians
NASA Astrophysics Data System (ADS)
Adomian, G.; Efinger, H. J.
1994-10-01
The decomposition method is applied to the time-dependent Schrödinger equation for linear or nonlinear Hamiltonian operators, without linearization, perturbation, or numerical methods, to obtain a rapidly converging analytic solution
Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential
NASA Astrophysics Data System (ADS)
Alfimov, G. L.; Avramenko, A. I.
2013-07-01
We study nonlinear states for the NLS-type equation with additional periodic potential U(x), also called the Gross-Pitaevskii equation, GPE, in theory of Bose-Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N. These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.
Solving nonlinear or stiff differential equations by Laplace homotopy analysis method(LHAM)
NASA Astrophysics Data System (ADS)
Chong, Fook Seng; Lem, Kong Hoong; Wong, Hui Lin
2015-10-01
The initial value problems of nonlinear or stiff ordinary differential equation appear in many fields of engineering science, particularly in the studies of electrical circuits, chemical reactions, wave vibration and so on. In this research, the standard homotopy analysis method hybrids with Laplace transform method to solve nonlinear and stiff differential equations. Using this modification, the problems solved by LHAM successfully yield good solutions. Some examples are examined to highlight the convenience and effectiveness of LHAM.
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.
H[alpha]-stability of modified Runge-Kutta methods for nonlinear neutral pantograph equations
NASA Astrophysics Data System (ADS)
Ma, S. F.; Yang, Z. W.; Liu, M. Z.
2007-11-01
In this paper, we investigate H[alpha]-stability of algebraically stable Runge-Kutta methods with a variable stepsize for nonlinear neutral pantograph equations. As a result, the Radau IA, Radau IIA, Lobatto IIIC method, the odd-stage Gauss-Legendre methods and the one-leg [theta]-method with are H[alpha]-stable for nonlinear neutral pantograph equations. Some experiments are given.
Exact solutions of a generalized nonlinear Schrödinger equation.
Zhang, Shaowu; Yi, Lin
2008-08-01
Exact chirped soliton solutions of a generalized nonlinear Schrödinger equation with the cubic-quintic nonlinearities as well as the self-steeping were obtained using a variable parametric method. It was found that the formation of solutions is determined by the sign of a joint parameter solely. By performing numerical simulations, the chirped solutions are stable under perturbations.
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.
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.
An Evolution Operator Solution for a Nonlinear Beam Equation
1990-12-01
uniqueness for the parabolic problem Ug + (-A) m u+ I I- u = f (14) on RN X (0, 1). Again, certain restrictions apply. The Schr ~ dinger equation , [68:pg 823...evolution equation because of the time dependence in the definition of the operator A. He identifies conditions for the existence of a unique solution. In...The arguments for the adjoint and dissipativity are not repeated. Because of the explicit time dependence , (71) is called an evolution equation . For
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks.
Smadbeck, Patrick; Kaznessis, Yiannis N
2014-08-01
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.
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)
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.
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.
Solitons in coupled nonlinear Schrödinger equations with variable coefficients
NASA Astrophysics Data System (ADS)
Han, Lijia; Huang, Yehui; Liu, Hui
2014-09-01
We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright-Dark and Bright-Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.
Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations
NASA Astrophysics Data System (ADS)
Hansen, Eskil
2007-08-01
Global error bounds are derived for full Galerkin/Runge-Kutta discretizations of nonlinear parabolic problems, including the evolution governed by the p-Laplacian with p[greater-or-equal, slanted]2. The analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and an extended B-convergence theory. The global error is bounded in L2 by [Delta]xr/2+[Delta]tq, where r is the convergence order of the Galerkin method applied to the underlying stationary problem and q is the stiff order of the algebraically stable Runge-Kutta method.
The Poincaré-Bendixson Theorem and the non-linear Cauchy-Riemann equations
NASA Astrophysics Data System (ADS)
van den Berg, J. B.; Munaò, S.; Vandervorst, R. C. A. M.
2016-11-01
Fiedler and Mallet-Paret (1989) prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
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.
Similarity solution to fractional nonlinear space-time diffusion-wave equation
NASA Astrophysics Data System (ADS)
Costa, F. Silva; Marão, J. A. P. F.; Soares, J. C. Alves; de Oliveira, E. Capelas
2015-03-01
In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.
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.
A comparison between the propagators method and the decomposition method for nonlinear equations
Azmy, Y.Y.; Protopopescu, V. ); Cacuci, D.G. . Dept. of Chemical and Nuclear Engineering)
1990-01-01
Recently, a new formalism for solving nonlinear problems has been formulated. The formalism is based on the construction of advanced and retarded propagators that generalize the customary Green's functions in linear theory. One of the main advantages of this formalism is the possibility of transforming nonlinear differential equations into nonlinear integral equations that are usually easier to handle theoretically and computationally. The aim of this paper is to compare, on an example, the performances of the propagator method with other methods used for nonlinear equations, in particular, the decomposition method. The propagator method is stable, accurate, and efficient for all initial values and time intervals considered, while the decomposition method is unstable at large time intervals, even for very conveniently chosen initial conditions. 5 refs., 4 tabs.
Forward-backward equations for nonlinear propagation in axially invariant optical systems.
Ferrando, Albert; Zacarés, Mario; Fernández de Córdoba, Pedro; Binosi, Daniele; Montero, Alvaro
2005-01-01
We present a general framework to deal with forward and backward components of the electromagnetic field in axially invariant nonlinear optical systems, which include those having any type of linear or nonlinear transverse inhomogeneities. With a minimum amount of approximations, we obtain a system of two first-order equations for forward and backward components, explicitly showing the nonlinear couplings among them. The modal approach used allows for an effective reduction of the dimensionality of the original problem from 3 + 1 (three spatial dimensions plus one time dimension) to 1 + 1 (one spatial dimension plus one frequency dimension). The new equations can be written in a spinor Dirac-like form, out of which conserved quantities can be calculated in an elegant manner. Finally, these equations inherently incorporate spatiotemporal couplings, so that they can be easily particularized to deal with purely temporal or purely spatial effects. Nonlinear forward pulse propagation and nonparaxial evolution of spatial structures are analyzed as examples.
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.
Statistical mechanics of a discrete Schrödinger equation with saturable nonlinearity.
Samuelsen, Mogens R; Khare, Avinash; Saxena, Avadh; Rasmussen, Kim Ø
2013-04-01
We study the statistical mechanics of the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. 84, 3740 (2000)] regarding the statistical mechanics of the one-dimensional DNLS equation with a cubic nonlinearity. As in this earlier study, we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is nontrivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS, in fact, becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.
Proposal for detection of QED vacuum nonlinearities in Maxwell's equations by the use of waveguides.
Brodin, G; Marklund, M; Stenflo, L
2001-10-22
We present a novel method for detecting nonlinearities, due to quantum electrodynamics through photon-photon scattering, in Maxwell's equation. The photon-photon scattering gives rise to self-interaction terms which are similar to the nonlinearities due to the polarization in nonlinear optics. These self-interaction terms vanish in the limit of parallel propagating waves, but if, instead of parallel propagating waves, the modes generated in waveguides are used, there will be a nonzero total effect. Based on this idea, we calculate the nonlinear excitation of new modes and estimate the strength of this effect. Furthermore, we suggest a principal experimental setup.
Parametric autoresonant excitation of the nonlinear Schrödinger equation.
Friedland, L; Shagalov, A G
2016-10-01
Parametric excitation of autoresonant solutions of the nonlinear Schrodinger (NLS) equation by a chirped frequency traveling wave is discussed. Fully nonlinear theory of the process is developed based on Whitham's averaged variational principle and its predictions verified in numerical simulations. The weakly nonlinear limit of the theory is used to find the threshold on the amplitude of the driving wave for entering the autoresonant regime. It is shown that above the threshold, a flat (spatially independent) NLS solution can be fully converted into a traveling wave. A simplified, few spatial harmonics expansion approach is also developed for studying this nonlinear mode conversion process, allowing interpretation as autoresonant interaction within triads of spatial harmonics.
NASA Astrophysics Data System (ADS)
Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua
2016-12-01
Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205
Grima, Ramon
2011-11-01
The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.
Yan, Zhenya; Konotop, V V
2009-09-01
It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrödinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3) space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
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.
Nonlinear Instability of the Incoherent State for the Kuramoto-Sakaguchi-Fokker-Plank Equation
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua
2015-07-01
We study the nonlinear instability of the incoherent solution to the Kuramoto-Sakaguchi-Fokker-Plank (KSFP) equation in a large coupling strength regime. For our instability analysis, we construct an approximate, exponentially growing perturbation mode using an elementary energy method. This method does not require spectral information from the linearized KSFP equation or an explicit growing solution for the corresponding linear equation. When the distribution function of oscillator's natural frequencies is either a Dirac measure or a bounded function with a compact support (in a small interval around the origin), the incoherent solution is nonlinearly unstable depending on the relative sizes of the coupling strength and diffusion coefficient.
Mártin, Daniel A; Hoyuelos, Miguel
2009-11-01
We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.
NASA Astrophysics Data System (ADS)
Mártin, Daniel A.; Hoyuelos, Miguel
2009-11-01
We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.
Continuous Dependence on Modeling in the Cauchy Problem for Nonlinear Elliptic Equations.
1987-04-01
parameter 4. AMON INTRODUCTION A problem in ordinary or partial differential equations is said to properly posed if it has a unique solution in the...problem for second-order nonlinear partial differential equations , Doctoral thesis, Cornell University, Ithaca, N.Y., 1986. [6] J. Conlan and G. N. Trytten...IModeling in the Cauchy Problem for Nonlinear Elliptic Equations by Allan Bennett DT1C A z1t17n m (It C ltd n Inttt " CENTER.FOR.NAVAL.ANALYSFS 4401
Bright and dark soliton solutions for some nonlinear fractional differential equations
NASA Astrophysics Data System (ADS)
Ozkan, Guner; Ahmet, Bekir
2016-03-01
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona-Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann-Liouville sense.
NASA Astrophysics Data System (ADS)
Cui-Cui, Liao; Jin-Chao, Cui; Jiu-Zhen, Liang; Xiao-Hua, Ding
2016-01-01
In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space. Project supported by the National Natural Science Foundation of China (Grant No. 11401259) and the Fundamental Research Funds for the Central Universities, China (Grant No. JUSRR11407).
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.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
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.
Soliton solution and other solutions to a nonlinear fractional differential equation
NASA Astrophysics Data System (ADS)
Guner, Ozkan; Unsal, Omer; Bekir, Ahmet; Kadem, Abdelouahab
2017-01-01
In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the ansatz method and the functional variable method are used to construct exact solutions for (3+1)-dimensional time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation. This fractional equation is turned into another nonlinear ordinary differential equation by fractional complex transform then these methods are applied to solve it. As a result, some new exact solutions obtained.
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
Nonlinear Localized Dissipative Structures for Long-Time Solution of Wave Equation
2009-07-01
Fatemi, E., Engquist, B., and Osher, S., " Numerical Solution of the High Frequency Asymptotic Expansion for the Scalar Wave Equation ", Journal of...FINAL REPORT Grant Title: Nonlinear Localized Dissipative Structures for Long-Time Solution of Wave Equation By Dr. John Steinhoff Grant number... numerical method, "Wave Confinement" (WC), is developed to efficiently solve the linear wave equation . This is similar to the originally developed
Parallel Methods for Solving Nonlinear Block Bordered Systems of Equations
1989-12-31
pendix A. It is the 741 op-amp circuit (see e.g. Sedra and Smith [1982]), which was introduced in 1966 and is currently produced by almost every analog...Computing, edited by R. Wilhelmson, University of Illinois Press. A. Sedra , K. Smith [1982], Microelectronic Circuits, CBS College Publishing. J. Smith ...741 op-amp circuits (see e.g. Smith [1971], Valkenburg [1982]). This circuit leads to a 2-level block-bordered nonlinear system, as follows. The
Existence of solutions to nonlinear Hammerstein integral equations and applications
NASA Astrophysics Data System (ADS)
Li, Fuyi; Li, Yuhua; Liang, Zhanping
2006-11-01
In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L2(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2013-11-01
Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.
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.
The quadratically damped oscillator: A case study of a non-linear equation of motion
NASA Astrophysics Data System (ADS)
Smith, B. R.
2012-09-01
The equation of motion for a quadratically damped oscillator, where the damping is proportional to the square of the velocity, is a non-linear second-order differential equation. Non-linear equations of motion such as this are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions. Like all second-order ordinary differential equations, it has a corresponding first-order partial differential equation, whose independent solutions constitute the constants of the motion. These constants readily provide an approximate solution correct to first order in the damping constant. They also reveal that the quadratically damped oscillator is never critically damped or overdamped, and that to first order in the damping constant the oscillation frequency is identical to the natural frequency. The technique described has close ties to standard tools such as integral curves in phase space and phase portraits.
Approximation and Numerical Analysis of Nonlinear Equations of Evolution.
1980-01-31
les Espaces d’ Interpolation; Dualitg", Math. Scand., 9, 1961, pp. 147-177. 9. __ "Equations Diff~rentielles Op ~ rationnelles dan les Espaces de Hilbert...relaxation," Revue Francaise d’automatique, informatique, recherche operationnelle, R3, 1973, p. 5-32. Ill DOUGLAS, J. and GALLIE, T.MI. "On the Numerical
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.
Existence of Forced Oscillations for Some Nonlinear Differential Equations.
1982-11-01
groups of level sets of the functional associated with the system are ", -t4 . I not trivial. Some more general results concerning systems of the type, f... general non autonomous systems of the type (1.3) 9 + v;(t,x) - 0 There is a vast literature devoted to the subject of nonlinear oscillations in systems...g(t,x) - 0 (x(t) .3) quite general results on the existence of periodic solutions have been obtained by Hartman 114] and Jacobovitz (151 (by using
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).
Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria
Frieman, E.A.; Chen, L.
1981-10-01
A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magentic field geometries. The specific case of axisymmetric tokamaks is then considered, and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating sceme, it is found that nonlinear ion Landau damping of kinetic shear-Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.
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.
Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2014-09-01
We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.
Initial value problem solution of nonlinear shallow water-wave equations.
Kânoğlu, 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.
Freezing of nonlinear Bloch oscillations in the generalized discrete nonlinear Schrödinger equation.
Cao, F J
2004-09-01
The dynamics in a nonlinear Schrödinger chain in a homogeneous electric field is studied. We show that discrete translational invariant integrability-breaking terms can freeze the Bloch nonlinear oscillations and introduce new faster frequencies in their dynamics. These phenomena are studied by direct numerical integration and through an adiabatic approximation. The adiabatic approximation allows a description in terms of an effective potential that greatly clarifies the phenomena.
Balescu, R.; Wang, H. ); Misguich, J.H. )
1994-12-01
The running diffusion coefficient [ital D]([ital t]) is evaluated for a system of charged particles undergoing the effect of a fluctuating magnetic field and of their mutual collisions. The latter coefficient can be expressed either in terms of the mean square displacement (MSD) of a test particle, or in terms of a correlation between a fluctuating distribution function and the magnetic field fluctuation. In the first case a stochastic differential equation of Langevin type for the position of a test particle must be solved; the second problem requires the determination of the distribution function from a kinetic equation. Using suitable simplifications, both problems are amenable to exact analytic solution. The conclusion is that the equivalence of the two approaches is by no means automatically guaranteed. A new type of object, the hybrid kinetic equation'' is constructed: it automatically ensures the equivalence with the Langevin results. The same conclusion holds for the generalized Fokker--Planck equation. The (Bhatnagar--Gross--Krook) (BGK) model for the collisions yields a completely wrong result. A linear approximation to the hybrid kinetic equation yields an inexact behavior, but represents an acceptable approximation in the strongly collisional limit.
Long time behavior of some nonlinear dispersive equations
NASA Astrophysics Data System (ADS)
Deng, Yu
This thesis is divided into two parts. The first part consists of Chapters 2 and 3, in which we study the random data theory for the Benjamin-Ono equation on the periodic domain. In Chapter 2 we shall prove the invariance of the Gibbs measure associated to the Hamiltonian E1 of the equation, which was constructed in [49]. Despite the fact that the support of the Gibbs measure contains very rough functions that are not even in L2, we have successfully established the global dynamics by combining probabilistic arguments, Xs,b type estimates and the hidden structure of the equation. In Chapter 3, which is joint work with N. Tzvetkov and N. Visciglia, we extend this invariance result to the weighted Gaussian measures associated with the higher order conservation laws E2 and E3, thus completing the collection of invariant measures (except for the white noise), given the result of [51]. The second part concerns the global behavior of solutions to quasilinear dispersive systems in Rd with suitably small data. In Chapter 4 we shall prove global existence and scattering for small data solutions to systems of quasilinear Klein-Gordon equations with arbitrary speed and mass in 3 D, which extends the results in [20] and [32]. Moreover, the methods introduced here are quite general, and can be applied in a number of different situations. In Chapter 5, we briefly discuss how these methods, together with other techniques, are used in recent joint work with A. Ionescu and B. Pausader to study the 2D Euler-Maxwell system.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
NASA Astrophysics Data System (ADS)
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2017-02-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkina, Oxana; Rouvinskaya, Ekaterina; Talipova, Tatiana; Kurkin, Andrey; Pelinovsky, Efim
2016-10-01
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg-de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Transition behavior of the discrete nonlinear Schrödinger equation.
Rumpf, Benno
2008-03-01
Many nonlinear lattice systems exhibit high-amplitude localized structures, or discrete breathers. Such structures emerge in the discrete nonlinear Schrödinger equation when the energy is above a critical threshold. This paper studies the statistical mechanics at the transition and constructs the probability distribution in the regime where breathers emerge. The entropy as a function of the energy is nonanalytic at the transition. The entropy is independent of the energy in the regime of breathers above the transition.
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.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkin, Andrey; Talipova, Tatiana; Kurkina, Oxana; Rouvinskaya, Ekaterina; Pelinovsky, Efim
2016-04-01
Nonlinear disintegration of sine wave is studied in the framework of the Gardner equation (extended version of the Korteweg - de Vries equation with both quadratic and cubic nonlinear terms). Undular bores appear here as an intermediate stage of wave evolution. Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative solitary-like pulses. It is shown that nonlinear interaction of waves happens according to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k4/3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Conditional probability calculations for the nonlinear Schrödinger equation with additive noise.
Terekhov, I S; Vergeles, S S; Turitsyn, S K
2014-12-05
The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.
Optical rogue waves for the inhomogeneous generalized nonlinear Schrödinger equation.
Loomba, Shally; Kaur, Harleen
2013-12-01
We present optical rogue wave solutions for a generalized nonlinear Schrodinger equation by using similarity transformation. We have predicted the propagation of rogue waves through a nonlinear optical fiber for three cases: (i) dispersion increasing (decreasing) fiber, (ii) periodic dispersion parameter, and (iii) hyperbolic dispersion parameter. We found that the rogue waves and their interactions can be tuned by properly choosing the parameters. We expect that our results can be used to realize improved signal transmission through optical rogue waves.
The Painlevé test for nonlinear system of differential equations with complex chaotic behavior
NASA Astrophysics Data System (ADS)
Tsegel’nik, V.
2017-01-01
The Painlevé-analysis was performed for solutions of nonlinear third-order autonomous system of differential equations with quadratic nonlinearities on their right-hand sides. At certain values of two constant parameters incorporated into the system, the latter exhibits complex chaotic behavior. When the parameters attain the values corresponding to complex chaotic behavior, the system was found not to possess the Painlevé property.
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.
1987-08-01
solution of the Korteweg-de Vries equation ( KdV ), working our way up to the derivation of the multi-soliton solution of the sine-Gordon equation (sG...SOLITARY WAVE SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS j DiS~~Uj~l. _’UDistribution/Willy Hereman AvaiiLi -itY Codes Technical Summary Report...Key Words: soliton theory, solitary waves, coupled KdV , evolution equations , direct methods, Harry Dym, sine-Gordon Mathematics Department, University
Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients.
Zhong, Wei-Ping; Belić, Milivoj R; Huang, Tingwen
2013-06-01
A similarity transformation is utilized to reduce the generalized nonlinear Schrödinger (NLS) equation with variable coefficients to the standard NLS equation with constant coefficients, whose rogue wave solutions are then transformed back into the solutions of the original equation. In this way, Ma breathers, the first- and second-order rogue wave solutions of the generalized equation, are constructed. Properties of a few specific solutions and controllability of their characteristics are discussed. The results obtained may raise the possibility of performing relevant experiments and achieving potential applications.