Intermittency and solitons in the driven dissipative nonlinear Schroedinger equation
NASA Technical Reports Server (NTRS)
Moon, H. T.; Goldman, M. V.
1984-01-01
The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
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.
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
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.
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.
Pulov, Vladimir I.
2011-04-07
This paper is devoted to finding an optimal system of one-dimensional subalgebras of an eight-dimensional Lie algebra of point symmetry transformations, admitted by a system of two coupled nonlinear Schroedinger equations.
Zhou Ruguang
2007-01-15
A procedure of nonlinearization of spectral problem that allows to impose reality conditions or restriction conditions on potentials is presented. As applications, integrable decompositions of the nonlinear Schroedinger equation and the real-valued modified Korteweg-de Vries equation are obtained.
Cylindrical nonlinear Schroedinger equation versus cylindrical Korteweg-de Vries equation
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2008-10-15
A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) equations and the one of cylindrical Korteweg-de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS equation can be associated with non-stationary soliton-like solutions of cKdV equation.
Yang Xiao; Du Dianlou
2010-08-15
The Poisson structure on C{sup N}xR{sup N} is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
Brugarino, Tommaso; Sciacca, Michele
2010-09-15
In this paper, we investigate the integrability of an inhomogeneous nonlinear Schroedinger equation, which has several applications in many branches of physics, as in Bose-Einstein condensates and fiber optics. The main issue deals with Painleve property (PP) and Liouville integrability for a nonlinear Schroedinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose-Einstein condensates are proposed (including Bose-Einstein condensates in three-dimensional in cylindrical symmetry).
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.
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.
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.
Vortex Solutions of the Defocusing Discrete Nonlinear Schroedinger Equation
Cuevas, J.; Kevrekidis, P. G.; Law, K. J. H.
2009-09-09
We consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing DNLS equation, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization-destabilization windows for any finite lattice.
Travelling-wave and separated variable solutions of a nonlinear Schroedinger equation
NASA Astrophysics Data System (ADS)
Bountis, Tassos; Nobre, Fernando D.
2016-08-01
Some interesting nonlinear generalizations have been proposed recently for the linear Schroedinger, Klein-Gordon, and Dirac equations of quantum and relativistic physics. These novel equations involve a real parameter q and reduce to the corresponding standard linear equations in the limit q → 1. Their main virtue is that they possess plane-wave solutions expressed in terms of a q-exponential function that can vanish at infinity, while preserving the Einstein energy-momentum relation for all q. In this paper, we first present new travelling wave and separated variable solutions for the main field variable Ψ ( x → , t ) , of the nonlinear Schroedinger equation (NLSE), within the q-exponential framework, and examine their behavior at infinity for different values of q. We also solve the associated equation for the second field variable Φ ( x → , t ) , derived recently within the context of a classical field theory, which corresponds to Ψ ∗ ( x → , t ) for the linear Schroedinger equation in the limit q → 1. For x ∈ ℜ, we show that certain perturbations of these q-exponential solutions Ψ(x, t) and Φ(x, t) are unbounded and hence would lead to divergent probability densities over the full domain -∞ < x < ∞. However, we also identify ranges of q values for which these solutions vanish at infinity, and may therefore be physically important.
Collision Dynamics of Polarized Solitons in Linearly Coupled Nonlinear Schroedinger Equations
Todorov, Michail D.; Christov, Christo I.
2011-04-07
The system of linearly coupled nonlinear Schroedinger equations is solved by a conservative difference scheme in complex arithmetic. The initial condition represents a superposition of two one-soliton solutions of linear polarizations. The head-on and takeover interaction of the solitons and their quasi-particle (QP) behavior is examined in conditions of rotational polarization and gain. We found that the polarization angle of a quasi-particle can change independently of the collision.
Computing wave functions of nonlinear Schroedinger equations: A time-independent approach
Chang, S.-L.; Chien, C.-S. Jeng, B.-W.
2007-09-10
We present a novel algorithm for computing the ground-state and excited-state solutions of M-coupled nonlinear Schroedinger equations (MCNLS). First we transform the MCNLS to the stationary state ones by using separation of variables. The energy level of a quantum particle governed by the Schroedinger eigenvalue problem (SEP) is used as an initial guess to computing their counterpart of a nonlinear Schroedinger equation (NLS). We discretize the system via centered difference approximations. A predictor-corrector continuation method is exploited as an iterative method to trace solution curves and surfaces of the MCNLS, where the chemical potentials are treated as continuation parameters. The wave functions can be easily obtained whenever the solution manifolds are numerically traced. The proposed algorithm has the advantage that it is unnecessary to discretize or integrate the partial derivatives of wave functions. Moreover, the wave functions can be computed for any time scale. Numerical results on the ground-state and excited-state solutions are reported, where the physical properties of the system such as isotropic and nonisotropic trapping potentials, mass conservation constraints, and strong and weak repulsive interactions are considered in our numerical experiments.
Vanvincq, O.; Travers, J. C.; Kudlinski, A.
2011-12-15
We reexamine the derivation of the generalized nonlinear Schroedinger equation in the case of nonaxially uniform optical fibers, taking into account the longitudinal and spectral evolutions of all pertinent linear parameters. Our theory leads to an improved form of this equation that highlights an additional term, which ensures the conservation of the total photon number in nonuniform optical fibers in the absence of attenuation. Numerical simulations confirm the validity of this theory in the context of a Raman-induced soliton self-frequency shift, emission of Cherenkov radiation, and a soliton blue shift.
Universal Critical Power for Nonlinear Schroedinger Equations with a Symmetric Double Well Potential
Sacchetti, Andrea
2009-11-06
Here we consider stationary states for nonlinear Schroedinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.
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.
Zhao Dun; Zhang Yujuan; Lou Weiwei; Luo Honggang
2011-04-15
By constructing nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schroedinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose-Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLS systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.
Sun Zhiyuan; Yu Xin; Liu Ying; Gao Yitian
2012-12-15
We investigate the dynamics of the bound vector solitons (BVSs) for the coupled nonlinear Schroedinger equations with the nonhomogenously stochastic perturbations added on their dispersion terms. Soliton switching (besides soliton breakup) can be observed between the two components of the BVSs. Rate of the maximum switched energy (absolute values) within the fixed propagation distance (about 10 periods of the BVSs) enhances in the sense of statistics when the amplitudes of stochastic perturbations increase. Additionally, it is revealed that the BVSs with enhanced coherence are more robust against the perturbations with nonhomogenous stochasticity. Diagram describing the approximate borders of the splitting and non-splitting areas is also given. Our results might be helpful in dynamics of the BVSs with stochastic noises in nonlinear optical fibers or with stochastic quantum fluctuations in Bose-Einstein condensates.
Fedele, Renato; Jovanovic, Dusan
2004-12-01
Charged-particle beams are employed for a number of scientific and technological applications. The conventional description of their collective behavior is usually given in terms of the Vlasov equation. In the last 15 years some alternative descriptions have been developed in terms of a nonlinear Schroedinger equation governing the collective dynamics of the beam while interacting with the surrounding medium. This approach gives new insights, providing an alternative 'key of reading' of the charged-particle beam dynamics, and have been applied to a number of physical problems concerning conventional particle accelerating machines as well as plasma-based accelerator schemes. Remarkably, it is based on a mathematical formalism fully similar to those used for the propagation of e.m. radiation beams in nonlinear media a well as the nonlinear dynamics of the Bose-Einsten condensates.In this paper, a presentation of some significant nonlinear collective effects of a charged-particle beam in particle accelerators, that have been recently investigated in the framework of the above Schroedinger-like descriptions, is given.
Emami, F.; Hatami, M.; Keshavarz, A. R.; Jafari, A. H.
2009-08-13
Using a combination of Runge-Kutta and Jacobi iterative method, we could solve the nonlinear Schroedinger equation describing the pulse propagation in FBGs. By decomposing the electric field to forward and backward components in fiber Bragg grating and utilizing the Fourier series analysis technique, the boundary value problem of a set of coupled equations governing the pulse propagation in FBG changes to an initial condition coupled equations which can be solved by simple Runge-Kutta method.
Al Khawaja, U.
2010-05-15
We derive the integrability conditions of nonautonomous nonlinear Schroedinger equations using the Lax pair and similarity transformation methods. We present a comparative analysis of these integrability conditions with those of the Painleve method. We show that while the Painleve integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space independent and the external potential to be only a quadratic function of position, the Lax Pair and the similarity transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painleve method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schroedinger equations for two- and three-spacial dimensions.
Stationary localized modes of the quintic nonlinear Schroedinger equation with a periodic potential
Alfimov, G. L.; Konotop, V. V.; Pacciani, P.
2007-02-15
We consider localized modes (bright solitons) of the one-dimensional quintic nonlinear Schroedinger equation with a periodic potential, describing several mean-field models of low-dimensional condensed gases. In the case of attractive nonlinearity we deduce sufficient conditions for collapse. We show that there exist spatially localized modes with arbitrarily large numbers of particles. We study such solutions in the semi-infinite gap (attractive case) and in the first gap (attractive and repulsive cases), and show that a nonzero minimum value of the number of particles is necessary for a localized mode to be created. In the limit of large negative frequencies (attractive case) we observe quantization of the number of particles of the stationary modes. Such solutions can be interpreted as coupled Townes solitons and appear to be stable. The modes in the first gap have numbers of particles infinitely growing with frequencies approaching one of the gap edges, which is explained by the power decay of the modes. Stability of the localized modes is discussed.
Koller, Andrew; Olshanii, Maxim
2011-12-15
We present a case demonstrating the connection between supersymmetric quantum mechanics (SUSYQM), reflectionless scattering, and soliton solutions of integrable partial differential equations. We show that the members of a class of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected via supersymmetric chains to a potential-free Hamiltonian, explaining their reflectionless nature. While the reflectionless property in question has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. Our results indicate that the multisoliton solutions of the sine-Gordon and nonlinear Schroedinger equations can be systematically generated via the supersymmetric chains connecting Akulin's Hamiltonians. Our findings also explain a well-known but little-understood effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t)=(n({h_bar}/2{pi})/{tau})/cosh(t/{tau}), with n being an integer and {tau} being the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning.
Nonlinearity, PT Symmetry, Twist, and Disorder in Discrete Nonlinear Schroedinger Equation
NASA Astrophysics Data System (ADS)
Castro-Castro, Claudia K.
The study of optical fiber arrays has drawn a great deal of attention in the field of nonlinear physics during the past few years since they provide spatially inhomogeneous structures for guiding light signals. We analyze the management and control of light transfer in nonlinear multi-core fibers. We utilize mathematical modeling and numerical simulations to specifically show how nonlinearity, coupling, geometric twist, and balanced gain/loss relate to existence and stability of nonlinear optical modes modeled by the Discrete Nonlinear Schrodinger Equation (DNLS). In addition, we explore the effects of the inherent variability on the fiber core diameter (disorder) by building a statistical understanding of the formation of low or high-amplitude (localized/breather) states, and the long-time asymptotics of DNLS with low-amplitude initial conditions.
Compactons in Nonlinear Schroedinger Lattices with Strong Nonlinearity Management
Abdullaev, F. Kh.; Kevrekidis, P. G.; Salerno, M.
2010-09-10
The existence of compactons in the discrete nonlinear Schroedinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. In the averaged discrete nonlinear Schroedinger equation, the resulting effective interwell tunneling depends on the modulation parameters and on the field amplitude. This introduces nonlinear dispersion in the system and can lead to a prototypical realization of single- or multisite stable discrete compactons in nonlinear optical waveguide and Bose-Einstein condensate arrays. These structures can dynamically arise out of Gaussian or compactly supported initial data.
Du, Dianlou; Geng, Xue
2013-05-15
In this paper, the relationship between the classical Dicke-Jaynes-Cummings-Gaudin (DJCG) model and the nonlinear Schroedinger (NLS) equation is studied. It is shown that the classical DJCG model is equivalent to a stationary NLS equation. Moreover, the standard NLS equation can be solved by the classical DJCG model and a suitably chosen higher order flow. Further, it is also shown that classical DJCG model can be transformed into the classical Gaudin spin model in an external magnetic field through a deformation of Lax matrix. Finally, the separated variables are constructed on the common level sets of Casimir functions and the generalized action-angle coordinates are introduced via the Hamilton-Jacobi equation.
Brazhnyi, V.A.; Konotop, V.V.
2005-08-01
The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schroedinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.
Wang, C.; Theocharis, G.; Kevrekidis, P. G.; Whitaker, N.; Law, K. J. H.; Frantzeskakis, D. J.; Malomed, B. A.
2009-10-15
We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schroedinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.
He Junrong; Li Huamei
2011-06-15
A large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schroedinger equation with time- and space-dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique. We use the cubic nonlinearity as an independent parameter function, where a simple procedure is established to obtain different classes of potentials and solutions. The solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion, cubic and quintic nonlinearities, and gain (or loss). We investigate the space-quadratic potential, optical lattice potential, flying bird potential, and potential barrier (well). Some interesting periodic solitary wave solutions corresponding to these potentials are then studied. Also, properties of a few solutions and physical applications of interest to the field are discussed. Finally, the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically; the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise.
Lue Xing Zhu Hongwu; Yao Zhenzhi; Meng Xianghua; Zhang Cheng; Zhang Chunyi; Tian Bo
2008-08-15
In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schroedinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Baecklund transformation transforms between (N - 1)- and N-soliton solutions.
Keanini, R.G.
2011-04-15
Research Highlights: > Systematic approach for physically probing nonlinear and random evolution problems. > Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. > Organization of near-molecular scale vorticity mediated by hydrodynamic modes. > Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion
A Parallel Time-Propagation Solver for the Non-Linear Schroedinger Equation.
NASA Astrophysics Data System (ADS)
Nygaard, Nicolai; Simula, Tapio; Schneider, Barry I.
2007-06-01
We describe a powerful numerical method for solving the time-dependent non-linear Schr"odinger equation. Our method is based on the finite-element discrete variable representation. The time-propagation is facilitated either by the Lanczos-Arnoldi method or by split-operator formulas of different orders. The ground-state solution is found by propagation in imaginary time using an adaptive time stepping algorithm, and the absolute convergence of the propagation is faithfully characterized by a positive-definite error norm. Parallelization of this method is transparent, and we have utilized an MPI implementation demonstrating linear scaling of wall-clock computation time with the number of processors used.
Misra, Amar P.; Roy Chowdhury, K.; Roy Chowdhury, A.
2007-01-15
Using the standard reductive perturbation technique, a nonlinear Schroedinger equation (NLSE) with complex coefficients is derived in a dusty plasma consisting of positive ions, nonthermal electrons, and charged dust grains. The effect of ion kinematic viscosity is taken into consideration, which makes the coefficients of NLSE complex. By means of a matching approach, the appearance mechanism of static pulses through a saddle-node bifurcation in the complex nonlinear Schroedinger equation is studied analytically. The analytical results are in good agreement with the direct numerical simulation. The modulational instability analysis is carried out for the dust ion-acoustic envelope solitary waves. The important role of the real part of the complex group velocity in the propagation of the one-dimensional wave packets in homogeneous active medium is predicted.
Hernandez Tenorio, C; Villagran Vargas, E; Serkin, Vladimir N; Aguero Granados, M; Belyaeva, T L; Pena Moreno, R; Morales Lara, L
2005-10-31
The dynamics of dark solitons is studied within the framework of the mathematical model of nonlinear Schroedinger equation (NSE) with an external harmonic potential. A comparative analysis of the solutions of nonstationary problems is performed for a linear harmonic oscillator and the NSE model with a harmonic potential for different signs of the self-action potential. It is shown that the main specific feature of the dynamics of dark NSE solitons in a parabolic trap is the formation of solitons with dynamically changing form factors producing the periodic variation in the modulation depth (the degree of 'blackness') of dark solitons. The oscillation period of the dark soliton does not coincide with the oscillation period of a linear quantum-mechanical oscillator, which is caused by the periodic transformation of the black soliton to the grey one and vice versa. The conditions of applicability of the method of inverse scattering problem are presented, the generalised Lax pair is found, and exact soliton solutions are given for the mathematical NSE model with an external harmonic potential. (solitons)
Solitons in strongly driven discrete nonlinear Schroedinger-type models
Garnier, Josselin; Abdullaev, Fatkhulla Kh.; Salerno, Mario
2007-01-15
Discrete solitons in the Ablowitz-Ladik (AL) and discrete nonlinear Schroedinger (DNLS) equations with damping and strong rapid drive are investigated. The averaged equations have the forms of the parametric AL and DNLS equations. An additional type of parametric bright discrete soliton and cnoidal waves are found and the stability properties are analyzed. The analytical predictions of the perturbed inverse scattering transform are confirmed by the numerical simulations of the AL and DNLS equations with rapidly varying drive and damping.
Alka,; Goyal, Amit; Gupta, Rama; Kumar, C. N.; Raju, Thokala Soloman
2011-12-15
We demonstrate that the competing cubic-quintic nonlinearity induces propagating solitonlike dark(bright) solitons and double-kink solitons in the nonlinear Schroedinger equation with self-steepening and self-frequency shift. Parameter domains are delineated in which these optical solitons exist. Also, fractional-transform solitons are explored for this model. It is shown that the nonlinear chirp associated with each of these optical pulses is directly proportional to the intensity of the wave and saturates at some finite value as the retarded time approaches its asymptotic value. We further show that the amplitude of the chirping can be controlled by varying the self-steepening term and self-frequency shift.
Ando, Taro; Fujimoto, Masatoshi
2005-08-01
We develop an accurate and efficient method for calculating evolution due to the extended nonlinear Schroedinger equation, which describes the propagation behavior of a femtosecond light pulse in a nonlinear medium. Applying Suzuki's exponential operator expansion to the evolution operator based on the finite-differential formulation, we realize the accurate and fast calculation that can be performed without large-scale computing systems even for (3+1)-dimensional problems. To study the correspondence between experiments and calculations, we calculate the propagation behavior of a femtosecond light pulse that is weakly focused in nitrogen gas of various pressures and compare the calculation results to the experimental ones. The calculation results reproduce the relative behavior of the spatial light pattern observed during the propagation. Additionally, the multiple-cone formation and interaction between two collimated pulses in nitrogen gas are also demonstrated as applications of the developed method.
Dynamical stabilization of solitons in cubic-quintic nonlinear Schroedinger model
Abdullaev, Fatkhulla Kh.; Garnier, Josselin
2005-09-01
We consider the existence of a dynamically stable soliton in the one-dimensional cubic-quintic nonlinear Schroedinger model with strong cubic nonlinearity management for periodic and random modulations. We show that the predictions of the averaged cubic-quintic nonlinear Schroedinger (NLS) equation and modified variational approach for the arrest of collapse coincide. The analytical results are confirmed by numerical simulations of a one-dimensional cubic-quintic NLS equation with a rapidly and strongly varying cubic nonlinearity coefficient.
A Chebychev propagator for inhomogeneous Schroedinger equations
Ndong, Mamadou; Koch, Christiane P.; Tal-Ezer, Hillel; Kosloff, Ronnie
2009-03-28
A propagation scheme for time-dependent inhomogeneous Schroedinger equations is presented. Such equations occur in time dependent optimal control theory and in reactive scattering. A formal solution based on a polynomial expansion of the inhomogeneous term is derived. It is subjected to an approximation in terms of Chebychev polynomials. Different variants for the inhomogeneous propagator are demonstrated and applied to two examples from optimal control theory. Convergence behavior and numerical efficiency are analyzed.
Lue Xing Tian Bo; Xu Tao; Cai Kejie; Liu Wenjun
2008-10-15
Under investigation in this paper is a nonlinear Schroedinger equation with an arbitrary linear time-dependent potential, which governs the soliton dynamics in quasi-one-dimensional Bose-Einstein condensates (quasi-1DBECs). With Painleve analysis method performed to this model, its integrability is firstly examined. Then, the distinct treatments based on the truncated Painleve expansion, respectively, give the bilinear form and the Painleve-Baecklund transformation with a family of new exact solutions. Furthermore, via the computerized symbolic computation, a direct method is employed to easily and directly derive the exact analytical dark- and bright-solitonic solutions. At last, of physical and experimental interests, these solutions are graphically discussed so as to better understand the soliton dynamics in quasi-1DBECs.
Nonpolynomial Schroedinger equation for resonantly absorbing gratings
Shabtay, Lior; Malomed, Boris A.
2011-02-15
We derive a nonlinear Schroedinger equation with a radical term, {approx}{radical}(1-|V|{sup 2}), as an asymptotic model of the resonantly absorbing Bragg reflector (RABR), i.e., a periodic set of thin layers of two-level atoms, resonantly interacting with the electromagnetic field and inducing the Bragg reflection. A family of bright solitons is found, which splits into stable and unstable parts, exactly obeying the Vakhitov-Kolokolov criterion. The soliton with the largest amplitude, (|V|){sub max}=1, is a ''quasipeakon,'' i.e., a solution with a discontinuity of the third derivative at the center. Families of exact cnoidal waves, built as periodic chains of quasipeakons, are found too. The ultimate solution belonging to the family of dark solitons, with the background level V=1, is a dark compacton. Those bright solitons that are unstable destroy themselves (if perturbed) attaining the critical amplitude, |V|=1. The dynamics of the wave field around this critical point is studied analytically, revealing a switch of the system into an unstable phase, in terms of the RABR model. Collisions between bright solitons are investigated too. The collisions between fast solitons are quasielastic, while slowly moving ones merge into breathers, which may persist or perish (in the latter case, also by attaining |V|=1).
The Schroedinger equation with friction from the quantum trajectory perspective
Garashchuk, Sophya; Dixit, Vaibhav; Gu Bing; Mazzuca, James
2013-02-07
Similarity of equations of motion for the classical and quantum trajectories is used to introduce a friction term dependent on the wavefunction phase into the time-dependent Schroedinger equation. The term describes irreversible energy loss by the quantum system. The force of friction is proportional to the velocity of a quantum trajectory. The resulting Schroedinger equation is nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of reactions proceeding with energy transfer from the system to the environment. Examples of dynamics are given for single and symmetric and asymmetric double well potentials.
Hidden Statistics of Schroedinger Equation
NASA Technical Reports Server (NTRS)
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Exponential Methods for the Time Integration of Schroedinger Equation
Cano, B.; Gonzalez-Pachon, A.
2010-09-30
We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.
Lee, Nam C.
2012-08-15
The reductive perturbation method is used to derive a generic form of nonlinear Schroedinger equation (NLSE) that describes the nonlinear evolution of electrostatic (ES)/electromagnetic (EM) waves in fully relativistic two-fluid plasmas. The matrix eigenvector analysis shows that there are two mutually exclusive modes of waves, each mode involving only either one of two electric potentials, A and {phi}. The general result is applied to the electromagnetic mode in electron-ion plasmas with relativistically high electron temperature (T{sub e} Much-Greater-Than m{sub e}c{sup 2}). In the limit of high frequency (ck Much-Greater-Than {omega}{sub e}), the NLSE predicts bump type electromagnetic soliton structures having width scaling as {approx}kT{sub e}{sup 5/2}. It is shown that, in electron-positron pair plasmas with high temperature, dip type electromagnetic solitons can exist. The NLSE is also applied to electrostatic (Langmuir) wave and it is shown that dip type solitons can exist if k{lambda}{sub D} Much-Less-Than 1, where {lambda}{sub D} is the electron's Debye length. For the k{lambda}{sub D} Much-Greater-Than 1, however, the solution is of bump type soliton with width scaling as {approx}1/(k{sup 5}T{sub e}). It is also shown that dip type solitons can exist in cold plasmas having relativistically high streaming speed.
Discrete transparent boundary conditions for Schroedinger-type equations
Schmidt, F.; Yevick, D.
1997-06-01
We present a general technique for constructing nonlocal transparent boundary conditions for one-dimensional Schroedinger-type equations. Our method supplies boundary conditions for the {theta}-family of implicit one-step discretizations of Schroedinger`s equation in time. The use of Mikusinski`s operator approach in time avoids direct and inverse transforms between time and frequency domains and thus implements the boundary conditions in a direct manner. 14 refs., 9 figs.
NASA Technical Reports Server (NTRS)
Bondeson, A.; Ott, E.; Antonsen, T. M., Jr.
1985-01-01
Certain first-order nonlinear ordinary differential equations exemplified by strongly damped, quasiperiodically driven pendula and Josephson junctions are isomorphic to Schroedinger equations with quasiperiodic potentials. The implications of this equivalence are discussed. In particular, it is shown that the transition to Anderson localization in the Schroedinger problem corresponds to the occurrence of a novel type of strange attractor in the pendulum problem. This transition should be experimentally observable in the frequency spectrum of the pendulum of Josephson junction.
Stable explicit schemes for equations of Schroedinger type
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.
1989-01-01
A method for constructing explicit finite-difference schemes which can be used to solve Schroedinger-type partial-differential equations is presented. A forward Euler scheme that is conditionally stable is given by the procedure. The results presented are based on the analysis of the simplest Schroedinger type equation.
The Liouville equation for singular ergodic magnetic Schroedinger operators
Kang Yang; Klein, Abel
2010-03-15
We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schroedinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schroedinger equation and solve a Liouville equation in an appropriate Hilbert space.
Bueyuekasik, Sirin A.; Pashaev, Oktay K.
2010-12-15
We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schroedinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schroedinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schroedinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola-Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
Spectrum of the ballooning Schroedinger equation
Dewar, R.L.
1997-01-01
The ballooning Schroedinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel-Kramers-Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second order differential equation with coefficients periodic in the independent variable {theta}{sub k} is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods: e.g. the apparent discontinuity between quantization formulae for {open_quotes}trapped{close_quotes} and {open_quotes}passing{close_quotes} modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over {theta}{sub k}.
The phase space of the focused cubic Schroedinger equation: A numerical study
Burlakov, Yuri O.
1998-05-01
In a paper of 1988 [41] on statistical mechanics of the nonlinear Schroedinger equation, it was observed that a Gibbs canonical ensemble associated with the nonlinear Schroedinger equation exhibits behavior reminiscent of a phase transition in classical statistical mechanics. The existence of a phase transition in the canonical ensemble of the nonlinear Schroedinger equation would be very interesting and would have important implications for the role of this equation in modeling physical phenomena; it would also have an important bearing on the theory of weak solutions of nonlinear wave equations. The cubic Schroedinger equation, as will be shown later, is equivalent to the self-induction approximation for vortices, which is a widely used equation of motion for a thin vortex filament in classical and superfluid mechanics. The existence of a phase transition in such a system would be very interesting and actually very surprising for the following reasons: in classical fluid mechanics it is believed that the turbulent regime is dominated by strong vortex stretching, while the vortex system described by the cubic Schroedinger equation does not allow for stretching. In superfluid mechanics the self-induction approximation and its modifications have been used to describe the motion of thin superfluid vortices, which exhibit a phase transition; however, more recently some authors concluded that these equations do not adequately describe superfluid turbulence, and the absence of a phase transition in the cubic Schroedinger equation would strengthen their argument. The self-induction approximation for vortices takes into account only very localized interactions, and the existence of a phase transition in such a simplified system would be very unexpected. In this thesis the authors present a numerical study of the phase transition type phenomena observed in [41]; in particular, they find that these phenomena are strongly related to the splitting of the phase space into
Dynamical system related to quasiperiodic Schroedinger equations in one dimension
Kohmoto, Mahito )
1992-02-01
A dynamical map is obtained from a class of quasiperiodic discrete Schroedinger equations in one dimension which include the Fibonacci system. The potentials are constant except for steps at special points.
Schroedinger-equation formalism for a dissipative quantum system
Anisimovas, E.; Matulis, A.
2007-02-15
We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role of a dissipative element. The coupling between the two--quantum and classical--parts of the compound system is treated in the spirit of the mean-field approximation and justification of the validity of such an approach is given. The equations of motion of the classical subsystem are solved explicitly and an effective dissipative Schroedinger equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its application to two basic problems: the decay of the quasistationary state and the calculation of the nonlinear resonance line shape.
Wigner function and Schroedinger equation in phase-space representation
Chruscinski, Dariusz; Mlodawski, Krzysztof
2005-05-15
We discuss a family of quasidistributions (s-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schroedinger equation. It turns out that although Wigner functions satisfy the Schroedinger equation in phase space, they have a completely different interpretation.
Strongly interacting bumps for the Schroedinger-Newton equations
Wei Juncheng; Winter, Matthias
2009-01-15
We study concentrated bound states of the Schroedinger-Newton equations h{sup 2}{delta}{psi}-E(x){psi}+U{psi}=0, {psi}>0, x(set-membership sign)R{sup 3}; {delta}U+(1/2)|{psi}|{sup 2}=0, x(set-membership sign)R{sup 3}; {psi}(x){yields}0, U(x){yields}0 as |x|{yields}{infinity}. Moroz et al. [''An analytical approach to the Schroedinger-Newton equations,'' Nonlinearity 12, 201 (1999)] proved the existence and uniqueness of ground states of {delta}{psi}-{psi}+U{psi}=0, {psi}>0, x(set-membership sign)R{sup 3}; {delta}U+(1/2)|{psi}|{sup 2}=0, x(set-membership sign)R{sup 3}; {psi}(x){yields}0, U(x){yields}0 as |x|{yields}{infinity}. We first prove that the linearized operator around the unique ground state radial solution ({psi}{sub 0},U{sub 0}) with {psi}{sub 0}(r)=(Ae{sup -r}/r)(1+o(1)) as r=|x|{yields}{infinity}, U{sub 0}(r)=(B/r)(1+o(1)) as r=|x|{yields}{infinity} for some A,B>0 has a kernel whose dimension is exactly 3 (corresponding to the translational modes). Using this result we further show that if for some positive integer K the points P{sub i}(set-membership sign)R{sup 3}, i=1,2...,K, with P{sub i}{ne}P{sub j} for i{ne}j are all local minimum or local maximum or nondegenerate critical points of E(P), then for h small enough there exist solutions of the Schroedinger-Newton equations with K bumps which concentrate at P{sub i}. We also prove that given a local maximum point P{sub 0} of E(P) there exists a solution with K bumps which all concentrate at P{sub 0} and whose distances to P{sub 0} are at least O(h{sup 1/3})
An Explicitly Correlated Wavelet Method for the Electronic Schroedinger Equation
Bachmayr, Markus
2010-09-30
A discretization for an explicitly correlated formulation of the electronic Schroedinger equation based on hyperbolic wavelets and exponential sum approximations of potentials is described, covering mathematical results as well as algorithmic realization, and discussing in particular the potential of methods of this type for parallel computing.
Stochastic Schroedinger equations with general complex Gaussian noises
Bassi, Angelo
2003-06-01
Within the framework of non-Markovian stochastic Schroedinger equations, we generalize the results of [W. T. Strunz, Phys. Lett. A 224, 25 (1996)] to the case of general complex Gaussian noises; we analyze the two important cases of purely real and purely imaginary stochastic processes.
Reformulating the Schroedinger equation as a Shabat-Zakharov system
Boonserm, Petarpa; Visser, Matt
2010-02-15
We reformulate the second-order Schroedinger equation as a set of two coupled first-order differential equations, a so-called 'Shabat-Zakharov system' (sometimes called a 'Zakharov-Shabat' system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an 'auxiliary condition' or 'gauge condition' that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schroedinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an 'elementary' process, then this represents complete quadrature, albeit formal, of the second-order linear ordinary differential equation.
A parallel algorithm for solving the 3d Schroedinger equation
Strickland, Michael; Yager-Elorriaga, David
2010-08-20
We describe a parallel algorithm for solving the time-independent 3d Schroedinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t {proportional_to} (N{sub nodes}){sup -0.95} {sup {+-} 0.04}. This makes it possible to solve the 3d Schroedinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.
Efficient and accurate numerical methods for the Klein-Gordon-Schroedinger equations
Bao, Weizhu . E-mail: bao@math.nus.edu.sg; Yang, Li . E-mail: yangli@nus.edu.sg
2007-08-10
In this paper, we present efficient, unconditionally stable and accurate numerical methods for approximations of the Klein-Gordon-Schroedinger (KGS) equations with/without damping terms. The key features of our methods are based on: (i) the application of a time-splitting spectral discretization for a Schroedinger-type equation in KGS (ii) the utilization of Fourier pseudospectral discretization for spatial derivatives in the Klein-Gordon equation in KGS (iii) the adoption of solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The numerical methods are either explicit or implicit but can be solved explicitly, unconditionally stable, and of spectral accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant when there is no damping terms in KGS, conserve (or keep the same decay rate of) the wave energy as that in KGS without (or with a linear) damping term, keep the same dynamics of the mean value of the meson field, and give exact results for the plane-wave solution. Extensive numerical tests are presented to confirm the above properties of our numerical methods for KGS. Finally, the methods are applied to study solitary-wave collisions in one dimension (1D), as well as dynamics of a 2D problem in KGS.
Xue Jukui
2006-02-15
In a recent paper, V. A. Brazhnyi and V. V. Konoto [Phys. Rev. E 72, 026616 (2005)] investigated the dynamics of vector dark solitons in two-component Bose-Einstein condensates. In the small amplitude limit, they deduced a coupled Korteweg-de Vries equation from the coupled Gross-Pitaevskii equations. They found that two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves exist. The slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into the stable fast solitons (corresponding to the upper branch of the dispersion law). However, our discussion shows that these results are incorrect.
Fractional Schroedinger equation for a particle moving in a potential well
Luchko, Yuri
2013-01-15
In this paper, the fractional Schroedinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, 'On the nonlocality of the fractional Schroedinger equation,' J. Math. Phys. 51, 062102 (2010)] and [S. S. Bayin, 'On the consistency of the solutions of the space fractional Schroedinger equation,' J. Math. Phys. 53, 042105 (2012)] published in this journal, controversial opinions regarding solutions to the fractional Schroedinger equation for a particle moving in the infinite potential well that were derived by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schroedinger equation for a particle moving in the infinite potential well given by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. ['On the nonlocality of the fractional Schroedinger equation,' J. Math. Phys. 51, 062102 (2010)].
Finite-difference scheme for the numerical solution of the Schroedinger equation
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Ramadhani, Issa
1992-01-01
A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.
Suarez, J.; Stamatiadis, S.; Farantos, S. C.; Lathouwers, L.
2009-08-13
Reproducing molecular dynamics is at the root of the basic principles of chemical change and physical properties of the matter. New insight on molecular encounters can be gained by solving the Schroedinger equation in cartesian coordinates, provided one can overcome the massive calculations that it implies. We have developed a parallel code for solving the molecular Time Dependent Schroedinger Equation (TDSE) in cartesian coordinates. Variable order Finite Difference methods result in sparse Hamiltonian matrices which can make the large scale problem solving feasible.
An Efficient and Accurate Quantum Lattice-Gas Model for the Many-Body Schroedinger Wave Equation
2002-01-01
CONTRACT NUMBER AN EFFICIENT AND ACCURATE QUANTUM LATTICE-GAS MODEL FOR THE MANY-BODY SCHROEDINGER WAVE EQUATION 5b. GRANT NUMBER SC. PROGRAM ELEMENT...for simulating the time-dependent evolution of a many-body jiiantum mechanical system of particles governed by the non-relativistic Schroedinger " wave...the numerical dispersion of the simulated wave packets is compared with the analytical solutions. 15. SUBJECT TERM: Schroedinger wave equation
Chithiika Ruby, V.; Senthilvelan, M.
2010-05-15
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schroedinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schroedinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.
The source-driven dissipative nonlinear Schroedinger model of resonance absorption
Larroche, O.; Pesme, D. )
1990-08-01
A source-driven dissipative nonlinear Schroedinger (NLS) equation is numerically studied, characterized by a nonlinearity parameter and a dissipative length, governing the generation of finite-amplitude, localized electrostatic plasma waves by resonance absorption of light in an inhomogeneous plasma. It is shown that as the nonlinearity parameter is increased a transition to chaos occurs through a quasiperiodic scenario. In the chaotic regime, it is shown from statistical diagnostics that as the dissipation length is increased, the system shifts from a convective regime governed by the competition between pumping and convection of the waves due to the inhomogeneity to a dissipative regime governed by the competition between pumping and a scale-length-dependent absorption mechanism, which approximately models Landau damping. The scaling laws obtained show that the turbulent state can be described in both regimes as a set of NLS solitons, interacting through the pumping and damping mechanisms.For a vanishing density gradient, the system admits a homogeneous limit, which is found to be chaotic and dissipative.
Eulerian Gaussian beams for Schroedinger equations in the semi-classical regime
Leung, Shingyu Qian Jianliang
2009-05-01
We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schroedinger equation. Traditional Gaussian beam type methods for the Schroedinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schroedinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schroedinger equation. For instance, in the one-dimensional case the proposed algorithm requires only O(sNm{sup 2}) operations to compute s different solutions with s different initial wave functions under the influence of the same potential, where N=O(1/h),h is the Planck constant, and m<
Comment on ''Solution of the Schroedinger equation for the time-dependent linear potential''
Bekkar, H.; Maamache, M.; Benamira, F.
2003-07-01
We present the correct way to obtain the general solution of the Schroedinger equation for a particle in a time-dependent linear potential following the approach used in the paper of Guedes [Phys. Rev. A 63, 034102 (2001)]. In addition, we show that, in this case, the solutions (wave packets) are described by the Airy functions.
Intertwining relations and Darboux transformations for Schroedinger equations in (n+1) dimensions
Schulze-Halberg, Axel
2010-03-15
We evaluate the intertwining relation for Schroedinger equations in (n+1) dimensions. The conditions for the existence of a Darboux transformation are analyzed and compared to their (1+1) dimensional counterparts. A complete solution of the conditions is given for (2+1) dimensions, and a Darboux transformation is constructed.
Morales, J.; Ovando, G.; Pena, J. J.
2010-12-23
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis a vis the standard Schroedinger equation with constant mass [1]. However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners. To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schroedinger-like standard equation is applied to the PDMSE. In that case, the former potential associated to the PDMSE and the potential involved in the Schroedinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD)m(x). Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.
Quasiseparation of variables in the Schroedinger equation with a magnetic field
Charest, F.; Hudon, C.; Winternitz, P.
2007-01-15
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schroedinger equation. We introduce the concept of 'quasiseparation of variables' and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra.
Cuevas, J.; Palmero, F.
2009-11-15
We propose analytical lower and upper estimates on the excitation threshold for breathers (in the form of spatially localized and time periodic solutions) in discrete nonlinear Schroedinger (DNLS) lattices with power nonlinearity. The estimation, depending explicitly on the lattice parameters, is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument. The numerical studies verify that the analytical estimates can be of particular usefulness, as a simple analytical detection of the activation energy for breathers in DNLS lattices.
Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential
NASA Technical Reports Server (NTRS)
Campbell, Joel
2009-01-01
The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.
A method of solving simple harmonic oscillator Schroedinger equation
NASA Technical Reports Server (NTRS)
Maury, Juan Carlos F.
1995-01-01
A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.
The thermal-wave model: A Schroedinger-like equation for charged particle beam dynamics
NASA Technical Reports Server (NTRS)
Fedele, Renato; Miele, G.
1994-01-01
We review some results on longitudinal beam dynamics obtained in the framework of the Thermal Wave Model (TWM). In this model, which has recently shown the capability to describe both longitudinal and transverse dynamics of charged particle beams, the beam dynamics is ruled by Schroedinger-like equations for the beam wave functions, whose squared modulus is proportional to the beam density profile. Remarkably, the role of the Planck constant is played by a diffractive constant epsilon, the emittance, which has a thermal nature.
A New Approach to Solve the Low-lying States of the Schroedinger Equation
NASA Astrophysics Data System (ADS)
Lee, T. D.
2005-12-01
We review a new iterative procedure to solve the low-lying states of the Schroedinger equation, done in collaboration with Richard Friedberg. For the groundstate energy, the nth order iterative energy is bounded by a finite limit, independent of n; thereby it avoids some of the inherent difficulties faced by the usual perturbative series expansions. For a fairly large class of problems, this new procedure can be proved to give convergent iterative solutions. These convergent solutions include the long standing difficult problem of a quartic potential with either symmetric or asymmetric minima
A center-of-mass principle for the multiparticle Schroedinger equation
Mohlenkamp, Martin J.
2010-02-15
The center-of-mass principle is the key to the rapid computation of the interaction of a large number of classical particles. Electrons governed by the multiparticle Schroedinger equation have a much more complicated interaction mainly due to their spatial extent and the antisymmetry constraint on the total wave function of the combined electron system. We present a center-of-mass principle for quantum particles that accounts for this spatial extent, the antisymmetry constraint, and the potential operators. We use it to construct an algorithm for computing a size-consistent approximate wave function for large systems with simple geometries.
Gumjudpai, Burin E-mail: buring@nu.ac.th
2008-09-15
Aspects of the non-linear Schroedinger (NLS) type of formulation of scalar (phantom) field cosmology for slow-roll, acceleration, the Wentzel-Kramers-Brillouin (WKB) approximation and the big rip singularity are presented. Slow-roll parameters for the curvature and barotropic density terms are introduced. We re-express all slow-roll parameters, slow-roll conditions and the acceleration condition in NLS form. The WKB approximation in the NLS formulation is also discussed while simplifying to the linear case. Most of the Schroedinger potentials in the NLS formulation are very slowly varying; hence the WKB approximation is valid in some ranges. In the NLS form of the big rip singularity, two quantities are infinite instead of three. We also found that approaching the big rip, w{sub eff}{yields}-1+2/3q (q<0), which is the same as the effective phantom equation of state in the flat case.
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.
A new fundamental model of moving particle for reinterpreting Schroedinger equation
Umar, Muhamad Darwis
2012-06-20
The study of Schroedinger equation based on a hypothesis that every particle must move randomly in a quantum-sized volume has been done. In addition to random motion, every particle can do relative motion through the movement of its quantum-sized volume. On the other way these motions can coincide. In this proposed model, the random motion is one kind of intrinsic properties of the particle. The every change of both speed of randomly intrinsic motion and or the velocity of translational motion of a quantum-sized volume will represent a transition between two states, and the change of speed of randomly intrinsic motion will generate diffusion process or Brownian motion perspectives. Diffusion process can take place in backward and forward processes and will represent a dissipative system. To derive Schroedinger equation from our hypothesis we use time operator introduced by Nelson. From a fundamental analysis, we find out that, naturally, we should view the means of Newton's Law F(vector sign) = ma(vector sign) as no an external force, but it is just to describe both the presence of intrinsic random motion and the change of the particle energy.
Integrable nonlinear relativistic equations
NASA Astrophysics Data System (ADS)
Hadad, Yaron
This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied. It is also proven that it is equivalent to a novel equation that poses a fascinating similarity to the Sinh-Gordon equation. For the 1+1 Einstein equation the Belinski-Zakharov transformation is explored. It is used to derive explicit formula for N gravitational solitons on arbitrary diagonal background. In particular, the method is used to derive gravitational solitons on the Einstein-Rosen background. The similarities and differences between the attributes of the solitons of the symmetric Chiral field equation and those of the 1+1 Einstein equation are emphasized, and their origin is pointed out. For the 1+2 Einstein equation, new equations describing diagonal metrics are derived and their compatibility is proven. Different gravitational waves are studied that naturally extend the class of Bondi-Pirani-Robinson waves. It is further shown that the Bondi-Pirani-Robinson waves are stable with respect to perturbations of the spacetime. Their stability is closely related to the stability of the Schwarzschild black hole and the relation between the two allows to conjecture about the stability of a wide range of gravitational phenomena. Lastly, a new set of equations that describe weak gravitational waves is derived. This new system of equations is closely and fundamentally connected with the nonlinear Schrodinger equation and can be properly called the nonlinear Schrodinger-Einstein equations. A few preliminary solutions are constructed.
Rajan, M. S. Mani; Mahalingam, A.
2013-04-15
In this paper, we introduce a system of the nonlinear Schroedinger-Maxwell-Bloch equation with variable coefficients which represents the propagation of optical pulses in an inhomogeneous erbium doped fiber with loss/gain driven by an external potential. The one and two soliton solutions in explicit forms are generated by using the Darboux transformation and the associated Lax pair. We consider the distributed amplification system, and some main features of the solutions are demonstrated graphically. We also consider the concept of soliton propagation in a dispersion managed erbium doped fiber and through symbolic computation, we have carried out our study from an analytic viewpoint.
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.
Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation
Pryce, J.D. )
1994-06-01
We present a numerical method, implemented in a Fortran code RESON, for computing resonance of the radial one-dimensional Schroedinger equation, for an underlying potential that decays sufficiently fast at infinity. The basic approach is to maximize the time-delay function [tau]([lambda]) as in the LeRoy program TDELAY. We present some theory that allows a preliminary bracketing of the resonance and various ways of reducing the total work. Together with automatic meshsize selection this leads to a method that has proved efficient, robust, and extremely trouble-free in numerical tests. The code makes use of Marletta's Sturm-Liouville solver, SLO2F, due to go into the NAG library. 24 refs., 4 figs., 3 tabs.
Leung Shingyu; Qian Jianliang
2010-11-20
We propose the backward phase flow method to implement the Fourier-Bros-Iagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schroedinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in . In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to compute beam ingredients rapidly. Numerical examples demonstrate the efficiency and accuracy of the proposed algorithms.
Blow-up profile to the solutions of two-coupled Schroedinger equations
Chen Jianqing; Guo Boling
2009-02-15
The model of the following two-coupled Schroedinger equations, i{sub t}+(1/2){delta}u=(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})uu, (t,x)(set-membership sign)R{sub +}xR{sup N}, and iv{sub t}+(1/2){delta}v=(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, (t,x)(set-membership sign)R{sub +}xR{sup N}, is proposed in the study of the Bose-Einstein condensates [Mitchell, et al., ''Self-traping of partially spatially incoherent light,'' Phys. Rev. Lett. 77, 490 (1996)]. We prove that for suitable initial data and p the solution blows up exactly like {delta} function. As a by-product, we prove that similar phenomenon occurs for the critical two-coupled Schroedinger equations with harmonic potential [Perez-Garcia, V. M. and Beitia, T. B., ''Sybiotic solitons in heteronuclear multicomponent Bose-Einstein condensates,'' Phys. Rev. A 72, 033620 (2005)], iu{sub t}+(1/2){delta}u=({omega}/2)|x|{sup 2}u+(g{sub 11}|u|{sup 2p}+g|u|{sup p-1}|v|{sup p+1})u, x(set-membership sign)R{sup N}, and iv{sub t}+(1/2){delta}v=({omega}/2)|x|{sup 2}v+(g|u|{sup p+1}|v|{sup p-1}+g{sub 22}|v|{sup 2p})v, x(set-membership sign)R{sup N}.
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.
Phase integral approximation for coupled ordinary differential equations of the Schroedinger type
Skorupski, Andrzej A.
2008-05-15
Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u{sub j}{sup ''}(x)+{sigma}{sub k=1}{sup N}R{sub jk}(x)u{sub k}(x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[{identical_to}(R{sub jk}(x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u{sup ''}(x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph].
Two routes to the one-dimensional discrete nonpolynomial Schroedinger equation
Gligoric, G.; Hadzievski, Lj.; Maluckov, A.; Salasnich, L.; Malomed, B. A.
2009-12-15
The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schroedinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce 'model 1' (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. 'Model 2,' which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2--in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.
Dubrovsky, V. G.; Topovsky, A. V.; Basalaev, M. Yu.
2010-09-15
The classes of exactly solvable multiline soliton potentials and corresponding wave functions of two-dimensional stationary Schroedinger equation via {partial_derivative}-dressing method are constructed and their physical interpretation is discussed.
Anastassi, Z. A.; Simos, T. E.
2010-09-30
We develop a new family of explicit symmetric linear multistep methods for the efficient numerical solution of the Schroedinger equation and related problems with oscillatory solution. The new methods are trigonometrically fitted and have improved intervals of periodicity as compared to the corresponding classical method with constant coefficients and other methods from the literature. We also apply the methods along with other known methods to real periodic problems, in order to measure their efficiency.
Interpretation of non-Markovian stochastic Schroedinger equations as a hidden-variable theory
Gambetta, Jay; Wiseman, H.M.
2003-12-01
Do diffusive non-Markovian stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system 'conditioned' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
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.
Tremblay, Jean Christophe; Carrington, Tucker Jr.
2004-12-15
If the Hamiltonian is time dependent it is common to solve the time-dependent Schroedinger equation by dividing the propagation interval into slices and using an (e.g., split operator, Chebyshev, Lanczos) approximate matrix exponential within each slice. We show that a preconditioned adaptive step size Runge-Kutta method can be much more efficient. For a chirped laser pulse designed to favor the dissociation of HF the preconditioned adaptive step size Runge-Kutta method is about an order of magnitude more efficient than the time sliced method.
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)
Corrêa Silva, E. V.; Monerat, G. A.; de Oliveira Neto, G.; Ferreira Filho, L. G.
2014-01-01
The Galerkin spectral method can be used for approximate calculation of eigenvalues and eigenfunctions of unidimensional Schroedinger-like equations such as the Wheeler-DeWitt equation. The criteria most commonly employed for checking the accuracy of results is the conservation of norm of the wave function, but some other criteria might be used, such as the orthogonality of eigenfunctions and the variation of the spectrum with varying computational parameters, e.g. the number of basis functions used in the approximation. The package Spectra, which implements the spectral method in Maple language together with a number of testing tools, is presented. Alternatively, Maple may interact with the Octave numerical system without the need of Octave programming by the user.
Non-linear generalization of the relativistic Schrödinger equations.
NASA Astrophysics Data System (ADS)
Ochs, U.; Sorg, M.
1996-09-01
The theory of the relativistic Schrödinger equations is further developped and extended to non-linear field equations. The technical advantage of the relativistic Schroedinger approach is demonstrated explicitly by solving the coupled Einstein-Klein-Gordon equations including a non-linear Higgs potential in case of a Robertson-Walker universe. The numerical results yield the effect of dynamical self-diagonalization of the Hamiltonian which corresponds to a kind of quantum de-coherence being enabled by the inflation of the universe.
Basko, D.M.
2011-07-15
Research Highlights: > In a one-dimensional disordered chain of oscillators all normal modes are localized. > Nonlinearity leads to chaotic dynamics. > Chaos is concentrated on rare chaotic spots. > Chaotic spots drive energy exchange between oscillators. > Macroscopic transport coefficients are obtained. - Abstract: The subject of this study is the long-time equilibration dynamics of a strongly disordered one-dimensional chain of coupled weakly anharmonic classical oscillators. It is shown that chaos in this system has a very particular spatial structure: it can be viewed as a dilute gas of chaotic spots. Each chaotic spot corresponds to a stochastic pump which drives the Arnold diffusion of the oscillators surrounding it, thus leading to their relaxation and thermalization. The most important mechanism of equilibration at long distances is provided by random migration of the chaotic spots along the chain, which bears analogy with variable-range hopping of electrons in strongly disordered solids. The corresponding macroscopic transport equations are obtained.
Hoang-Do, Ngoc-Tram; Hoang, Van-Hung; Le, Van-Hoang
2013-05-15
The Feranchuk-Komarov operator method is developed by combining with the Levi-Civita transformation in order to construct analytical solutions of the Schroedinger equation for a two-dimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wave-functions in strong magnetic field. The results could be used for various physical analyses and the method used here could also be applied to other atomic systems.
Cooper, J. D.; Valavanis, A.; Ikonic, Z.; Harrison, P.; Cunningham, J. E.
2010-12-01
The nonparabolic Schroedinger equation for electrons in quantum cascade lasers (QCLs) is a cubic eigenvalue problem (EVP) which cannot be solved directly. While a method for linearizing this cubic EVP has been proposed in principle for quantum dots [Hwang et al., Math. Comput. Modell., 40, 519 (2004)] it was deemed too computationally expensive because of the three-dimensional geometry under consideration. We adapt this linearization approach to the one-dimensional geometry of QCLs, and arrive at a direct and exact solution to the cubic EVP. The method is then compared with the well established shooting method, and it is shown to be more accurate and reliable for calculating the bandstructure of mid-infrared QCLs.
NASA Astrophysics Data System (ADS)
Stroud, Carlos
1993-07-01
Recent work at the University of Rochester that was supported by the Army Research Office through the University Research Initiative Program was featured in a recent book Taming the Atom by Hans Christian von Baeyer. An excerpt from that book is presented that shows that the work in Rochester is the realization of Erwin Schroedinger's hope in the earliest days of quantum theory that a solution to his equation could be found in the form of a localized wave packet moving along the elliptical orbit predicted by classical theory. In a series of calculations and experiments the group in Rochester was shown that a short laser pulse can be used to excite such a wave packet state and that as the wave packet moves many times around the orbit it undergoes a complicated time evolution in which it spreads all the way around the orbit, and then repeatedly relocalizes in the form of a single wave packet or a series of identical sub-wave packets equally spaced around the orbit. This work sheds new light on the boundary between microscopic quantum systems and macroscopic classical systems.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.
1991-05-15
dependent Schr ~ dinger equation , and a Dis~tribut ion/ numrerical implementation of absorbing boundary conditions for propagating wavefunctions atf the...mesoscopic systems with open boundaries using the multidirnensiji.j* time - dependent Schrodinger equation 12. PERSONAL AUTHOR(S) Register, Leonard F...if necessary and identify by block number) A numerical procedure based on the time - dependent Schroainger equation for the modelin~g of multidimensional
Dimensional interpolation for nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Fred
2005-09-01
Dimensional interpolation has been used successfully by physicists and chemists to solve the Schroedinger equation for atoms and complex molecules. The same basic idea can be used to solve the Fokker-Planck equation for nonlinear filters. In particular, it is well known (by physicists) that two Schroedinger equations are equivalent to two Fokker-Planck equations. Moreover, we can avoid the Schroedinger equation altogether and use dimensional interpolation directly on the Fokker-Planck equation. Dimensional interpolation sounds like a crazy idea, but it works. We will attempt to make this paper accessible to normal engineers who do not have quantum mechanics for breakfast.
Dubrovsky, V. G.; Topovsky, A. V.
2013-03-15
New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, Horizontal-Ellipsis , N are constructed via Zakharov and Manakov {partial_derivative}-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 Less-Than-Or-Slanted-Equal-To k{sub 1} < k{sub 2} < Horizontal-Ellipsis < k{sub m} Less-Than-Or-Slanted-Equal-To N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.
Vega, Ines de; Alonso, Daniel; Gaspard, Pierre
2005-02-01
It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schroedinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schroedinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.
NASA Astrophysics Data System (ADS)
Landau, Rubin H.
2000-03-01
A simple and explicit technique for the numerical solution of the two-particle, time-dependent Schrödinger equation is assembled and used to generate animations of two wavepackets scattering from each other. This is an extension of the work of Goldberg et. al [1] and Visscher [2], the former having been used to produce the classic one-particle animations in Schiff's Quantum Mechanics textbook. The technique can handle interparticle potentials that are arbitrary functions of the coordinates of each particle, arbitrary initial and boundary conditions, and multi-dimensional equations. Animations of both the two particle probability density and of the single particle densities for each particle will be given in the talk as well as on the World Wide Web. Physical effects, such as pickup, fission, and exchange, will be shown to occur even with square-well potential. [1] Goldberg, H. M. Schey, and J.L. Schwartz, Amer. J. Phys., 35, 177-186 (1967). [2] P.B. Visscher, Computers in Phys, 596-598 (Nov/Dec 1991).
NASA Technical Reports Server (NTRS)
Costiner, Sorin; Taasan, Shlomo
1994-01-01
This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.
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.
Eskin, G.
2008-02-15
We consider the inverse boundary value problem for the Schroedinger operator with time-dependent electromagnetic potentials in domains with obstacles. We extend the resuls of the author's works [Inverse Probl. 19, 49 (2003); 19, 985 (2003); 20, 1497 (2004)] to the case of time-dependent potentials. We relate our results to the Aharonov-Bohm effect caused by magnetic and electric fluxes.
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
Additivity of nonlinear biomass equations
Bernard R. Parresol
2001-01-01
Two procedures that guarantee the property of additivity among the components of tree biomass and total tree biomass utilizing nonlinear functions are developed. Procedure 1 is a simple combination approach, and procedure 2 is based on nonlinear joint-generalized regression (nonlinear seemingly unrelated regressions) with parameter restrictions. Statistical theory is...
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 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 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.
Robust iterative method for nonlinear Helmholtz equation
NASA Astrophysics Data System (ADS)
Yuan, Lijun; Lu, Ya Yan
2017-08-01
A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr nonlinearity. In the strongly nonlinear regime, the nonlinear Helmholtz equation could have multiple solutions related to phenomena such as optical bistability and symmetry breaking. The new method exhibits a much more robust convergence behavior than existing iterative methods, such as frozen-nonlinearity iteration, Newton's method and damped Newton's method, and it can be used to find solutions when good initial guesses are unavailable. Numerical results are presented for the scattering of light by a nonlinear circular cylinder based on the exact nonlocal boundary condition and a pseudospectral method in the polar coordinate system.
Michel, N.
2008-02-15
Demonstrating the completeness of wave function solutions of the radial Schroedinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schroedinger potentials. One can mention, in particular, unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided that the non-Coulomb part of potentials vanishes after a finite radius. Nonlocality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k=0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.
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.
Hybrid approach for nonlinear wave equation
NASA Astrophysics Data System (ADS)
Bogdanov, Alexander; Mareev, Vladimir
2017-07-01
The solution of nonintegrable nonlinear equations is very difficult even numerically and practically impossible by standard analytical technic. New view, offered by heterogeneous computational systems, gives some new possibilities, but also need novel approaches for numerical realization of pertinent algorithms. We shall give some examples of such analysis on the base of nonlinear wave's evolution study in multiphase media with chemical reaction.
Nonlinear quantum equations: Classical field theory
Rego-Monteiro, M. A.; Nobre, F. D.
2013-10-15
An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schrödinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q→ 1. The main characteristic of this field theory consists on the fact that besides the usual Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 4}, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.
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.
NASA Astrophysics Data System (ADS)
Kovarik, M. D.; Barnes, T.
We describe a Monte Carlo simulation of a dynamical fermion problem in two spatial dimensions on an Intel iPSC/860 hypercube. The problem studied is the determination of the dispersion relation of a dynamical hole in the t-J model of the high temperature superconductors. Since this problem involves the motion of many fermions in more than one spatial dimension, it is representative of the class of systems that suffer from the 'minus sign problem' of dynamical fermions which has made Monte Carlo simulation very difficult. We demonstrate that for small values of the hole hopping parameter one can extract the entire hole dispersion relation using the GRW Monte Carlo algorithm, which is a simulation of the Euclidean time Schroedinger equation, and present results on 4 x 4 and 6 x 6 lattices. Generalization to physical hopping parameter values will only require use of an improved trial wavefunction for importance sampling.
Tao Liang; Rescigno, T. N.; Vanroose, W.; Reps, B.; McCurdy, C. W.
2009-12-15
We demonstrate that exterior complex scaling (ECS) can be used to impose outgoing wave boundary conditions exactly on solutions of the time-dependent Schroedinger equation for atoms in intense electromagnetic pulses using finite grid methods. The procedure is formally exact when applied in the appropriate gauge and is demonstrated in a calculation of high-harmonic generation in which multiphoton resonances are seen for long pulse durations. However, we also demonstrate that while the application of ECS in this way is formally exact, numerical error can appear for long-time propagations that can only be controlled by extending the finite grid. A mathematical analysis of the origins of that numerical error, illustrated with an analytically solvable model, is also given.
Kovarik, M.D.; Barnes, T. |
1993-10-01
We describe a Monte Carlo simulation of a dynamical fermion problem in two spatial dimensions on an Intel iPSC/860 hypercube. The problem studied is the determination of the dispersion relation of a dynamical hole in the t-J model of the high temperature superconductors. Since this problem involves the motion of many fermions in more than one spatial dimensions, it is representative of the class of systems that suffer from the ``minus sign problem`` of dynamical fermions which has made Monte Carlo simulation very difficult. We demonstrate that for small values of the hole hopping parameter one can extract the entire hole dispersion relation using the GRW Monte Carlo algorithm, which is a simulation of the Euclidean time Schroedinger equation, and present results on 4 {times} 4 and 6 {times} 6 lattices. Generalization to physical hopping parameter values wig only require use of an improved trial wavefunction for importance sampling.
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.
On implicit abstract neutral nonlinear differential equations
Hernández, Eduardo; O’Regan, Donal
2016-04-15
In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.
Deriving average soliton equations with a perturbative method
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P. )
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically.
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.
Vortex Nucleation in a Dissipative Variant of the Nonlinear Schroedinger Equation Under Rotation
2014-12-01
Einstein condensates. AMS subject classifications. 34A34, 35Q55, 76M23, 1. Introduction. Vortices are persistent circulating flow patterns that occur...examples in sunspots, dust devils [4], and plant propulsion [5]. The realm of atomic Bose- Einstein condensates (BECs) [6, 7, 8] has produced a novel and...Smith, Bose- Einstein condensation in dilute gases, (Cambridge University Press, Cambridge, 2002). [8] L.P. Pitaevskii and S. Stringari, Bose- Einstein
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.
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.
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.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
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.
1981-04-01
Contract No. DAAG29- 10-(’-w4l and by an A.M.S. Postdoctoral Research Fellowship. I SIGNIFICANCE AND EXPLANATION The Korteweg - deVries equation (KdV for...decomposition of the solution resembling the use of Fourier transforms in solving constant coefficient equations . For the linearization about the zero...1.3) to eliminate (f 2)I(x,k), it is more convenient not to do so.) We prove the theorem by solving the equation - 4Q’ - 2Q’ + 4k 2 ’ for ’ and
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}
Nonlinear progressive wave equation for stratified atmospheres.
Edward McDonald, B; Piacsek, Andrew A
2011-11-01
The nonlinear progressive wave equation (NPE) [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406-1417 (1987)] is expressed in a form to accommodate changes in the ambient atmospheric density, pressure, and sound speed as the time-stepping computational window moves along a path possibly traversing significant altitude differences (in pressure scale heights). The modification is accomplished by the addition of a stratification term related to that derived in the 1970s for linear range-stepping calculations and later adopted into Khokhlov-Zabolotskaya-Kuznetsov-type nonlinear models. The modified NPE is shown to preserve acoustic energy in a ray tube and yields analytic similarity solutions for vertically propagating N waves in isothermal and thermally stratified atmospheres.
Exact and explicit solitary wave solutions to some nonlinear equations
Jiefang Zhang
1996-08-01
Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative {Phi}{sup 4}-model equation, the generalized Fisher equation, and the elastic-medium wave equation.
Ada Programming for Solving Nonlinear Equations
NASA Astrophysics Data System (ADS)
Wu, Trong
This paper introduces the Ada programming for solving non-linear equations over a new class of real numbers which are based on the concepts of model numbers and rough numbers for a given computer system. We will study structures of Ada interval computation over model numbers and rough numbers. To do these, we must revise commonly interval computation from compact intervals to closed-open intervals for their initial intervals. This way, we can promise that the final resulting interval will always be a shorter than the result from the ordinary interval computation. Two examples are presented, one is use Newton method and the other is apply iterative method for solving non-line equations. The Ada programs and their the approximated solutions are given in both decimal and binary values.
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
ERIC Educational Resources Information Center
Ozdemir, Burhanettin
2017-01-01
The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…
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
Greenman, Loren; Mazziotti, David A.
2011-05-07
Direct computation of energies and two-electron reduced density matrices (2-RDMs) from the anti-Hermitian contracted Schroedinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)], it is shown, recovers both single- and multi-reference electron correlation in the chemiluminescent reaction of dioxetanone especially in the vicinity of the conical intersection where strong correlation is important. Dioxetanone, the light-producing moiety of firefly luciferin, efficiently converts chemical energy into light by accessing its excited-state surface via a conical intersection. Our previous active-space 2-RDM study of dioxetanone [L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010)] concluded that correlating 16 electrons in 13 (active) orbitals is required for realistic surfaces without correlating the remaining (inactive) orbitals. In this paper we pursue two complementary goals: (i) to correlate the inactive orbitals in 2-RDMs along dioxetanone's reaction coordinate and compare these results with those from multireference second-order perturbation theory (MRPT2) and (ii) to assess the size of the active space--the number of correlated electrons and orbitals--required by both MRPT2 and ACSE for accurate energies and surfaces. While MRPT2 recovers very different amounts of correlation with (4,4) and (16,13) active spaces, the ACSE obtains a similar amount of correlation energy with either active space. Nevertheless, subtle differences in excitation energies near the conical intersection suggest that the (16,13) active space is necessary to determine both energetic details and properties. Strong electron correlation is further assessed through several RDM-based metrics including (i) total and relative energies, (ii) the von Neumann entropy based on the 1-electron RDM, as well as the (iii) infinity and (iv) squared Frobenius norms based on the cumulant 2-RDM.
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 integral equations for the sausage model
NASA Astrophysics Data System (ADS)
Ahn, Changrim; Balog, Janos; Ravanini, Francesco
2017-08-01
The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to ‘sausage’ shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between ν and λ. Dedicated to the memory of Petr Petrovich Kulish.
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.
Localized Nonlinear Waves in Systems with Time- and Space-Modulated Nonlinearities
Belmonte-Beitia, Juan; Perez-Garcia, Victor M.; Vekslerchik, Vadym; Konotop, Vladimir V.
2008-04-25
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schroedinger equations with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly nontrivial solutions such as periodic (breathers), resonant, or quasiperiodically oscillating solitons. Some implications to the field of matter waves are also discussed.
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.
NASA Technical Reports Server (NTRS)
Kis, Z.; Janszky, J.; Vinogradov, An. V.; Kobayashi, T.
1996-01-01
The optical Schroedinger cat states are simple realizations of quantum states having nonclassical features. It is shown that vibrational analogues of such states can be realized in an experiment of double pulse excitation of vibrionic transitions. To track the evolution of the vibrational wave packet we derive a non-unitary time evolution operator so that calculations are made in a quasi Heisenberg picture.
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.
Nonlinear equation for Farley–Buneman waves in multispecies plasma
Litt, S. K.; Smolyakov, A. I. Bains, A. S.; Pokhotelov, O. A.; Onishchenko, O. G.; Horton, W.
2016-05-15
The nonlinear equation describing the Farley–Buneman (FB) waves in multispecies collisional plasmas is derived by employing the multiple-scale reduction analysis. It is shown that the presence of several ion species with different collisionalities and different ion masses removes the degeneracy of the nonlinear equation and generates the nonlinear terms resulting in wave steepening and wave breaking. This effect may be responsible for formation of one-dimensional coherent FB waves of a finite amplitude.
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.
Exploring the nonlinear cloud and rain equation
NASA Astrophysics Data System (ADS)
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.
Nonlinear Evolution Equations in Banach Spaces.
relationship to the evolution equation is studied. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases. (Author)
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.
On a generalized Kirchhoff equation with sublinear nonlinearities
NASA Astrophysics Data System (ADS)
Santos Júnior, João R.; Siciliano, Gaetano
2017-07-01
In this paper we consider a generalized Kirchhoff? equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools.
Additive nonlinear biomass equations: A likelihood-based approach
David L. R. Affleck; Ulises Dieguez-Aranda
2016-01-01
Since Parresolâs (Can. J. For. Res. 31:865-878, 2001) seminal article on the topic, it has become standard to develop nonlinear tree biomass equations to ensure compatibility among total and component predictions and to fit these equations using multistep generalized least-squares methods. In particular, many studies have specified equations for total tree...
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.
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.
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.
Linearized oscillation theory for a nonlinear delay impulsive equation
NASA Astrophysics Data System (ADS)
Berezansky, Leonid; Braverman, Elena
2003-12-01
For a scalar nonlinear impulsive delay differential equationwith rk(t)≥0,hk(t)≤t, limj-->∞ τj=∞, such an auxiliary linear impulsive delay differential equationis constructed that oscillation (nonoscillation) of the nonlinear equation can be deduced from the corresponding properties of the linear equation. Coefficients rk(t) and delays are not assumed to be continuous. Explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics, such as the impulsive logistic equation and the impulsive generalized Lasota-Wazewska equation which describes the survival of red blood cells. It is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative.
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.
An integrable shallow water equation with linear and nonlinear dispersion.
Dullin, H R; Gottwald, G A; Holm, D D
2001-11-05
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.
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
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.
Differentiability at lateral boundary for fully nonlinear parabolic equations
NASA Astrophysics Data System (ADS)
Ma, Feiyao; Moreira, Diego R.; Wang, Lihe
2017-09-01
For fully nonlinear uniformly parabolic equations, the first derivatives regularity of viscosity solutions at lateral boundary is studied under new Dini type conditions for the boundary, which is called Reifenberg Dini conditions and is weaker than usual Dini conditions.
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
NASA Astrophysics Data System (ADS)
Ley, Olivier; Nguyen, Vinh Duc
2017-10-01
Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.
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).
A Nonlinear Hyperbolic Volterra Equation in Viscoelasticity.
1980-06-01
35L55, 35L67, 47H10, 47H15 Key Words: nonlinear viscoelastic motion, materials with memory, stress- strain relaxation functions, nonlinear Volterra...homogeneous body. Here the dissipation mechanism which is induced by memory effects of the viscoelastic materials (stress-strain relaxation function - the...GREENBERG, J. M., A priori estimates for flows in dissipative materials , J. Math. Anal. Appl. 60 (1977), 617-630. CMD/JAN/scr Ii -31- SECURITY
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.
The Fluid Dynamic Limit of the Nonlinear Boltzmann Equation,
1980-02-01
dynamics is ’ . strongly nonlinear. Previously, Glikson [4] and Kaniel and Shinbrot [10 showed existence locally in time. Global existence of solutions... Glikson , A., On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off, Arch. Rational
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.
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
NASA Astrophysics Data System (ADS)
Abdel-Salam, Emad A.-B.; Hassan, Gamal F.
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag-Leffler function methods. The obtained results recover the well-know solutions when α = 1.
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.
Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang—Mills Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Hon-Wah, Tam
2014-02-01
With the help of some reductions of the self-dual Yang Mills (briefly written as sdYM) equations, we introduce a Lax pair whose compatibility condition leads to a set of (2 + 1)-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new (2 + 1)-dimensional nonlinear integrable systems, in particular, obtains a kind of (2 + 1)-dimensional integrable couplings of a new (2 + 1)-dimensional integrable nonlinear equation.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
Cylindrical Pulsons in Nonlinear Relativistic Wave Equations
NASA Astrophysics Data System (ADS)
Geicke, J.
1984-05-01
Numerical results to the Higgs scalar equation and the sine-Gordon equation with cylindrical symmetry are reported. Two separated energy regions are found where a Higgs kink develops into pulsons when reaching the origin r = 0, while only for higher energies a reflection is observed. The pulsons to both equations are studied in detail by modifying the initial kink shapes. In comparison with the spherical pulsons the cylindrical ones are extremely long-lived. For amplitudes slightly below half the distance between two (neighboured) vacua of the theories no decrease of the amplitudes and no perceivable radiation have been obtained by the numerical solution, examined during a time of order 1000. On the other hand, "heavy" sine-Gordon pulsons (with amplitudes 3π ~ 4π) are found to decay fast during a time t approx 100 in cylindrical symmetry.
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.
Nonlinear evolution of Alfven waves in a finite beta plasma
Som, B.K. ); Dasgupta, B.; Patel, V.L. ); Gupta, M.R. )
1989-12-01
A general form of the derivative nonlinear Schroedinger (DNLS) equation, describing the nonlinear evolution of Alfven waves propagating parallel to the magnetic field, is derived by using two-fluid equations with electron and ion pressure tensors obtained from Braginskii (in {ital Reviews} {ital of} {ital Plasma Physics} (Consultants Bureau, New York, 1965), Vol. 1, p. 218). This equation is a mixed version of the nonlinear Schroedinger (NLS) equation and the DNLS, as it contains an additional cubic nonlinear term that is of the same order as the derivative of the nonlinear terms, a term containing the product of a quadratic term, and a first-order derivative. It incorporates the effects of finite beta, which is an important characteristic of space and laboratory plasmas.
Nonlinear Resonance and Duffing's Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…
Nonlinear Resonance and Duffing's Spring Equation II
ERIC Educational Resources Information Center
Fay, T. H.; Joubert, Stephan V.
2007-01-01
The paper discusses the boundary in the frequency-amplitude plane for boundedness of solutions to the forced spring Duffing type equation x[umlaut] + x + [epsilon]x[cubed] = F cos[omega]t. For fixed initial conditions and for representative fixed values of the parameter [epsilon], the results are reported of a systematic numerical investigation…
Non-Linear Spring Equations and Stability
ERIC Educational Resources Information Center
Fay, Temple H.; Joubert, Stephan V.
2009-01-01
We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…
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.
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.
NASA Astrophysics Data System (ADS)
Chen, Junchao; Li, Biao
2011-12-01
In this paper, the generalized sub-equation method is extended to investigate localized nonlinear waves of the one-dimensional nonlinear Schrödinger equation (NLSE) with potentials and nonlinearities depending on time and on spatial coordinates. With the help of symbolic computation, three families of analytical solutions of this NLS-type equation are presented. Based on these solutions, periodically and quasiperiodically oscillating solitons (dark and bright) and moving solitons are observed. Some implications to Bose-Einstein condensates are also discussed
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.
Asymptotic solutions of weakly nonlinear, dispersive wave-propagation problems by Fourier analysis
Srinivasan, R.
1989-01-01
A perturbation method based on Fourier analysis and multiple scales is introduced for solving weakly nonlinear, dispersive wave propagation problems with Fourier transformable initial conditions. Asymptotic solutions are derived for the weakly nonlinear cubic Schroedinger (NLS) equation with variable coefficients and the weakly nonlinear Kortewegde-Vries (KdV) equation; the results for the NLS equation are verified by comparison with numerical solutions. In the special case of constant coefficients, the asymptotic solution for the weakly nonlinear NLS equation agrees to leading order with previously derived results in the literature; in general, this is not true to higher orders. Therefore previous asymptotic results for the strongly nonlinear Schroedinger equation can be valid only for restricted initial conditions. Similar conclusions apply to the KdV equation.
Periodic orbits in nonlinear wave equations on networks
NASA Astrophysics Data System (ADS)
Caputo, J. G.; Khames, I.; Knippel, A.; Panayotaros, P.
2017-09-01
We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits exist: modes without soft nodes and others. For the former which are the Goldstone and the bivalent modes, the linearized equations decouple. A Floquet analysis was conducted systematically for chains; it indicates that the Goldstone mode is usually stable and the bivalent mode is always unstable. The linearized equations for the second type of modes are coupled, they indicate which modes will be excited when the orbit destabilizes. Numerical results for the second class show that modes with a single eigenvalue are unstable below a threshold amplitude. Conversely, modes with multiple eigenvalues always seem unstable. This study could be applied to coupled mechanical systems.
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.
An Efficient Numerical Solution of Nonlinear Hunter-Saxton Equation
NASA Astrophysics Data System (ADS)
Parand, Kourosh; Delkhosh, Mehdi
2017-05-01
In this paper, the nonlinear Hunter-Saxton equation, which is a famous partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.
Elliptically polarised cnoidal waves in a medium with spatial dispersion of cubic nonlinearity
Makarov, Vladimir A; Perezhogin, I A; Petnikova, V M; Potravkin, N N; Shuvalov, Vladimir V
2012-02-28
We present new specific analytic solutions of a system of nonlinear Schroedinger equations, corresponding to elliptically polarised cnoidal waves in an isotropic gyrotropic medium with spatial dispersion of cubic nonlinearity and second-order frequency dispersion under the conditions of formation of the waveguides of the same type for each of the circularly polarised components of the light field.
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.
Burgers' equation and the evolution of nonlinear second sound
NASA Astrophysics Data System (ADS)
Davidowitz, Hananel; L'vov, Yuri; Steinberg, Victor
A systematic, experimental and numerical search for subharmonic generation and/or amplification was conducted at intermediate times and moderate Reynolds numbers in nonlinear second sound near the superfluid transition. We found that the nonlinear acoustic waves are dynamically monotonic in the sense that only energy cascades to smaller and smaller scales (until the dissipation scale) exist. There is no indication of a decay of monochromatic waves to waves of lower wave numbers. This precludes the existence of a decay instability in Burgers' equation as has been discussed in the literature. We thus extend the theoretical proof of Sinai concerning the absence of subharmonics in the solutions of Burger's equation to intermediate times.
Philosophy of Erwin Schroedinger: a diachronic view of Schroedinger's thoughts
Melgar, M.F.
1988-03-01
There is no agreement within the scientific community about the philosophy of Schroedinger. Some people think that he was a realist, while others defend him as an idealist. In this paper we study a number of Schroedinger's works and we show that the epithets of realist and idealist do not do him justice. Toward the end we conclude that it would be more adequate to place him in the trend known as the philosophy of immanence.
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.
Multiply scaled constrained nonlinear equation solvers. [for nonlinear heat conduction problems
NASA Technical Reports Server (NTRS)
Padovan, Joe; Krishna, Lala
1986-01-01
To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.
Conservation laws of inviscid Burgers equation with nonlinear damping
NASA Astrophysics Data System (ADS)
Abdulwahhab, Muhammad Alim
2014-06-01
In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).
Existence of stationary states for nonlinear Dirac equations
NASA Astrophysics Data System (ADS)
Merle, F.
We generalize the previous result of Cazenave and Vasquez on the existence of stationary states for nonlinear Dirac equations of the form i∑ μ3 = 0 γμ∂μΨ - mΨ + L( ΨΨ) Ψ = 0. We seek solutions which are separable in spherical coordinates and we then make use of a shooting method to solve the associated problem for ordinary differential equations.
Topological horseshoes in travelling waves of discretized nonlinear wave equations
Chen, Yi-Chiuan; Chen, Shyan-Shiou; Yuan, Juan-Ming
2014-04-15
Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.
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…
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 propagation of a wave packet in a hard-walled circular duct
NASA Technical Reports Server (NTRS)
Nayfeh, A. H.
1975-01-01
The method of multiple scales is used to derive a nonlinear Schroedinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cutoff frequencies.
Nonlinear propagation of a wave packet in a hard-walled circular duct
NASA Technical Reports Server (NTRS)
Nayfeh, A. H.
1974-01-01
The method of multiple scales is used to derive a nonlinear Schroedinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cut off frequencies.
A new perturbative approach to nonlinear partial differential equations
Bender, C.M.; Boettcher, S. ); Milton, K.A. )
1991-11-01
This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation {ital u}{sub {ital t}}+{ital uu}{sub {ital x}}={ital u}{sub {ital xx}}, the general nonlinear equation {ital u}{sub {ital t}}+{ital u}{sup {delta}}{ital u}{sub {ital x}}={ital u}{sub {ital xx}} is considered and expanded in powers of {delta}. The coefficients of the {delta} series to sixth order in powers of {delta} is determined and Pade summation is used to evaluate the perturbation series for large values of {delta}. The numerical results are accurate and the method is very general; it applies to other well-studied partial differential equations such as the Korteweg--de Vries equation, {ital u}{sub {ital t}}+{ital uu}{sub {ital x}} ={ital u}{sub {ital xxx}}.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
NASA Astrophysics Data System (ADS)
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
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.
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.
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.
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.
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
Explicit blow-up solutions to the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2}
Ding Qing
2009-10-15
In this article, we prove that the equation of the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schroedinger-type system of three unknown complex functions p, q, r, and a real function u: iq{sub t}+q{sub zz}-2uq+2(pq){sub z}-2pq{sub z}-4|p|{sup 2}q=0, ir{sub t}-r{sub zz}+2ur+2(pr){sub z}-2pr{sub z}+4|p|{sup 2}r=0, ip{sub t}+(qr){sub z}-u{sub z}=0, p{sub z}+p{sub z}=-|q|{sup 2}+|r|{sup 2}, -r{sub z}+q{sub z}=-2(pr+pq), where z is a complex coordinate of the plane R{sup 2} and z is the complex conjugate of z. Although this nonlinear Schroedinger-type system looks complicated, it admits a class of explicit blow-up smooth solutions: p=0, q=(e{sup i(bzz/2(a+bt))}/a+bt){alpha}z, r=e{sup -i(bzz/2(a+bt))}/(a+bt){alpha}z, u=2{alpha}{sup 2}zz/(a+bt){sup 2}, where a and b are real numbers with ab<0 and {alpha} satisfies {alpha}{sup 2}=b{sup 2}/16. From these facts, we explicitly construct smooth solutions to the Schroedinger maps from R{sup 2} to the hyperbolic 2-space H{sup 2} by using the gauge transformations such that the absolute values of their gradients blow up in finite time. This reveals some blow-up phenomenon of Schroedinger maps.
Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two
NASA Astrophysics Data System (ADS)
Chiron, David; Scheid, Claire
2016-02-01
We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and "one-dimensional spreading" phenomena.
Modulational instability in fractional nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Zhang, Lifu; He, Zenghui; Conti, Claudio; Wang, Zhiteng; Hu, Yonghua; Lei, Dajun; Li, Ying; Fan, Dianyuan
2017-07-01
Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrödinger equation. We derive the MI gain spectrum in terms of the Lévy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Lévy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.
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.
Unleashing Empirical Equations with "Nonlinear Fitting" and "GUM Tree Calculator"
NASA Astrophysics Data System (ADS)
Lovell-Smith, J. W.; Saunders, P.; Feistel, R.
2017-10-01
Empirical equations having large numbers of fitted parameters, such as the international standard reference equations published by the International Association for the Properties of Water and Steam (IAPWS), which form the basis of the "Thermodynamic Equation of Seawater—2010" (TEOS-10), provide the means to calculate many quantities very accurately. The parameters of these equations are found by least-squares fitting to large bodies of measurement data. However, the usefulness of these equations is limited since uncertainties are not readily available for most of the quantities able to be calculated, the covariance of the measurement data is not considered, and further propagation of the uncertainty in the calculated result is restricted since the covariance of calculated quantities is unknown. In this paper, we present two tools developed at MSL that are particularly useful in unleashing the full power of such empirical equations. "Nonlinear Fitting" enables propagation of the covariance of the measurement data into the parameters using generalized least-squares methods. The parameter covariance then may be published along with the equations. Then, when using these large, complex equations, "GUM Tree Calculator" enables the simultaneous calculation of any derived quantity and its uncertainty, by automatic propagation of the parameter covariance into the calculated quantity. We demonstrate these tools in exploratory work to determine and propagate uncertainties associated with the IAPWS-95 parameters.
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.
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.
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.
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.
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.
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.
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.
Loss of Energy Concentration in Nonlinear Evolution Beam Equations
NASA Astrophysics Data System (ADS)
Garrione, Maurizio; Gazzola, Filippo
2017-05-01
Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation u_{tt} + u_{xxxx} + f(u)= g(x, t) in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities f and for some forcing terms g, highlighting some of their structural properties and performing some numerical simulations.
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.
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.
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.
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.
Nonlinear Landau damping of transverse electromagnetic waves in dusty plasmas
Tsintsadze, N. L.; Chaudhary, Rozina; Shah, H. A.; Murtaza, G.
2009-04-15
High-frequency transverse electromagnetic waves in a collisionless isotropic dusty plasma damp via nonlinear Landau damping. Taking into account the latter we have obtained a generalized set of Zakharov equations with local and nonlocal terms. Then from this coupled set of Zakharov equations a kinetic nonlinear Schroedinger equation with local and nonlocal nonlinearities is derived for special cases. It is shown that the modulation of the amplitude of the electromagnetic waves leads to the modulation instability through the nonlinear Landau damping term. The maximum growth rate is obtained for the special case when the group velocity of electromagnetic waves is close to the dust acoustic velocity.
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.
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.
Nonlinear Optical Wave Equation for Micro- and Nano-Structured Media and Its Application
2013-03-01
AFRL-AFOSR-UK-TR-2013-0012 Nonlinear Optical Wave Equation for Micro - and Nano - Structured Media and Its Application Dr...September 2012 4. TITLE AND SUBTITLE Nonlinear Optical Wave Equation for Micro - and Nano - Structured Media and Its...Equation, Nano -structed Media, Nonlinear Fiber Lasers 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT SAR 18, NUMBER OF PAGES 12
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.
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.
Complete integrability of nonlocal nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Gerdjikov, V. S.; Saxena, A.
2017-01-01
Based on the completeness relation for the squared solutions of the Lax operator L, we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schrödinger equations (NLS) is a completely integrable system. The spectral properties of the Lax operator indicate that there are two types of soliton solutions. The relevant action-angle variables are parametrized by the scattering data of the Lax operator. The notion of the symplectic basis, which directly maps the variations of the potential of L to the variations of the action-angle variables has been generalized to the nonlocal case. We also show that the inverse scattering method can be viewed as a generalized Fourier transform. Using the trace identities and the symplectic basis, we construct the hierarchy Hamiltonian structures for the nonlocal NLS equations.
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.
Evaluation of model fit in nonlinear multilevel structural equation modeling
Schermelleh-Engel, Karin; Kerwer, Martin; Klein, Andreas G.
2013-01-01
Evaluating model fit in nonlinear multilevel structural equation models (MSEM) presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are non-normally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of non-normality, they have not yet been investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated. PMID:24624110
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 → ± ∞.
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
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
Robust fast controller design via nonlinear fractional differential equations.
Zhou, Xi; Wei, Yiheng; Liang, Shu; Wang, Yong
2017-07-01
A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
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
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.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
Khare, Avinash; Saxena, Avadh
2014-03-15
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
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.
New Analytical Solution for Nonlinear Shallow Water-Wave Equations
NASA Astrophysics Data System (ADS)
Aydin, Baran; Kânoğlu, Utku
2017-08-01
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.
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.
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
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. PMID:24737972
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.
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.
New variable separation solutions for the generalized nonlinear diffusion equations
NASA Astrophysics Data System (ADS)
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
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.
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}.
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
Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices
Bludov, Yu. V.; Konotop, V. V.
2010-01-15
It is shown that the one-dimensional nonlinear Schroedinger equation with a dissipative periodic potential, nonlinear losses, and a linear pump allow for the existence of stable nonlinear Bloch states which are attractors. The model describes a Bose-Einstein condensate with inelastic two- and three-body interactions loaded in an optical lattice with losses due to inelastic interactions of the atoms with photons.
Tao, Liang; Vanroose, Wim; Reps, Brian; Rescigno, Thomas N.; McCurdy, C. William
2009-09-08
We demonstrate that exterior complex scaling (ECS) can be used to impose outgoing wave boundary conditions exactly on solutions of the time-dependent Schrodinger equation for atoms in intense electromagnetic pulses using finite grid methods. The procedure is formally exact when applied in the appropriate gauge and is demonstrated in a calculation of high harmonic generation in which multiphoton resonances are seen for long pulse durations. However, we also demonstrate that while the application of ECS in this way is formally exact, numerical error can appear for long time propagations that can only be controlled by extending the finite grid. A mathematical analysis of the origins of that numerical error, illustrated with an analytically solvable model, is also given.
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.
Nonlinear periodic waves solutions of the nonlinear self-dual network equations
Laptev, Denis V. Bogdan, Mikhail M.
2014-04-15
The new classes of periodic solutions of nonlinear self-dual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.
Charged anisotropic matter with linear or nonlinear equation of state
Varela, Victor; Rahaman, Farook; Ray, Saibal; Chakraborty, Koushik; Kalam, Mehedi
2010-08-15
Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplifications achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or nonlinear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge (10{sup 19}C) and maximum electric field intensities are very large (10{sup 23}-10{sup 24} statvolt/cm) even in the case of zero net charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations with larger net charges and electric field intensities than those found in studies of charged isotropic fluids. Links of these results with charged strange quark stars as well as models of dark matter including massive charged particles are highlighted. The van der Waals equation of state leading to matter densities constrained by cubic polynomial equations is briefly considered. The fundamental question of stability is left open.
An effective analytic approach for solving nonlinear fractional partial differential equations
NASA Astrophysics Data System (ADS)
Ma, Junchi; Zhang, Xiaolong; Liang, Songxin
2016-08-01
Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.
Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations
NASA Astrophysics Data System (ADS)
Zhang, Linghai
2017-10-01
The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 < 2 (1 + αγ) θ < αβγ; the existence and stability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and γ2 ε > 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 <γ2 ε < 1; the existence and instability of an upside down standing pulse solution if 0 < (1 + αγ) θ < αβγ < 2 (1 + αγ) θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 < 2 θ < β; the existence and stability of two standing wave fronts if 2 θ = β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 < θ < β < 2 θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 < 2 θ < β. To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear
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.
Chaos in the fractional order nonlinear Bloch equation with delay
NASA Astrophysics Data System (ADS)
Baleanu, Dumitru; Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha
2015-08-01
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ = 0 , we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α = 0.8548 with subsequent period doubling that leads to chaos at α = 0.9436 . A periodic window is observed for the range 0.962 < α < 0.9858 , with chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value α = 0.8532 , and the transition from two to four cycles at α = 0.9259 . With further increases in the fractional order, period doubling continues until at α = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α = 0.8441 , and α = 0.8635 , respectively. However, the system exhibits chaos at much lower values of α (α = 0.8635). A periodic window is observed in the interval 0.897 < α < 0.9341 , with chaos again appearing for larger values of α . In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in
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
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…
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.
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.
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.
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.
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.
Soliton solutions to coupled nonlinear wave equations in (2 + 1)-dimensions
NASA Astrophysics Data System (ADS)
Jawad, A. J. M.; Johnson, S.; Yildirim, A.; Kumar, S.; Biswas, A.
2013-03-01
This paper implemented the tanh method to solve a few coupled nonlinear wave equations in (2 + 1)-dimensions. They are the Konopelchenko-Dubrovsky equation, dispersive long wave equation and the Riemann wave equation. Additionally, the traveling wave hypothesis is used to extract a few more solutons to some of these equations. Finally, the numerical simulations supplement these analytical results.
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
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.
Yan, Zhenya; Chen, Yong
2017-07-01
We investigate the nonlinear Schrödinger (NLS) equation with generalized nonlinearities and complex non-Hermitian potentials and present the novel parity-time-( PT-) symmetric potentials for the NLS equation with power-law nonlinearities supporting some bright solitons. For distinct types of PT-symmetric potentials including Scarf-II, Hermite-Gaussian, and asymptotically periodic potentials, we, respectively, explore the phase transitions for the linear Hamiltonian operators. Moreover, we analytically find stable bright solitons in the generalized NLS equations with several types of PT-symmetric potentials, and their stability is corroborated by the linear stability spectrum and direct wave-propagation simulations. Interactions of two solitons are also explored. More interestingly, we find that the nonlinearity can excite the unstable linear modes (i.e., possessing broken linear PT-symmetric phase) to stable nonlinear modes. The results may excite potential applications in nonlinear optics, Bose-Einstein condensates, and relevant fields.
NASA Astrophysics Data System (ADS)
Yan, Zhenya; Chen, Yong
2017-07-01
We investigate the nonlinear Schrödinger (NLS) equation with generalized nonlinearities and complex non-Hermitian potentials and present the novel parity-time-( P T -) symmetric potentials for the NLS equation with power-law nonlinearities supporting some bright solitons. For distinct types of P T -symmetric potentials including Scarf-II, Hermite-Gaussian, and asymptotically periodic potentials, we, respectively, explore the phase transitions for the linear Hamiltonian operators. Moreover, we analytically find stable bright solitons in the generalized NLS equations with several types of P T -symmetric potentials, and their stability is corroborated by the linear stability spectrum and direct wave-propagation simulations. Interactions of two solitons are also explored. More interestingly, we find that the nonlinearity can excite the unstable linear modes (i.e., possessing broken linear P T -symmetric phase) to stable nonlinear modes. The results may excite potential applications in nonlinear optics, Bose-Einstein condensates, and relevant fields.
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.
Propagation of ultrashort polarized light pulses in a nonlinear medium
Maimistov, A.I.
1995-03-01
Propagation of ultrashort optical pulses in a medium with degenerate resonance levels with respect to the angular momentum projections is considered. Under the assumption that the Rabi frequency is much smaller than the transition frequency and without using the slowly varying envelope approximation, a new nonlinear equation is obtained for describing this pulse dynamics. In the particular case when the pulse polarization is not changed, this is the modified Korteweg-de Vries equation. In the approximation of slowly varying envelopes, the reduced wave equation transforms into the vector nonlinear Schroedinger equation. 13 refs.
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.
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.
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.
Bifurcations in a Mathieu equation with cubic nonlinearities: Part II
NASA Astrophysics Data System (ADS)
Ng, Leslie; Rand, Richard
2002-09-01
In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation: d2x /dt 2+(δ+ɛ cost)x+ɛ Ax 3+Bx 2dx /dt +Cx dx /dt 2+D dx /dt 3=0 We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.
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.
Solitons and other solutions to the coupled nonlinear Schrödinger type equations
NASA Astrophysics Data System (ADS)
El-Borai, M. M.; El-Owaidy, H. M.; Ahmed, Hamdy M.; Arnous, A. H.; Mirzazadeh, M.
2017-06-01
Nonlinear Schrödinger type equations arise from a wide variety of fields, such as fluids, nonlinear optics, the theory of deep water waves, plasma physics, and so on. In this paper, two integration schemes are employed to obtain solitons, periodic waves and other forms of solutions of the coupled nonlinear Schrödinger type equations. The two schemes that are studied in this paper are the Bäcklund transformation of Riccati equation and the trial solution method.
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
Gosse, Laurent . E-mail: mauser@univie.ac.at
2006-01-01
This work is concerned with the semiclassical approximation of the Schroedinger-Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a N-body quantum problem, then K-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for Pauli's exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects.
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.
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.
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 Schroedinger's equation using random walks
NASA Astrophysics Data System (ADS)
Aspuru-Guzik, Alan
Exact and almost exact solutions for energies and properties of atoms and molecules can be obtained by quantum Monte Carlo (QMC) methods. This thesis is composed of different contributions to various QMC methodologies, as well as applications to electronic excitations of biological systems. We propose a wave function optimization functional that is robust regarding the presence of outliers. Our work, and subsequent applications by others, has shown the convergence properties and robustness of the absolute deviation (AD) functional as compared to the variance functional (VF). We apply the method to atoms from the second row of the periodic table, as well as third-row transition metal atoms, including an all-electron calculation of Sc. In all cases, the AD functional converges faster than the VF. Soft effective core potentials (ECPs) with no divergence at the origin are constructed and validated for second- an third-row atoms of the periodic table. The ECPs we developed have been used by others in several successful studies. As an application of the DMC approach to biochemical problems, we studied the electronic excitations of free-base porphyrin and obtained results in excellent agreement with experiment. These findings validate the use of the DMC approach for these kinds of systems. A study of the role of spheroidene in the photo-protection mechanism of Rhodobacter sphaeroides is described. At the time of writing, calculations for the estimation of excitation energies for the bacteriochlorophyll and spheroidene molecules as well as storage of the random walkers for future prediction of the excitation energy transfer rate are being performed. To date, the calculations mentioned above are the largest all-electron studies on molecules. For the computation of these systems, a sparse linear-scaling DMC algorithm was developed. This algorithm provides a speedup of at least a factor of ten over previously published methods. The method is validated on systems up to 390 electrons. A summary of the Fermion Monte Carlo (FMC) algorithm as well as an application to the Be atom are discussed. The Zori package, a linear-scaling massively-parallel open-source program that uses modern programming libraries, was developed. The program is made available to the public under the GNU/General Public License (GPL). The capabilities of the Zori program are summarized.
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Tadmor, Eitan
1989-01-01
Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.
Nonlinear 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.
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.
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.
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
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.
NASA Astrophysics Data System (ADS)
Wu, Hong-Yu; Jiang, Li-Hong
2017-09-01
From a generic transformation, a (3+1)-dimensional nonlinear Schrödinger equation with spatially modulated nonlinearity is studied and exact spatiotemporal soliton cluster solutions are derived. When the azimuthal parameter m = 0, Gaussian solitons are constructed. For the modulation depth q = 1 and the azimuthal parameter m\
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.
Maximum Likelihood Analysis of Nonlinear Structural Equation Models with Dichotomous Variables
ERIC Educational Resources Information Center
Song, Xin-Yuan; Lee, Sik-Yum
2005-01-01
In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the…
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.
Localized solutions of extended discrete nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Umarov, B. A.; Ismail, Nazmi Hakim Bin; Hadi, Muhammad Salihi Abdul; Hassan, T. H.
2017-09-01
We consider the extended discrete nonlinear Schrödinger (EDNLSE) equation which includes the nearest neighbor nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. This equation describes nonlinear excitations in dipolar Bose-Einstein condensate in a periodic optical lattice. We are particularly interested with the existence and stability conditions of localized nonlinear excitations of different types. The problem is tackled numerically, by application of Newton methods and by solving the eigenvalue problem for linearized system near the exact solution. Also the modulational instability of plane wave solution is discussed.
NASA Astrophysics Data System (ADS)
Ndzana, Fabien, II; Mohamadou, Alidou; Kofané, Timoléon Crépin
2007-07-01
The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE's) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE's, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.
Linear homotopy solution of nonlinear systems of equations in geodesy
NASA Astrophysics Data System (ADS)
Paláncz, Béla; Awange, Joseph L.; Zaletnyik, Piroska; Lewis, Robert H.
2010-01-01
A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton-Raphson.
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}.
Analytical approximate solution for nonlinear space—time fractional Klein—Gordon equation
NASA Astrophysics Data System (ADS)
Khaled, A. Gepreel; Mohamed, S. Mohamed
2013-01-01
The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space—time fractional derivatives Klein—Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space—time fractional derivatives Klein—Gordon equation. This method introduces a promising tool for solving many space—time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.
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
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.
A family of nonlinear Schrödinger equations admitting q-plane wave solutions
NASA Astrophysics Data System (ADS)
Nobre, F. D.; Plastino, A. R.
2017-08-01
Nonlinear Schrödinger equations with power-law nonlinearities have attracted considerable attention recently. Two previous proposals for these types of equations, corresponding respectively to the Gross-Pitaievsky equation and to the one associated with nonextensive statistical mechanics, are here unified into a single, parameterized family of nonlinear Schrödinger equations. Power-law nonlinear terms characterized by exponents depending on a real index q, typical of nonextensive statistical mechanics, are considered in such a way that the Gross-Pitaievsky equation is recovered in the limit q → 1. A classical field theory shows that, due to these nonlinearities, an extra field Φ (x → , t) (besides the usual one Ψ (x → , t)) must be introduced for consistency. The new field can be identified with Ψ* (x → , t) only when q → 1. For q ≠ 1 one has a pair of coupled nonlinear wave equations governing the joint evolution of the complex valued fields Ψ (x → , t) and Φ (x → , t). These equations reduce to the usual pair of complex-conjugate ones only in the q → 1 limit. Interestingly, the nonlinear equations obeyed by Ψ (x → , t) and Φ (x → , t) exhibit a common, soliton-like, traveling solution, which is expressible in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics.
A simple and direct method for generating travelling wave solutions for nonlinear equations
Bazeia, D. Das, Ashok; Silva, A.
2008-05-15
We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.
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.
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.
NASA Astrophysics Data System (ADS)
Xie, Xi-Yang; Tian, Bo; Liu, Lei; Guan, Yue-Yang; Jiang, Yan
2017-06-01
In this paper, we investigate a generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. Under certain integrable constraints, bilinear forms, bright one- and two-soliton solutions are obtained. Via certain transformation, we investigate the properties of the solitons with the first-order dispersion parameter σ1(x, t), second-order dispersion parameter σ2(x, t), third-order dispersion parameter σ3(x, t), phase modulation and gain (loss) v(x, t). Soliton propagation and collision are graphically presented and analyzed: One soliton is shown to maintain its amplitude and width during the propagation. When we choose σ1(x, t), σ2(x, t) and σ3(x, t) differently, travelling direction of the soliton is found to alter. v(x, t) is observed to affect the amplitude of the soliton. Head-on collision between the two solitons is presented with σ1(x, t), σ2(x, t), σ3(x, t) and v(x, t) as the constants, and solitons' amplitudes are the same before and after the collision. When σ1(x, t), σ2(x, t) and σ3(x, t) are chosen as certain functions, the solitons' traveling directions change during the collision. v(x, t) can influence the amplitudes of the two solitons.
Erwin Schroedinger, Francis Crick and epigenetic stability
Ogryzko, Vasily V
2008-01-01
Schroedinger's book 'What is Life?' is widely credited for having played a crucial role in development of molecular and cellular biology. My essay revisits the issues raised by this book from the modern perspective of epigenetics and systems biology. I contrast two classes of potential mechanisms of epigenetic stability: 'epigenetic templating' and 'systems biology' approaches, and consider them from the point of view expressed by Schroedinger. I also discuss how quantum entanglement, a nonclassical feature of quantum mechanics, can help to address the 'problem of small numbers' that led Schroedinger to promote the idea of a molecular code-script for explaining the stability of biological order. PMID:18419815
Erwin Schroedinger, Francis Crick and epigenetic stability.
Ogryzko, Vasily V
2008-04-17
Schroedinger's book 'What is Life?' is widely credited for having played a crucial role in development of molecular and cellular biology. My essay revisits the issues raised by this book from the modern perspective of epigenetics and systems biology. I contrast two classes of potential mechanisms of epigenetic stability: 'epigenetic templating' and 'systems biology' approaches, and consider them from the point of view expressed by Schroedinger. I also discuss how quantum entanglement, a nonclassical feature of quantum mechanics, can help to address the 'problem of small numbers' that led Schroedinger to promote the idea of a molecular code-script for explaining the stability of biological order.
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.
Linear and nonlinear propagation of water wave groups
NASA Technical Reports Server (NTRS)
Pierson, W. J., Jr.; Donelan, M. A.; Hui, W. H.
1992-01-01
Results are presented from a study of the evolution of waveforms with known analytical group shapes, in the form of both transient wave groups and the cloidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schroedinger equation. The waveforms were generated in a long wind-wave tank of the Canada Centre for Inland Waters. It was found that the low-amplitude transients behaved as predicted by the linear theory and that the cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schroedinger equation. Some of the results did not fit into any of the available theories for waves on water, but they provide important insight on how actual groups of waves propagate and on higher-order effects for a transient waveform.
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.
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.
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.
A nonlinear Klein–Gordon equation for relativistic superfluidity
NASA Astrophysics Data System (ADS)
Waldron, Oliver; Van Gorder, Robert A.
2017-10-01
Many neutron star features can be accurately modeled only if one assumes that a significant portion of the neutron star interior is in a superfluid state and if relativitic effects are considered, and possible solutions to the underlying mathematical models include vortex solutions. It was recently shown that vorticity in relativistic superfluids can be studied under the framework of a nonlinear Klein–Gordon (NLKG) model in general curvilinear coordinates where the phase dynamics of solutions to this equation give rise to superfluidity (Xiong et al 2014 Phys. Rev. D 90 125019), and some numerical solutions were obtained. The aim of this paper will be to extract asymptotic solutions to obtain a better qualitative understanding of the possible relativistic superfluid dynamics possible under the NLKG model. We obtain asymptotic results for both spherically symmetric and cylindrically symmetric solutions, demonstrating that the solutions actually appear more regular in the relativistic regime compared to the non-relativistic limit. In fact, the asymptotic and numerical solutions actually show the best agreement in the relativistic case. We demonstrate that the relativistic effects actually tend to regularize or stabilize the solutions, relative to the non-relativistic solutions, which is an interesting finding. We then obtain a Thomas–Fermi-like perturbation result in the very large-mass limit where the kinetics become negligible relative to the self-interaction term (at leading order). We finally extend the NLKG model by assuming a curved spacetime with a metric generally used to model the space surrounding a neutron star, which is a novel generalization of the NLKG model to curved spacetime. We again obtain solutions in the large-mass limit for this case, and find that for such a spacetime non-stationary states (rather than simply stationary states) are possible in the large-mass limit.
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.
Equations for description of nonlinear standing waves in constant-cross-sectioned resonators.
Bednarik, Michal; Cervenka, Milan
2014-03-01
This work is focused on investigation of applicability of two widely used model equations for description of nonlinear standing waves in constant-cross-sectioned resonators. The investigation is based on the comparison of numerical solutions of these model equations with solutions of more accurate model equations whose validity has been verified experimentally in a number of published papers.
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.
NASA Astrophysics Data System (ADS)
Nazari, Farshid; Mohammadian, Abdolmajid; Zadra, Ayrton; Charron, Martin
2013-03-01
Stability concerns are always a factor in the numerical solution of nonlinear diffusion equations, which are a class of equations widely applicable in different fields of science and engineering. In this study, a modified extended backward differentiation formulae (ME BDF) scheme is adapted for the solution of nonlinear diffusion equations, with a special focus on the atmospheric boundary layer diffusion process. The scheme is first implemented and examined for a widely used nonlinear ordinary differential equation, and then extended to a system of two nonlinear diffusion equations. A new temporal filter which leads to significant improvement of numerical results is proposed, and the impact of the filter on the stability and accuracy of the results is investigated. Noteworthy improvements are obtained as compared to other commonly used numerical schemes. Linear stability analysis of the proposed scheme is performed for both systems, and analytical stability limits are presented.
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.
Eigenvalue cutoff in the cubic-quintic nonlinear Schrödinger equation.
Prytula, Vladyslav; Vekslerchik, Vadym; Pérez-García, Víctor M
2008-08-01
Using theoretical arguments, we prove the numerically well-known fact that the eigenvalues of all localized stationary solutions of the cubic-quintic (2+1) -dimensional nonlinear Schrödinger equation exhibit an upper cutoff value. The existence of the cutoff is inferred using Gagliardo-Nirenberg and Hölder inequalities together with Pohozaev identities. We also show that, in the limit of eigenvalues close to zero, the eigenstates of the cubic-quintic nonlinear Schrödinger equation behave similarly to those of the cubic nonlinear Schrödinger equation.
Finding all solutions of nonlinear equations using the dual simplex method
NASA Astrophysics Data System (ADS)
Yamamura, Kiyotaka; Fujioka, Tsuyoshi
2003-03-01
Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.
Nonexistence of small, odd breathers for a class of nonlinear wave equations
NASA Astrophysics Data System (ADS)
Kowalczyk, Michał; Martel, Yvan; Muñoz, Claudio
2017-05-01
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)
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.
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.
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.
Spectral Target Detection using Schroedinger Eigenmaps
NASA Astrophysics Data System (ADS)
Dorado-Munoz, Leidy P.
Applications of optical remote sensing processes include environmental monitoring, military monitoring, meteorology, mapping, surveillance, etc. Many of these tasks include the detection of specific objects or materials, usually few or small, which are surrounded by other materials that clutter the scene and hide the relevant information. This target detection process has been boosted lately by the use of hyperspectral imagery (HSI) since its high spectral dimension provides more detailed spectral information that is desirable in data exploitation. Typical spectral target detectors rely on statistical or geometric models to characterize the spectral variability of the data. However, in many cases these parametric models do not fit well HSI data that impacts the detection performance. On the other hand, non-linear transformation methods, mainly based on manifold learning algorithms, have shown a potential use in HSI transformation, dimensionality reduction and classification. In target detection, non-linear transformation algorithms are used as preprocessing techniques that transform the data to a more suitable lower dimensional space, where the statistical or geometric detectors are applied. One of these non-linear manifold methods is the Schroedinger Eigenmaps (SE) algorithm that has been introduced as a technique for semi-supervised classification. The core tool of the SE algorithm is the Schroedinger operator that includes a potential term that encodes prior information about the materials present in a scene, and enables the embedding to be steered in some convenient directions in order to cluster similar pixels together. A completely novel target detection methodology based on SE algorithm is proposed for the first time in this thesis. The proposed methodology does not just include the transformation of the data to a lower dimensional space but also includes the definition of a detector that capitalizes on the theory behind SE. The fact that target pixels and
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.
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)
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.
Equations of state and bump Cepheids. II - Non-linear results
NASA Astrophysics Data System (ADS)
Kanbur, Shashi M.
1992-06-01
The Hummer-Mihalas-Dappen (MHD, 1988) and a simple Saha type equation of state are used to obtain nonlinear pulsation characteristics of a grid of models spanning the Hertzsprung sequence (Cox, 1974). The grid of models is taken from Simon and Davis (1983) who used the Los Alamos equation of state in their computations. The result is a sensitivity analysis of theoretical nonlinear bump Cepheid models to the equation of state employed in the calculation. The results obtained with these equations of state are different, though not enough to resolve the Cepheid bump mass discrepancy (Stobie 1969; Simon and Schmidt, 1976; Simon, 1986).
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.
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}-\
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.
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
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).
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.
How to Solve Schroedinger Problems by Approximating the Potential Function
Ledoux, Veerle; Van Daele, Marnix
2010-09-30
We give a survey over the efforts in the direction of solving the Schroedinger equation by using piecewise approximations of the potential function. Two types of approximating potentials have been considered in the literature, that is piecewise constant and piecewise linear functions. For polynomials of higher degree the approximating problem is not so easy to integrate analytically. This obstacle can be circumvented by using a perturbative approach to construct the solution of the approximating problem, leading to the so-called piecewise perturbation methods (PPM). We discuss the construction of a PPM in its most convenient form for applications and show that different PPM versions (CPM,LPM) are in fact equivalent.
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.
NASA Astrophysics Data System (ADS)
Singh, Prince; Sharma, Dinkar
2017-07-01
Series solution is obtained on solving non-linear fractional partial differential equation using homotopy perturbation transformation method. First of all, we apply homotopy perturbation transformation method to obtain the series solution of non-linear fractional partial differential equation. In this case, the fractional derivative is described in Caputo sense. Then, we present the facts obtained by analyzing the convergence of this series solution. Finally, the established fact is supported by an example.
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.
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.
Nonlinear System Of Equations For Multicomponent Analysis Of Artificial Food Coloring
NASA Astrophysics Data System (ADS)
Santosa, I. E.; Budiasih, L. K.
2010-12-01
In multicomponent analysis of artificial food coloring (AFC), nonlinear relation of the absorbance and the concentration forms a nonlinear system of equations. The Newton's method based algorithm has been used to calculate individual AFC concentration in the mixture of two AFCs. The absorbance was measured using a spectrophotometer at two different wavelengths.
Garbow, B.S.; Hillstrom, K.E.; More, J.J.
1980-07-01
MINPACK-1 is a package of Fortran subprograms for the numerical solution of systems of nonlinear equations and nonlinear least-squares problems. This report describes how to implement the package from the tape on which it is transmitted. 3 tables.
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.
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.
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.
2014-01-01
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
Experiments with neutron Schroedinger waves
NASA Astrophysics Data System (ADS)
Arif, Muhammad
1986-09-01
Four neutron Schroedinger wave interference experiments are described: (1) null Fizeau effect for thermal neutrons in thermal matter; (2) precision measurements of the energy dependent scattering length of U-235; (3) gravitationally induced quantum interference; and (4) investigation of incident beam coherence questions in dynamical diffraction experiments. In the Fizeau effect experiment it was verified that a neutron de Broglie wave is not affected by the state of motion of matter within fixed boundaries through which the neutron is propagating. Upper limits for the energy dependence of the neutron optical potential and the wavelength dependence of s-wave neutron scattering lengths were deduced. Using the neutron interferometric technique the energy dependent bound coherent scattering lengths of U-235 were directly measured in the energy range of 30 to 92 meV with an error of 0.3%. In the gravity induced quantum interference experiments (COW), the results showed that the observed loss of contrast of the interference pattern is caused in part by the bending of the perfect crystal neutron interferometer. Lastly, mathematical and numerical calculation techniques are presented to investigate the effect of wavefront modulation and transverse coherence of an incident neutron beam in dynamical diffraction experiments. Experimental and theoretical pendelloesung interference patterns in perfect silicon crystal were obtained and compared.
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.
Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations
Hong Jialin . E-mail: lichun@lsec.cc.ac.cn
2006-01-20
In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law.
Linear Integro-differential Schroedinger and Plate Problems Without Initial Conditions
Lorenzi, Alfredo
2013-06-15
Via Carleman's estimates we prove uniqueness and continuous dependence results for the temporal traces of solutions to overdetermined linear ill-posed problems related to Schroedinger and plate equation. The overdetermination is prescribed in an open subset of the (space-time) lateral boundary.
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.
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.
Nonlinear waves: Dynamics and evolution
NASA Astrophysics Data System (ADS)
Gaponov-Grekhov, A. V.; Rabinovich, M. I.
Papers on nonlinear waves are presented, covering topics such as the history of studies on nonlinear dynamics since Poincare, attractors, pattern formation and the dynamics of two-dimensional structures in nonequilibirum dissipative media, the onset of spatial chaos in one-dimensional systems, and self-organization phenomena in laser thermochemistry. Additional topics include criteria for the existence of moving structures in two-component reaction-diffusion systems, space-time structures in optoelectronic devices, stimulated scattering and surface structures, and distributed wave collapse in the nonlinear Schroedinger equation. Consideration is also given to dimensions and entropies in multidimensional systems, measurement methods for correlation dimensions, quantum localization and dynamic chaos, self-organization in bacterial cells and populations, nonlinear phenomena in condensed matter, and the origin and evolutionary dynamics of Uranian rings.
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.
NASA Astrophysics Data System (ADS)
Kudryashov, Nikolay A.
2015-11-01
The fourth-order equation for description of nonlinear waves is considered. A few variants of this equation are studied. Painlevé test is applied to investigate integrability of these equations. We show that all these equations are not integrable, but some exact solutions of these equations exist. Analytic solutions in closed-form of the equations are found.
On the solution of the fractional nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Rida, S. Z.; El-Sherbiny, H. M.; Arafa, A. A. M.
2008-01-01
We present the nonlinear Schrödinger (NLS) equation of fractional order. The fractional derivatives are described in the Caputo sense. The Adomian decomposition method (ADM) in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution constructed in power series with easily computable components.
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.
NASA Astrophysics Data System (ADS)
Singla, Komal; Gupta, R. K.
2017-05-01
In Paper I [Singla, K. and Gupta, R. K., J. Math. Phys. 57, 101504 (2016)], Lie symmetry method is developed for time fractional systems of partial differential equations. In this article, the Lie symmetry approach is proposed for space-time fractional systems of partial differential equations and applied to study some well-known physically significant space-time fractional nonlinear systems successfully.
Initial study of Schroedinger eigenmaps for spectral target detection
NASA Astrophysics Data System (ADS)
Dorado-Munoz, Leidy P.; Messinger, David W.
2016-08-01
Spectral target detection refers to the process of searching for a specific material with a known spectrum over a large area containing materials with different spectral signatures. Traditional target detection methods in hyperspectral imagery (HSI) require assuming the data fit some statistical or geometric models and based on the model, to estimate parameters for defining a hypothesis test, where one class (i.e., target class) is chosen over the other classes (i.e., background class). Nonlinear manifold learning methods such as Laplacian eigenmaps (LE) have extensively shown their potential use in HSI processing, specifically in classification or segmentation. Recently, Schroedinger eigenmaps (SE), which is built upon LE, has been introduced as a semisupervised classification method. In SE, the former Laplacian operator is replaced by the Schroedinger operator. The Schroedinger operator includes by definition, a potential term V that steers the transformation in certain directions improving the separability between classes. In this regard, we propose a methodology for target detection that is not based on the traditional schemes and that does not need the estimation of statistical or geometric parameters. This method is based on SE, where the potential term V is taken into consideration to include the prior knowledge about the target class and use it to steer the transformation in directions where the target location in the new space is known and the separability between target and background is augmented. An initial study of how SE can be used in a target detection scheme for HSI is shown here. In-scene pixel and spectral signature detection approaches are presented. The HSI data used comprise various target panels for testing simultaneous detection of multiple objects with different complexities.
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.
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.
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.
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.
Distribution-valued initial data for the complex Ginzburg-Landau equation
Levermore, C.D.; Oliver, M.
1997-11-01
The generalized complex Ginzburg-Landau (CGL) equation with a nonlinearity of order 2{sigma} + 1 in d spatial dimensions has a unique local classical solution for distributional initial data in the Sobolev space H{sup q} provided that q > d/2 - 1/{sigma}. This result directly corresponds to a theorem for the nonlinear Schroedinger (NLS) equation which has been proved by Cazenave and Weissler in 1990. While the proof in the NLS case relies on Besov space techniques, it is shown here that for the CGL equation, the smoothing properties of the linear semigroup can be eased to obtain an almost optimal result by elementary means. 1 fig.
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.
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.
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
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.
Finite-difference schemes for nonlinear wave equation that inherit energy conservation property
NASA Astrophysics Data System (ADS)
Furihata, Daisuke
2001-09-01
We propose two general finite-difference schemes that inherit energy conservation property from nonlinear wave equations, such as the nonlinear Klein-Gordon equation (NLKGE). One of proposed schemes is implicit and another is explicit. Many studies exist on FDSs that inherit energy conservation property from NLKGE and we can derive all of their schemes from the proposed general schemes in this paper. The most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts, which implies that the inherited properties are satisfied exactly. Because of this the derived schemes are expected to be numerically stable and yield solutions converging to PDE solutions. We make new FDSs for Fermi-Pasta-Ulam equation, string vibration equation, Shimoji-Kawai equation (SKE) and Ebihara equation and verify numerically the inheritance of the energy conservation property for NLKGE and SKE.
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
Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order
NASA Astrophysics Data System (ADS)
Pradip, Roul
2013-09-01
The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy—perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.
Non-iterative solution of the Schroedinger Eq. in the presence of exchange terms.
NASA Astrophysics Data System (ADS)
Rawitscher, George H.; Kang, S.-Y.; Koltracht, I.
2000-06-01
In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schroedinger Eq. of an integro-differential form. We show that this equation can be solved non-iteratively by the same integral equation algorithm developed previously [1] for local potentials. This holds for non-localities of the exchange type, since a) the corresponding integration kernel is semi-separable, b) the convolution of the semi-separable exchange kernel with the semi-separable Green's function kernel is also of semi-separable form, and c) the integral equation method works well with semi-separable kernels. Numerical examples for electron-hydrogen scattering will be presented, and comparisons with existing iterative methods will be given. [1] R. A. Gonzales et. al., ''Integral Equation Method for Coupled Schroedinger Equations'', J. Comput. Phys., 153, 160 (1999).
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).
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.
Nonlinear heat conduction equations with memory: Physical meaning and analytical results
NASA Astrophysics Data System (ADS)
Artale Harris, Pietro; Garra, Roberto
2017-06-01
We study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell-Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell-Cattaneo law, based on the application of long-tail Mittag-Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations. We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case.
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.
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.
Belyi, V.V.; Kukharenko, Y.A.; Wallenborn, J. ||
1996-05-01
Taking into account the first non-Markovian correction to the Balescu-Lenard equation, we have derived an expression for the pair correlation function and a nonlinear kinetic equation valid for a nonideal polarized classical plasma. This last equation allows for the description of the correlational energy evolution and shows the global conservation of energy with dynamical polarization. {copyright} {ital 1996 The American Physical Society.}
Application of variational and Galerkin equations to linear and nonlinear finite element analysis
NASA Technical Reports Server (NTRS)
Yu, Y.-Y.
1974-01-01
The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.
Nonlinear 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
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.
NASA Astrophysics Data System (ADS)
Zhou, J. B.; Xu, J.; Wei, J. D.; Yang, X. Q.
2017-04-01
This paper is concerned with the existence of travelling wave solutions to a singularly perturbed generalized Gardner equation with nonlinear terms of any order. By using geometric singular perturbation theory and based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of solitary wave solutions of this equation is proved when the perturbation parameter is sufficiently small. The numerical simulations verify our theoretical analysis.
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 static and dynamic analysis of beam structures using fully intrinsic equations
NASA Astrophysics Data System (ADS)
Sotoudeh, Zahra
2011-07-01
Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis. This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables. In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research. This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.
Virtual charts of solutions for parametrized nonlinear equations
NASA Astrophysics Data System (ADS)
Vitse, Matthieu; Néron, David; Boucard, Pierre-Alain
2014-12-01
During many decades, engineers relied on experimental abaci to design and optimize mechanical structures. Virtual charts are based on numerical computations and aim at making the design and optimization of structures faster and cheaper as it requires less (or no) experimentation. The parametric problems are solved offline and the associated charts are used for online design. However, in this context, the cost associated to the resolution of parametric problems can be extremely prohibitive. Model reduction techniques are an answer to circumvent this issue. This paper provides the developments of an algorithm for solving nonlinear parametric problems, based on the LATIN method for the treatment of the nonlinear aspects and the PGD for the parameter dependency. A full time-space-parameter decomposition of the solution is introduced into the LATIN algorithm, numerical examples on academic problems are given so as to point out the advantages of this extension and leads on possible improvements to the strategy are given.
Shah, A A; Xing, W W; Triantafyllidis, V
2017-04-01
In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.
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.
Four-photon homoclinic instabilities in nonlinear highly birefringent media
De Angelis, C.; Santagiustina, M. ); Trillo, S. )
1995-01-01
We investigate the nonlinear dynamics of a nonconventional (i.e., pumped by a mixed-mode wave) modulational instability in a highly birefringent nonlinear dispersive medium. We find that the depleted regime of propagation beyond the linearized stage can be described analytically in a proper region of the parameter space. In this case the governing coupled nonlinear Schroedinger equations, which are not integrable, are reduced to an integrable one-dimensional nonlinear oscillator that rules the propagation of the pump wave and a single sideband pair. This approach permits us to predict the existence of stable and unstable manifolds of time-periodic solutions of the coupled nonlinear Schroedinger equations. The nonlinear dynamics governed by these equations mimics the period-doubling instabilities associated with the homoclinic separatrices in the reduced phase space. Moreover, our approach is also capable of describing the onset of spatial chaos that occurs when the parameter values are such that the additional degree of freedom represented by the conjugated sidebands becomes effective.
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
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.
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.
Polynomial elimination theory and non-linear stability analysis for the Euler equations
NASA Technical Reports Server (NTRS)
Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.
1986-01-01
Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.
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
NASA Astrophysics Data System (ADS)
Triki, Houria; Porsezian, K.; Choudhuri, Amitava
2017-06-01
A nonlinear Schrödinger equation that includes two terms with power-law nonlinearity and external potential modulated both on time and on the spatial coordinates is considered. The model appears in various branches of contemporary physics, especially in the case of lower values of the nonlinearity power. A significant generalization of the similarity transformations approach to construct explicit localized solutions for the model with arbitrary power-law nonlinearities is introduced. We obtain the exact analytical bright and kink soliton solutions of the governing equation for different nonlinearities and potentials that are of particular interest in applications to Bose-Einstein condensates and nonlinear optics. Necessary conditions on the physical parameters for propagating envelope formation are presented. The obtained results can be straightforwardly applied to a large variety of nonlinear Schrödinger models and hence would be of value to understand nonlinear phenomena in a diversity of nonlinear media.
Triki, Houria; Porsezian, K; Choudhuri, Amitava
2017-06-01
A nonlinear Schrödinger equation that includes two terms with power-law nonlinearity and external potential modulated both on time and on the spatial coordinates is considered. The model appears in various branches of contemporary physics, especially in the case of lower values of the nonlinearity power. A significant generalization of the similarity transformations approach to construct explicit localized solutions for the model with arbitrary power-law nonlinearities is introduced. We obtain the exact analytical bright and kink soliton solutions of the governing equation for different nonlinearities and potentials that are of particular interest in applications to Bose-Einstein condensates and nonlinear optics. Necessary conditions on the physical parameters for propagating envelope formation are presented. The obtained results can be straightforwardly applied to a large variety of nonlinear Schrödinger models and hence would be of value to understand nonlinear phenomena in a diversity of nonlinear media.
Non-linear corrections in orbital perturbation equations for CHAMP-like satellite
NASA Astrophysics Data System (ADS)
Zhu, Yongchao; Yu, Jinhai; Meng, Xiangchao
2016-04-01
Based on the theory of the dynamic approach which can estimate the gravity field models using the CHAMP-like satellite's SST data,the article discusses the linearization of the orbital perturbation equation in detail. It is analyzed that under the accuracy levels of the GRACE non-gravitational measurement 3.0×m/s2the residual orbits between the real orbit and reference orbit can't exceed 12 meters due to the linearization. As the omitted terms in the derivations can be expressed clearly, the observation equations of the spherical harmonic coefficients of gravitational field are established for the GRACE based on the solutions of orbital perturbation equations including the nonlinear corrections. Especially,it is analyzed and examined by numerical simulations for GRACE mission. Compared with the linearized orbital perturbation differential equations, the one with the nonlinear corrections has two improvements. Firstly,under the measurement accuracy of the non-gravitational accelerations reaching to 3×10-10m/s2, the residual orbit between the real orbit and reference orbit are not more than 3 km. It means the long time-arc can also recover the gravity field at the same accuracy level by the nonlinear equation,compared with the linearized equation using short arc. Secondly,the nonlinear method can improve two to three degrees using short arc relative to the linear method.
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.
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).
Dromion interactions of (2+1)-dimensional nonlinear evolution equations
Ruan; Chen
2000-10-01
Starting from two line solitons, the solution of integrable (2+1)-dimensional mKdV system and KdV system in bilinear form yields a dromion solution or a "Solitoff" solution. Such a dromion solution is localized in all directions and the Solitoff solution decays exponentially in all directions except a preferred one for the physical field or a suitable potential. The interactions between two dromions and between the dromion and Solitoff are studied by the method of figure analysis for a (2+1)-dimensional modified KdV equation and a (2+1)-dimensional KdV type equation. Our analysis shows that the interactions between two dromions may be elastic or inelastic for different forms of solutions.
Soliton solutions and conservation laws for lossy nonlinear transmission line equation
NASA Astrophysics Data System (ADS)
Tchier, Fairouz; Yusuf, Abdullahi; Aliyu, Aliyu Isa; Inc, Mustafa
2017-07-01
In this article, the Lie symmetry and Ricatti-Bernoulli (RB) sub-ODE method are applied to obtain soliton solutions for nonlinear transmission line equation (NLTLs). The NLTLs is defined to be a structure whereby a short-duration pulses known as electrical solitons can be invented and disseminated. We compute conservation laws (Cls) via a non-linear self-adjointness approach. A suitable substitution for NLTLs is found and the obtained substitution makes the NLTLs equation a non-linearly self-adjoint. We establish Cls for NLTLs equation by the new Cls theorem presented by Ibragimov. We obtain trigonometric, algebraic and soliton solutions. The obtained solutions can be useful for describing the concentrations of transmission lines problems, for NLTLs. The parameters of the transmission line play a significant role in managing the original form of the soliton.
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
Kanoglu, Utku; Synolakis, Costas
2006-10-06
The initial value problem solution of the nonlinear shallow water-wave equations is developed under initial waveforms with and without velocity. We present a solution method based on a hodograph-type transformation to reduce the nonlinear shallow water-wave equations into a second-order linear partial differential equation and we solve its initial value problem. The proposed solution method overcomes earlier limitation of small waveheights when the initial velocity is nonzero, and the definition of the initial conditions in the physical and transform spaces is consistent. Our solution not only allows for evaluation of differences in predictions when specifying an exact initial velocity based on nonlinear theory and its linear approximation, which has been controversial in geophysical practice, but also helps clarify the differences in runup observed during the 2004 and 2005 Sumatran tsunamigenic earthquakes.