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.
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.
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.
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.
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.
Blow up of solutions to the nonlinear Schroedinger equations on manifolds
Ma Li; Zhao Lin
2007-05-15
In this paper, we partially settle down the long-standing open problem of the finite time blow-up property about the nonlinear Schroedinger equations on some Riemannian manifolds such as the standard 2-sphere S{sup 2} and the hyperbolic 2-space H{sup 2}(-1). Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic n-space. Extensions to n-dimensional Riemannian warped product manifolds with n{>=}2 are also given.
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.
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.
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.
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.
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.
Vortex Nucleation in a Dissipative Variant of the Nonlinear Schroedinger Equation Under Rotation
2014-12-01
For this purpose we have pro- duced long-term simulations of the DGPE (2.3) starting from the stationary state bearing no vorticity. Two typical...address: http://nonlinear.sdsu.edu/ carreter/RotatingBEC.html [Movies#1 and #2]. The simulations were chosen for rotations that are slightly above critical...periphery. This is precisely what we observe in numerical simulations where smaller values of γ give rise to final configurations with a smaller
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.
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.
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.
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.
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})
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.
1989-01-01
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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
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.
Nonlinear Poisson equation for heterogeneous media.
Hu, Langhua; Wei, Guo-Wei
2012-08-22
The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.
Nonlinear Poisson Equation for Heterogeneous Media
Hu, Langhua; Wei, Guo-Wei
2012-01-01
The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects. PMID:22947937
Nonlinear gyrokinetic equations for tokamak microturbulence
Hahm, T.S.
1988-05-01
A nonlinear electrostatic gyrokinetic Vlasov equation, as well as Poisson equation, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.
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.
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.
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.
Algorithms For Integrating Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
Exact solutions for nonlinear foam drainage equation
NASA Astrophysics Data System (ADS)
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2017-02-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Exact solutions for nonlinear foam drainage equation
NASA Astrophysics Data System (ADS)
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2016-09-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G) -expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
A combination method for solving nonlinear equations
NASA Astrophysics Data System (ADS)
Silalahi, B. P.; Laila, R.; Sitanggang, I. S.
2017-01-01
This paper discusses methods for finding solutions of nonlinear equations: the Newton method, the Halley method and the combination of the Newton method, the Newton inverse method and the Halley method. Computational results in terms of the accuracy, the number of iterations and the running time for solving some given problems are presented.
The Stochastic Nonlinear Damped Wave Equation
Barbu, V. Da Prato, G.
2002-12-19
We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in R{sup n} of the Klein-Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied.
Taming the nonlinearity of the Einstein equation.
Harte, Abraham I
2014-12-31
Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically interesting cases where metrics become approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations that directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well.
Explicit integration of Friedmann's equation with nonlinear equations of state
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong E-mail: gwg1@damtp.cam.ac.uk
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
NASA Technical Reports Server (NTRS)
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
Forces Associated with Nonlinear Nonholonomic Constraint Equations
NASA Technical Reports Server (NTRS)
Roithmayr, Carlos M.; Hodges, Dewey H.
2010-01-01
A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.
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
Numerical solutions of nonlinear wave equations
Kouri, D.J.; Zhang, D.S.; Wei, G.W.; Konshak, T.; Hoffman, D.K.
1999-01-01
Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equation are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds. Earlier local, grid-based numerical studies have encountered difficulties, including numerically induced chaos for such initial conditions. The present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple, explicit method for the time evolution. The numerical solutions are chaos-free for the same conditions employed in previous work that encountered chaos. Moreover, stable results that are free of homoclinic-orbit crossing are obtained even when initial conditions are within 10{sup {minus}7} of the phase space separatrix value {pi}. It also is found that the present approach yields extremely accurate solutions for the Korteweg{endash}de Vries and nonlinear Schr{umlt o}dinger equations. Our results support Ablowitz and co-workers{close_quote} conjecture that ensuring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations. {copyright} {ital 1999} {ital The American Physical Society}
Invariant metrics, contractions and nonlinear matrix equations
NASA Astrophysics Data System (ADS)
Lee, Hosoo; Lim, Yongdo
2008-04-01
In this paper we consider the semigroup generated by the self-maps on the open convex cone of positive definite matrices of translations, congruence transformations and matrix inversion that includes symplectic Hamiltonians and show that every member of the semigroup contracts any invariant metric distance inherited from a symmetric gauge function. This extends the results of Bougerol for the Riemannian metric and of Liverani-Wojtkowski for the Thompson part metric. A uniform upper bound of the Lipschitz contraction constant for a member of the semigroup is given in terms of the minimum eigenvalues of its determining matrices. We apply this result to a variety of nonlinear equations including Stein and Riccati equations for uniqueness and existence of positive definite solutions and find a new convergence analysis of iterative algorithms for the positive definite solution depending only on the least contraction coefficient for the invariant metric from the spectral norm.
Galerkin Methods for Nonlinear Elliptic Equations.
NASA Astrophysics Data System (ADS)
Murdoch, Thomas
Available from UMI in association with The British Library. Requires signed TDF. This thesis exploits in the nonlinear situation the optimal approximation property of the finite element method for linear, elliptic problems. Of particular interest are the steady state semiconductor equations in one and two dimensions. Instead of discretising the differential equations by the finite element method and solving the nonlinear algebraic equations by Newton's method, a Newton linearisation of the continuous problem is preferred and a sequence of linear problems solved until some convergence criterion is achieved. For nonlinear Poisson equations, this approach reduces to solving a sequence of linear, elliptic, self -adjoint problems, their approximation by the finite element being optimal in a suitably defined energy norm. Consequently, there is the potential to recover a smoother representation of the underlying solution at each step of the Newton iteration. When this approach is applied to the continuity equations for semiconductor devices, a sequence of linear problems of the form -_{nabla }(anabla u - bu) = f must be solved. The Galerkin method in its crude form does not adequately represent the true solution: however, generalising the framework to permit Petrov-Galerkin approximations remedies the situation. For one dimensional problems, the work of Barrett and Morton allows an optimal test space to be chosen at each step of the Newton iteration so that the resulting approximation is near optimal in a norm closely related to the standard L^2 norm. More detailed information about the underlying solution can then be obtained by recovering a solution of an appropriate form. For two-dimensional problems, since the optimal test functions are difficult to find in practice, an upwinding method due to Heinrich et.al. is used at each step of the Newton iteration. Also, a framework is presented in which various upwind methods may be compared. The thesis also addresses the
Strongly Nonlinear Integral Equations of Hammerstein Type
Browder, Felix E.
1975-01-01
This paper studies the solution of the nonlinear Hammerstein equation u(x) + ʃ k(x,y)f[y,u(y)]μ(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) ≥ 0, that K maps L1 into L1loc and is compact from L1 [unk] L∞ into L1loc, that f(y,s) has the same sign as s for ǀsǀ ≥ R, and that for each constant r > 0, ǀf(y,s)ǀ ≤ gr(y) for ǀsǀ ≤ r where g is bounded and summable. The proof is obtained by combining a priori bounds, a truncation procedure, and a convergence argument using the Dunford-Pettis theorem. PMID:16578727
Notes on the Modified Nonlinear Schrodinger Equation
NASA Astrophysics Data System (ADS)
Pizzo, N. E.; Melville, W. K.
2011-12-01
In this study, we present the derivation of a modified Nonlinear Schrodinger equation (MNLSE) based on variational calculus. Using weakly nonlinear theory we derive an averaged Lagrangian, which in turn yields a slightly modified version of the MNLSE that conserves wave action. We also explore ramifications of the MNLSE with respect to the coupling between mean currents and non-uniform radiation stresses. We present this in the context of breaking waves and the free long waves they generate (Kristian Dysthe, personal communication). It has been noted in laboratory experiments (Meza et al, 1999) that breaking waves transfer some energy to modes far below the peak frequency of the spectrum. The transfer mechanism is widely believed to be the result of nonlinear four wave resonant interactions; however, the coupling between breaking-induced non-uniform radiation stresses and long wave radiation suggests a potential alternative explanation. Through direct numerical simulations, along with the theory, we test the feasibility of this mechanism by comparing it to data from wave tank experiments (Drazen et al., 2008).
Nonlinear Dynamics of the Leggett Equation
NASA Astrophysics Data System (ADS)
Ragan, Robert J.
1995-01-01
We study the nonlinear dynamics of spin-polarized Fermi liquids. Our starting point is the equation of motion for the magnetization derived by Leggett and Rice, which accounts for spin-rotation effects in the limit of small polarization. We also include later modifications to the theory by Meyerovich, and Jeon and Mullin, which account for polarization dependences of the transport coefficients. In the analysis of NMR experiments the methods of current research can be summarized as follows: (a) to linearize the Leggett equation by considering small amplitude oscillations (small tip angles), (b) to use perturbation theory to account for small spin-rotation effects, (c) to exploit the simple helical solution which describes spin-echo experiments. In this thesis, we report progress in several directions: (1) We extend the linear theory to describe bounded spin diffusion with spin-rotation and finite-polarization effects. The analysis is valid for arbitrary tip angles and arbitrary degree of nonlinearity. (2) We show that because of the spin-rotation effect, the helical solution exhibits a Castiang instability for large tip angles. In the limit of small damping, we use the inverse scattering theory developed by Levy to display the full nonlinear evolution of the instabilities. (3) We use perturbation theory to show that anisotropy in the spin diffusion coefficients gives rise to multiple spin echoes, even in the absence of spin -rotation effects. This description applies to experiments on ^3He-^4He solutions at ^3He concentrations of 3-5%. This experiment provides a unique means of verifying the theory of Jeon and Mullin. We also report some exact results in the theory of anisotropic spin diffusion.
Nonlinear scalar field equations involving the fractional Laplacian
NASA Astrophysics Data System (ADS)
Byeon, Jaeyoung; Kwon, Ohsang; Seok, Jinmyoung
2017-04-01
In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.
Some new solutions of nonlinear evolution equations with variable coefficients
NASA Astrophysics Data System (ADS)
Virdi, Jasvinder Singh
2016-05-01
We construct the traveling wave solutions of nonlinear evolution equations (NLEEs) with variable coefficients arising in physics. Some interesting nonlinear evolution equations are investigated by traveling wave solutions which are expressed by the hyperbolic functions, the trigonometric functions and rational functions. The applied method will be used in further works to establish more entirely new solutions for other kinds of such nonlinear evolution equations with variable coefficients arising in physics.
Nonlinear Parabolic Equations Involving Measures as Initial Conditions.
1981-09-01
CHART N N N Afl4Uf’t 1N II Il MRC Technical Summary Report # 2277 0 NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS I Haim Brezis ...NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS Haim Brezis and Avner Friedman Technical Summary Report #2277 September 1981...with NRC, and not with the authors of this report. * s ’a * ’ 4| NONLINEAR PARABOLIC EQUATIONS INVOLVING MEASURES AS INITIAL CONDITIONS Haim Brezis
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.
Bifurcation and stability for a nonlinear parabolic partial differential equation
NASA Technical Reports Server (NTRS)
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
Exact solutions to nonlinear delay differential equations of hyperbolic type
NASA Astrophysics Data System (ADS)
Vyazmin, Andrei V.; Sorokin, Vsevolod G.
2017-01-01
We consider nonlinear delay differential equations of hyperbolic type, including equations with varying transfer coefficients and varying delays. The equations contain one or two arbitrary functions of a single argument. We present new classes of exact generalized and functional separable solutions. All the solutions involve free parameters and can be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for similar and more complex nonlinear differential-difference equations.
Forced nonlinear Schrödinger equation with arbitrary nonlinearity
NASA Astrophysics Data System (ADS)
Cooper, Fred; Khare, Avinash; Quintero, Niurka R.; Mertens, Franz G.; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction (g2)/(κ+1)(ψψ)κ+1 in the presence of the external forcing terms of the form re-i(kx+θ)-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where vk=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq˙(t)<0, where p(t) is the normalized canonical momentum p(t)=(1)/(M(t))(∂L)/(∂q˙), and q˙(t) is the solitary wave velocity. Here M(t)=∫dxψ(x,t)ψ(x,t). Stability is also studied using a “phase portrait” of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.
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.
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.
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.
Exploring the nonlinear cloud and rain equation.
Koren, Ilan; Tziperman, Eli; Feingold, Graham
2017-01-01
Marine stratocumulus cloud decks are regarded as the reflectors of the climate system, returning back to space a significant part of the income solar radiation, thus cooling the atmosphere. Such clouds can exist in two stable modes, open and closed cells, for a wide range of environmental conditions. This emergent behavior of the system, and its sensitivity to aerosol and environmental properties, is captured by a set of nonlinear equations. Here, using linear stability analysis, we express the transition from steady to a limit-cycle state analytically, showing how it depends on the model parameters. We show that the control of the droplet concentration (N), the environmental carrying-capacity (H0), and the cloud recovery parameter (τ) can be linked by a single nondimensional parameter (μ=N/(ατH0)), suggesting that for deeper clouds the transition from open (oscillating) to closed (stable fixed point) cells will occur for higher droplet concentration (i.e., higher aerosol loading). The analytical calculations of the possible states, and how they are affected by changes in aerosol and the environmental variables, provide an enhanced understanding of the complex interactions of clouds and rain.
Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2015-10-01
The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.
Forced nonlinear Schrödinger equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Quintero, Niurka R; Mertens, Franz G; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.
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.
A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects.
Xu, Zhengfu; Bao, Gang
2010-11-01
A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.
Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations
NASA Astrophysics Data System (ADS)
Ma, Wen-Xiu; Zhou, Yuan; Dougherty, Rachael
2016-08-01
Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on f are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on u, generated from taking a transformation of dependent variables u = 2(ln f)x.
Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities.
Khare, Avinash; Rasmussen, Kim Ø; Salerno, Mario; Samuelsen, Mogens R; Saxena, Avadh
2006-07-01
A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.
Kinetic equation for nonlinear resonant wave-particle interaction
NASA Astrophysics Data System (ADS)
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2016-09-01
We investigate the nonlinear resonant wave-particle interactions including the effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relationship between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations.
Nonlinear modes of the tensor Dirac equation and CPT violation
NASA Technical Reports Server (NTRS)
Reifler, Frank J.; Morris, Randall D.
1993-01-01
Recently, it has been shown that Dirac's bispinor equation can be expressed, in an equivalent tensor form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was also shown that the free tensor Dirac equation is a completely integrable Hamiltonian system with Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. The Yang-Mills equation for a finite coupling constant is investigated. It is shown that the nonlinear Yang-Mills equation has exact plane wave solutions in one-to-one correspondence with the plane wave solutions of Dirac's bispinor equation. The theory of nonlinear dispersive waves is applied to establish the existence of wave packets. The CPT violation of these nonlinear wave packets, which could lead to new observable effects consistent with current experimental bounds, is investigated.
Variable-coefficient extended mapping method for nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Zhang, Sheng; Xia, Tiecheng
2008-03-01
In this Letter, a variable-coefficient extended mapping method is proposed to seek new and more general exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients and ( 2+1)-dimensional Nizhnik-Novikov-Veselov equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the proposed method provides a very effective and powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.
Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation.
1980-06-01
Equation (1) may also be considered as an ordinary differential equation on a Banach space. This is the setting I prefer, as it usually seems much more... NONLINEAR WAVE EQUATION ~0 by gc~ Paul Massatt Lefschetz Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, Rhode...Interim -) DAMPED NONLINEAR WAVE EQUATION . 6. PERFORMING 0G. RMRT UMBER 7. AUTHOR(a) S. CONTRACT OR GRANT NUMBER(O) PAUL!MASSATT 47 -Xo AFdSR-76-3,992 / 9
Comparative study of homotopy continuation methods for nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
Late-time attractor for the cubic nonlinear wave equation
Szpak, Nikodem
2010-08-15
We apply our recently developed scaling technique for obtaining late-time asymptotics to the cubic nonlinear wave equation and explain the appearance and approach to the two-parameter attractor found recently by Bizon and Zenginoglu.
Integrable nonlocal nonlinear Schrödinger equation.
Ablowitz, Mark J; Musslimani, Ziad H
2013-02-08
A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed. A method to find pure soliton solutions is given. An explicit breathing one soliton solution is found. Key properties are discussed and contrasted with the classical nonlinear Schrödinger equation.
Invariant tori for a class of nonlinear evolution equations
Kolesov, A Yu; Rozov, N Kh
2013-06-30
The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.
Nonlinear acoustic wave equations with fractional loss operators.
Prieur, Fabrice; Holm, Sverre
2011-09-01
Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.
The nonlinear modified equation approach to analyzing finite difference schemes
NASA Technical Reports Server (NTRS)
Klopfer, G. H.; Mcrae, D. S.
1981-01-01
The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.
Slunyaev, A; Pelinovsky, E; Sergeeva, A; Chabchoub, A; Hoffmann, N; Onorato, M; Akhmediev, N
2013-07-01
The rogue wave solutions (rational multibreathers) of the nonlinear Schrödinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation, also known as the Dysthe equation. This numerical modeling allowed us to directly compare simulations with recent results of laboratory measurements in Chabchoub et al. [Phys. Rev. E 86, 056601 (2012)]. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near-breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.
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.
Entropy and convexity for nonlinear partial differential equations
Ball, John M.; Chen, Gui-Qiang G.
2013-01-01
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768
Nonlinear flap-lag axial equations of a rotating beam
NASA Technical Reports Server (NTRS)
Kaza, K. R. V.; Kvaternik, R. G.
1977-01-01
It is possible to identify essentially four approaches by which analysts have established either the linear or nonlinear governing equations of motion for a particular problem related to the dynamics of rotating elastic bodies. The approaches include the effective applied load artifice in combination with a variational principle and the use of Newton's second law, written as D'Alembert's principle, applied to the deformed configuration. A third approach is a variational method in which nonlinear strain-displacement relations and a first-degree displacement field are used. The method introduced by Vigneron (1975) for deriving the linear flap-lag equations of a rotating beam constitutes the fourth approach. The reported investigation shows that all four approaches make use of the geometric nonlinear theory of elasticity. An alternative method for deriving the nonlinear coupled flap-lag-axial equations of motion is also discussed.
Entropy and convexity for nonlinear partial differential equations.
Ball, John M; Chen, Gui-Qiang G
2013-12-28
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.
Model Predictive Control for Nonlinear Parabolic Partial Differential Equations
NASA Astrophysics Data System (ADS)
Hashimoto, Tomoaki; Yoshioka, Yusuke; Ohtsuka, Toshiyuki
In this study, the optimal control problem of nonlinear parabolic partial differential equations (PDEs) is investigated. Optimal control of nonlinear PDEs is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Model predictive control with a fast numerical solution method has been well established to solve the optimal control problem of nonlinear systems described by ordinary differential equations. In this study, we develop a design method of the model predictive control for nonlinear systems described by parabolic PDEs. Our approach is a direct infinite dimensional extension of the model predictive control method for finite-dimensional systems. The objective of this paper is to develop an efficient algorithm for numerically solving the model predictive control problem of nonlinear parabolic PDEs. The effectiveness of the proposed method is verified by numerical simulations.
The numerical dynamic for highly nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
Relations between nonlinear Riccati equations and other equations in fundamental physics
NASA Astrophysics Data System (ADS)
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Generalized nonlinear Proca equation and its free-particle solutions
NASA Astrophysics Data System (ADS)
Nobre, F. D.; Plastino, A. R.
2016-06-01
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schrödinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ ^{μ }(ěc {x},t), involves an additional field Φ ^{μ }(ěc {x},t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E2 = p2c2 + m2c4 for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed.
The Jeffcott equations in nonlinear rotordynamics
NASA Technical Reports Server (NTRS)
Zalik, R. A.
1987-01-01
The Jeffcott equations are a system of coupled differential equations representing the behavior of a rotating shaft. This is a simple model which allows investigation of the basic dynamic behavior of rotating machinery. Nolinearities can be introduced by taking into consideration deadband, side force, and rubbing, among others. The properties of the solutions of the Jeffcott equations with deadband are studied. In particular, it is shown how bounds for the solution of these equations can be obtained from bounds for the solutions of the linearized equations. By studying the behavior of the Fourier transforms of the solutions, we are also able to predict the onset of destructive vibrations. These conclusions are verified by means of numerical solutions of the equations, and of power spectrum density (PSD) plots. This study offers insight into a possible detection method to determine pump stability margins during flight and hot fire tests, and was motivated by the need to explain a phenomenon observed in the development phase of the cryogenic pumps of the Space Shuttle, during hot fire ground testing; namely, the appearance of vibrations at frequencies that could not be accounted for by means of linear models.
Nonlinear equations of dynamics for spinning paraboloidal antennas
NASA Technical Reports Server (NTRS)
Utku, S.; Shoemaker, W. L.; Salama, M.
1983-01-01
The nonlinear strain-displacement and velocity-displacement relations of spinning imperfect rotational paraboloidal thin shell antennas are derived for nonaxisymmetrical deformations. Using these relations with the admissible trial functions in the principle functional of dynamics, the nonlinear equations of stress inducing motion are expressed in the form of a set of quasi-linear ordinary differential equations of the undetermined functions by means of the Rayleigh-Ritz procedure. These equations include all nonlinear terms up to and including the third degree. Explicit expressions are given for the coefficient matrices appearing in these equations. Both translational and rotational off-sets of the axis of revolution (and also the apex point of the paraboloid) with respect to the spin axis are considered. Although the material of the antenna is assumed linearly elastic, it can be anisotropic.
A general non-linear multilevel structural equation mixture model
Kelava, Augustin; Brandt, Holger
2014-01-01
In the past 2 decades latent variable modeling has become a standard tool in the social sciences. In the same time period, traditional linear structural equation models have been extended to include non-linear interaction and quadratic effects (e.g., Klein and Moosbrugger, 2000), and multilevel modeling (Rabe-Hesketh et al., 2004). We present a general non-linear multilevel structural equation mixture model (GNM-SEMM) that combines recent semiparametric non-linear structural equation models (Kelava and Nagengast, 2012; Kelava et al., 2014) with multilevel structural equation mixture models (Muthén and Asparouhov, 2009) for clustered and non-normally distributed data. The proposed approach allows for semiparametric relationships at the within and at the between levels. We present examples from the educational science to illustrate different submodels from the general framework. PMID:25101022
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
Nonlinear waves described by the generalized Swift-Hohenberg equation
NASA Astrophysics Data System (ADS)
Ryabov, P. N.; Kudryashov, N. A.
2017-01-01
We study the wave processes described by the generalized Swift-Hohenberg equation. We show that the traveling wave reduction of this equation does not pass the Kovalevskaya test. Some solitary wave solutions and kink solutions of the generalized Swift-Hohenberg equation are found. We use the pseudo-spectral algorithm to perform the numerical simulation of the wave processes described by the mixed boundary value problem for the generalized Swift-Hohenberg equation. This algorithm was tested on the obtained solutions. Some features of the nonlinear waves evolution described by the generalized Swift-Hohenberg equation are studied.
Approximating a nonlinear advanced-delayed equation from acoustics
NASA Astrophysics Data System (ADS)
Teodoro, M. Filomena
2016-10-01
We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.
An iterative method for systems of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Scroggs, Jeffrey S.
1989-01-01
An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.
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.
Non-Linear Spring Equations and Stability
ERIC Educational Resources Information Center
Fay, Temple H.; Joubert, Stephan V.
2009-01-01
We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of…
Nonlinear Resonance and Duffing's Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and…
Nonlinear Resonance and Duffing's Spring Equation II
ERIC Educational Resources Information Center
Fay, T. H.; Joubert, Stephan V.
2007-01-01
The paper discusses the boundary in the frequency-amplitude plane for boundedness of solutions to the forced spring Duffing type equation x[umlaut] + x + [epsilon]x[cubed] = F cos[omega]t. For fixed initial conditions and for representative fixed values of the parameter [epsilon], the results are reported of a systematic numerical investigation…
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.
Exact solutions of the derivative nonlinear Schrödinger equation for a nonlinear transmission line.
Kengne, E; Liu, W M
2006-02-01
We consider the derivative nonlinear Schrödinger equation with constant potential as a model for wave propagation on a discrete nonlinear transmission line. This equation can be derived in the small amplitude and long wavelength limit using the standard reductive perturbation method and complex expansion. We construct some exact soliton and elliptic solutions of the mentioned equation by perturbation of its Stokes wave solutions. We find that for some values of the coefficients of the equation and for some parameters of solutions, the graphical representations show some kinds of symmetries such as mirror symmetry and rotational symmetry.
Transport equations for subdiffusion with nonlinear particle interaction.
Straka, P; Fedotov, S
2015-02-07
We show how the nonlinear interaction effects 'volume filling' and 'adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.
A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation
NASA Astrophysics Data System (ADS)
Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.
2014-02-01
This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.
NASA Astrophysics Data System (ADS)
Tchinang Tchameu, J. D.; Togueu Motcheyo, A. B.; Tchawoua, C.
2016-09-01
The discrete multi-rogue waves (DMRW) as solution of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearities is studied numerically. These biological rogue waves represent the complex probability amplitude of finding an amide-I vibrational quantum at a site. We observe that the growth in the higher order saturable nonlinearity implies the formation of DMRW including an increase in the short-living DMRW and a decrease in amplitude of the long-living DMRW.
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.
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1988-01-01
An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-08
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
Decay and stability for nonlinear hyperbolic equations
NASA Astrophysics Data System (ADS)
Marcati, Pierangelo
This paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under some restrictions, sufficient conditions for precompactness of the orbits and decay of solutions are given. An existence and uniqueness theorem is proved in the Appendix. Some examples are given.
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.
Nonlinear Landau damping, and nonlinear envelope equation, for a driven plasma wave
NASA Astrophysics Data System (ADS)
Benisti, Didier; Morice, Olivier; Gremillet, Laurent; Strozzi, David
2009-11-01
A nonlinear envelope equation for a laser-driven electron plasma wave (EPW) is derived in a 3-D geometry, starting from first principles. This equation accounts the nonlinear variations of the EPW Landau damping rate, frequency, and group velocity, as well as for the nonlinear variations of the coupling of the EPW to the electromagnetic waves. All these quantities are moreover shown to be nonlocal because of nonlocal variations of the electron distribution function. Each piece of our model is carefully tested against Vlasov simulations of stimulated Raman scattering (SRS), and very good agreement is found between the numerical and theoretical results. Our envelope equations for both, the electrostatic and electromagnetic waves, are solved numerically, and comparisons with Vlasov simulations regarding the growth of SRS are provided. Finally, from our theory we can straightforwardly deduce a nonlinear gain factor which provides an alternate, simpler and faster method to quantify the SRS reflectivity. First results using this method will be shown.
Case-Deletion Diagnostics for Nonlinear Structural Equation Models
ERIC Educational Resources Information Center
Lee, Sik-Yum; Lu, Bin
2003-01-01
In this article, a case-deletion procedure is proposed to detect influential observations in a nonlinear structural equation model. The key idea is to develop the diagnostic measures based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. An one-step pseudo approximation is proposed to reduce the…
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.
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.
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.
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.
Kedziora, D J; Ankiewicz, A; Chowdury, A; Akhmediev, N
2015-10-01
We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.
Numerical solution of control problems governed by nonlinear differential equations
Heinkenschloss, M.
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Physical dynamics of quasi-particles in nonlinear wave equations
NASA Astrophysics Data System (ADS)
Christov, Ivan; Christov, C. I.
2008-02-01
By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field.
On the Amplitude Equations for Weakly Nonlinear Surface Waves
NASA Astrophysics Data System (ADS)
Benzoni-Gavage, Sylvie; Coulombel, Jean-François
2012-09-01
Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185-202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3-4):1463-1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3-4):303-320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220-2240, 2011). In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.
Numerical Solution of a Nonlinear Integro-Differential Equation
NASA Astrophysics Data System (ADS)
Buša, Ján; Hnatič, Michal; Honkonen, Juha; Lučivjanský, Tomáš
2016-02-01
A discretization algorithm for the numerical solution of a nonlinear integrodifferential equation modeling the temporal variation of the mean number density a(t) in the single-species annihilation reaction A + A → 0 is discussed. The proposed solution for the two-dimensional case (where the integral entering the equation is divergent) uses regularization and then finite differences for the approximation of the differential operator together with a piecewise linear approximation of a(t) under the integral. The presented numerical results point to basic features of the behavior of the number density function a(t) and suggest further improvement of the proposed algorithm.
Connecting orbits for nonlinear differential equations at resonance
NASA Astrophysics Data System (ADS)
Kokocki, Piotr
We study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the Conley index of associated semiflow with respect to appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of the well-known in literature Landesman-Lazer and strong resonance conditions. Obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.
Solving nonlinear evolution equation system using two different methods
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Bekir, Ahmet; Ozer, Mehmet N.
2015-12-01
This paper deals with constructing more general exact solutions of the coupled Higgs equation by using the (G0/G, 1/G)-expansion and (1/G0)-expansion methods. The obtained solutions are expressed by three types of functions: hyperbolic, trigonometric and rational functions with free parameters. It has been shown that the suggested methods are productive and will be used to solve nonlinear partial differential equations in applied mathematics and engineering. Throughout the paper, all the calculations are made with the aid of the Maple software.
Inverse Problem of Variational Calculus for Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Ali, Sk. Golam; Talukdar, B.; Das, U.
2007-06-01
We couple a nonlinear evolution equation with an associated one and derive the action principle. This allows us to write the Lagrangian density of the system in terms of the original field variables rather than Casimir potentials. We find that the corresponding Hamiltonian density provides a natural basis to recast the pair of equations in the canonical form. Amongst the case studies presented the KdV and modified KdV pairs exhibit bi-Hamiltonian structure and allow one to realize the associated fields in physical terms.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Williams, L.R.; Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Singular Solutions of Fully Nonlinear Elliptic Equations and Applications
NASA Astrophysics Data System (ADS)
Armstrong, Scott N.; Sirakov, Boyan; Smart, Charles K.
2012-08-01
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of {R^n} , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.
Numerical study of fractional nonlinear Schrödinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Quadratic nonlinear Klein-Gordon equation in one dimension
NASA Astrophysics Data System (ADS)
Hayashi, Nakao; Naumkin, Pavel I.
2012-10-01
We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v - vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah ["Normal forms and quadratic nonlinear Klein-Gordon equations," Commun. Pure Appl. Math. 38, 685-696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort ["Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi-linéaire á données petites en dimension 1," Ann. Sci. Ec. Normale Super. 34(4), 1-61 (2001)].
The exotic conformal Galilei algebra and nonlinear partial differential equations
NASA Astrophysics Data System (ADS)
Cherniha, Roman; Henkel, Malte
2010-09-01
The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.
Nonlinear Generalized Hydrodynamic Wave Equations in Strongly Coupled Dusty Plasmas
Veeresha, B. M.; Sen, A.; Kaw, P. K.
2008-09-07
A set of nonlinear equations for the study of low frequency waves in a strongly coupled dusty plasma medium is derived using the phenomenological generalized hydrodynamic (GH) model and is used to study the modulational stability of dust acoustic waves to parallel perturbations. Dust compressibility contributions arising from strong Coulomb coupling effects are found to introduce significant modifications in the threshold and range of the instability domain.
Chaoticons described by nonlocal nonlinear Schrödinger equation.
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-30
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).
Chaoticons described by nonlocal nonlinear Schrödinger equation
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-01
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions). PMID:28134268
Fast neural solution of a nonlinear wave equation
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad; Barhen, Jacob
1992-01-01
A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.
Stabilisation of second-order nonlinear equations with variable delay
NASA Astrophysics Data System (ADS)
Berezansky, Leonid; Braverman, Elena; Idels, Lev
2015-08-01
For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.
Dispersion relation equation preserving FDTD method for nonlinear cubic Schrödinger equation
NASA Astrophysics Data System (ADS)
Sheu, Tony W. H.; Le Lin
2015-10-01
In this study we aim to solve the cubic nonlinear Schrödinger (CNLS) equation by the method of fractional steps. Over a time step from tn to tn+1, the linear part of the Schrödinger equation is solved firstly through four time integration steps. In this part of the simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term. The second-order spatial derivative term in the linear Schrödinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of the simulation, the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment. The proposed semi-discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution. Several problems have been solved to show that application of this new finite difference scheme for the calculation of one- and two-dimensional Schrödinger equations can deemed conserve Hamiltonian quantities and preserve dispersion relation equation (DRE).
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
NASA Astrophysics Data System (ADS)
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Xie, Xi-Yang; Tian, Bo Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-15
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation.
Bustamante, Miguel D; Nazarenko, Sergey
2015-11-01
We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines,H=κ(2)/8π∫(|s-s'|>ξ(*))(ds·ds')/|s-s'|,with cutoff length ξ(*)≈0.3416293/√(ρ(0)), where ρ(0) is the background condensate density far from the vortex lines and κ is the quantum of circulation.
Equations for Nonlinear MHD Convection in Shearless Magnetic Systems
Pastukhov, V.P.
2005-07-15
A closed set of reduced dynamic equations is derived that describe nonlinear low-frequency flute MHD convection and resulting nondiffusive transport processes in weakly dissipative plasmas with closed or open magnetic field lines. The equations obtained make it possible to self-consistently simulate transport processes and the establishment of the self-consistent plasma temperature and density profiles for a large class of axisymmetric nonparaxial shearless magnetic devices: levitated dipole configurations, mirror systems, compact tori, etc. Reduced equations that are suitable for modeling the long-term evolution of the plasma on time scales comparable to the plasma lifetime are derived by the method of the adiabatic separation of fast and slow motions.
Solovchuk, Maxim; Sheu, Tony W H; Thiriet, Marc
2013-11-01
This study investigates the influence of blood flow on temperature distribution during high-intensity focused ultrasound (HIFU) ablation of liver tumors. A three-dimensional acoustic-thermal-hydrodynamic coupling model is developed to compute the temperature field in the hepatic cancerous region. The model is based on the nonlinear Westervelt equation, bioheat equations for the perfused tissue and blood flow domains. The nonlinear Navier-Stokes equations are employed to describe the flow in large blood vessels. The effect of acoustic streaming is also taken into account in the present HIFU simulation study. A simulation of the Westervelt equation requires a prohibitively large amount of computer resources. Therefore a sixth-order accurate acoustic scheme in three-point stencil was developed for effectively solving the nonlinear wave equation. Results show that focused ultrasound beam with the peak intensity 2470 W/cm(2) can induce acoustic streaming velocities up to 75 cm/s in the vessel with a diameter of 3 mm. The predicted temperature difference for the cases considered with and without acoustic streaming effect is 13.5 °C or 81% on the blood vessel wall for the vein. Tumor necrosis was studied in a region close to major vessels. The theoretical feasibility to safely necrotize the tumors close to major hepatic arteries and veins was shown.
Optical similaritons in a tapered graded-index nonlinear-fiber amplifier with an external source
Raju, Thokala Soloman; Panigrahi, Prasanta K.
2011-09-15
We analytically explore a wide class of optical similariton solutions to the nonlinear Schroedinger equation appropriately modified to model beam propagation in a tapered, graded-index nonlinear-fiber amplifier with an external source. Under certain physical conditions, we reduce the coupled nonlinear Schroedinger equations to a single-wave equation that aptly describes similariton propagation through asymmetric twin-core fiber amplifiers. The asymmetric twin-core fiber is composed of two adjoining, closely spaced, single-mode fibers in which the active one is a tapered, graded-index nonlinear-fiber and the passive one is a step-index fiber. We obtain these self-similar waves for different choices of tapered index profile, using a Moebius transformation. Our procedure is applicable for both self-focusing and self-defocusing nonlinearities.
Coupled equations of electromagnetic waves in nonlinear metamaterial waveguides.
Azari, Mina; Hatami, Mohsen; Meygoli, Vahid; Yousefi, Elham
2016-11-01
Over the past decades, scientists have presented ways to manipulate the macroscopic properties of a material at levels unachieved before, and called them metamaterials. This research can be considered an important step forward in electromagnetics and optics. In this study, higher-order nonlinear coupled equations in a special kind of metamaterial waveguides (a planar waveguide with metamaterial core) will be derived from both electric and magnetic components of the transverse electric mode of electromagnetic pulse propagation. On the other hand, achieving the refractive index in this research is worthwhile. It is also shown that the coupled equations are not symmetric with respect to the electric and magnetic fields, unlike these kinds of equations in fiber optics and dielectric waveguides. Simulations on the propagation of a fundamental soliton pulse in a nonlinear metamaterial waveguide near the resonance frequency (a little lower than the magnetic resonant frequency) are performed to study its behavior. These pulses are recommended to practice in optical communications in controlled switching by external voltage, even in low power.
Approximate analytic solutions to coupled nonlinear Dirac equations
NASA Astrophysics Data System (ADS)
Khare, Avinash; Cooper, Fred; Saxena, Avadh
2017-03-01
We consider the coupled nonlinear Dirac equations (NLDEs) in 1 + 1 dimensions with scalar-scalar self-interactions g12 / 2 (ψ bar ψ) 2 + g22/2 (ϕ bar ϕ) 2 + g32 (ψ bar ψ) (ϕ bar ϕ) as well as vector-vector interactions of the form g1/22 (ψ bar γμ ψ) (ψ bar γμ ψ) + g22/2 (ϕ bar γμ ϕ) (ϕ bar γμ ϕ) + g32 (ψ bar γμ ψ) (ϕ bar γμ ϕ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ =e - iω1 t {R1 cos θ ,R1 sin θ }, ϕ =e - iω2 t {R2 cos η ,R2 sin η }, and assuming that θ (x) , η (x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri (x) which are valid for small values of g32 / g22 and g32 / g12. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ± ∞.
Approximate analytic solutions to coupled nonlinear Dirac equations
Khare, Avinash; Cooper, Fred; Saxena, Avadh
2017-01-30
Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g12/2(more » $$\\bar{ψ}$$ψ)2 + g22/2($$\\bar{Φ}$$Φ)2 + g23($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g12/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g22/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g23($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e–iω1tR1cosθ,R1sinθΦ=e–iω2tR2cosη,R2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g32/g22 and g32/g12. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less
Continuous symmetries of certain nonlinear partial difference equations and their reductions
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Nagavigneshwari, G.
2014-09-01
In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
NASA Astrophysics Data System (ADS)
Khare, Avinash; Saxena, Avadh
2014-03-01
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ4, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn2(x, m), it also admits solutions in terms of dn^2(x,m) ± sqrt{m} cn(x,m) dn(x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
On the nonlinear Schrodinger equation with nonzero boundary conditions
NASA Astrophysics Data System (ADS)
Fagerstrom, Emily
This thesis is concerned with the study of the nonlinear Schrodinger (NLS) equation, which is important both from a physical and a mathematical point of view. In physics, it is a universal model for the evolutions of weakly nonlinear dispersive wave trains. As such it appears in many physical contexts, such as optics, acoustics, plasmas, biology, etc. Mathematically, it is a completely integrable, infinite-dimensional Hamiltonian system, and possesses a surprisingly rich structure. This equation has been extensively studied in the last 50 years, but many important questions are still open. In particular, this thesis contains the following original contributions: NLS with real spectral singularities. First, the focusing NLS equation is considered with decaying initial conditions. This situation has been studied extensively before, but the assumption is almost always made that the scattering coefficients have no real zeros, and thus the scattering data had no poles on the real axis. However, it is easy to produce example potentials with this behavior. For example, by modifying parameters in Satsuma-Yajima's sech potential, or by choosing a "box" potential with a particular area, one can obtain corresponding scattering entries with real zeros. The inverse scattering transform can be implemented by formulating the modified Jost eigenfunctions and the scattering data as a Riemann Hilbert problem. But it can also be formulated by using integral kernels. Doing so produces the Gelf'and-Levitan-Marchenko (GLM) equations. Solving these integral equations requires integrating an expression containing the reflection coefficient over the real axis. Under the usual assumption, the reflection coefficient has no poles on the real axis. In general, the integration contour cannot be deformed to avoid poles, because the reflection coefficient may not admit analytic extension off the real axis. Here it is shown that the GLM equations may be (uniquely) solved using a principal value
Some existence results on nonlinear fractional differential equations.
Baleanu, Dumitru; Rezapour, Shahram; Mohammadi, Hakimeh
2013-05-13
In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η
Fourth order wave equations with nonlinear strain and source terms
NASA Astrophysics Data System (ADS)
Liu, Yacheng; Xu, Runzhang
2007-07-01
In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u0)[greater-or-equal, slanted]0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.
New Analytical Solution for Nonlinear Shallow Water-Wave Equations
NASA Astrophysics Data System (ADS)
Aydin, Baran; Kânoğlu, Utku
2017-03-01
We solve the nonlinear shallow water-wave equations over a linearly sloping beach as an initial-boundary value problem under general initial conditions, i.e., an initial wave profile with and without initial velocity. The methodology presented here is extremely simple and allows a solution in terms of eigenfunction expansion, avoiding integral transform techniques, which sometimes result in singular integrals. We estimate parameters, such as the temporal variations of the shoreline position and the depth-averaged velocity, compare with existing solutions, and observe perfect agreement with substantially less computational effort.
Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
NASA Astrophysics Data System (ADS)
Enkov, Svetoslav; Georgieva, Atanaska; Nikolla, Renato
2016-12-01
In this work we investigate nonlinear Hammerstein fuzzy functional integral equation. Our aim is to provide an efficient iterative method of successive approximations by optimal quadrature formula for classes of fuzzy number-valued functions of Lipschitz type to approximate the solution. We prove the convergence of the method by Banach's fixed point theorem and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, illustrative numerical experiment demonstrate the accuracy and the convergence of the proposed method.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
A procedure to construct exact solutions of nonlinear fractional differential equations.
Güner, Özkan; Cevikel, Adem C
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.
A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations
Güner, Özkan; Cevikel, Adem C.
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
NASA Astrophysics Data System (ADS)
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1982-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
Ergodicity for Nonlinear Stochastic Equations in Variational Formulation
Barbu, Viorel Da Prato, Giuseppe
2006-03-15
This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods.The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P{sub t}. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure {nu}. Moreover, we solve the Kolmogorov equation and prove the so-called 'identite du carre du champs'. This will be used to study the Sobolev space W{sup 1,2}(H,{nu}) and to obtain information on the domain of the infinitesimal generator of P{sub t}.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
NASA Astrophysics Data System (ADS)
Junaid, Ali Khan; Muhammad, Asif Zahoor Raja; Ijaz Mansoor, Qureshi
2011-02-01
We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Approximate symmetry and solutions of the nonlinear Klein-Gordon equation with a small parameter
NASA Astrophysics Data System (ADS)
Rahimian, Mohammad; Toomanian, Megerdich; Nadjafikhah, Mehdi
In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein-Gordon equation with a small parameter. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method.
NASA Astrophysics Data System (ADS)
Yao, Ruo-Xia; Wang, Wei; Chen, Ting-Hua
2014-11-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
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.
On the dynamics of approximating schemes for dissipative nonlinear equations
NASA Technical Reports Server (NTRS)
Jones, Donald A.
1993-01-01
Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.
Nonlinear Envelope Equation and Nonlinear Landau Damping Rate for a Driven Electron Plasma Wave
NASA Astrophysics Data System (ADS)
Bénisti, Didier; Morice, Olivier; Gremillet, Laurent; Strozzi, David J.
2011-10-01
In this article, we provide a theoretical description and calculate the nonlinear frequency shift, group velocity, and collionless damping rate, ν, of a driven electron plasma wave (EPW). All these quantities, whose physical content will be discussed, are identified as terms of an envelope equation allowing one to predict how efficiently an EPW may be externally driven. This envelope equation is derived directly from Gauss' law and from the investigation of the nonlinear electron motion, provided that the time and space rates of variation of the EPW amplitude, ?, are small compared to the plasma frequency or the inverse of the Debye length. ν arises within the EPW envelope equation as a more complicated operator than a plain damping rate and may only be viewed as such because [?]? remains nearly constant before abruptly dropping to zero. We provide a practical analytic formula for ν and show, without resorting to complex contour deformation, that in the limit ?0, ν is nothing but the Landau damping rate. We then term ν the "nonlinear Landau damping rate" of the driven plasma wave. As for the nonlinear frequency shift of the driven EPW, it is also derived theoretically and found to assume values significantly different from previously published ones, which were obtained by assuming that the wave was freely propagating. Moreover, we find no limitation in ?, ? being the plasma wavenumber and ? the Debye length, for a solution to the dispertion relation to exist, and want to stress here the importance of specifying how an EPW is generated to discuss its properties. Our theoretical predictions are in excellent agreement with results inferred from Vlasov simulations of stimulated Raman scattering (SRS), and an application of our theory to the study of SRS is presented.
NASA Astrophysics Data System (ADS)
Canoglu, Ahmet; Güldogan, Bahri; Salihoglu, Selâmi
We obtain new integrable coupled nonlinear partial differential equations by assuming the soliton connection having values in orthogonal-symplectic Lie superalgebras [B(m, n), C(n), D(m, n)]. These equations are coupled Nonlinear Schrödinger equations on various super symmetric spaces.
Exact Multisoliton Solutions of General Nonlinear Schrödinger Equation with Derivative
Li, Qi; Duan, Qiu-yuan; Zhang, Jian-bing
2014-01-01
Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions. PMID:25013858
Bayesian Analysis of Structural Equation Models with Nonlinear Covariates and Latent Variables
ERIC Educational Resources Information Center
Song, Xin-Yuan; Lee, Sik-Yum
2006-01-01
In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the…
Exact multisoliton solutions of general nonlinear Schrödinger equation with derivative.
Li, Qi; Duan, Qiu-yuan; Zhang, Jian-bing
2014-01-01
Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions.
2014-09-30
nonlinear Schrodinger equation. It is well known that dark solitons are exact solutions of such equation. In the present paper it has been shown that gray...in numerical computations of Nonlinear Schrodinger equation, and in the optical fibers experiments. In particular it has been shown that the
A new method for parameter estimation in nonlinear dynamical equations
NASA Astrophysics Data System (ADS)
Wang, Liu; He, Wen-Ping; Liao, Le-Jian; Wan, Shi-Quan; He, Tao
2015-01-01
Parameter estimation is an important scientific problem in various fields such as chaos control, chaos synchronization and other mathematical models. In this paper, a new method for parameter estimation in nonlinear dynamical equations is proposed based on evolutionary modelling (EM). This will be achieved by utilizing the following characteristics of EM which includes self-organizing, adaptive and self-learning features which are inspired by biological natural selection, and mutation and genetic inheritance. The performance of the new method is demonstrated by using various numerical tests on the classic chaos model—Lorenz equation (Lorenz 1963). The results indicate that the new method can be used for fast and effective parameter estimation irrespective of whether partial parameters or all parameters are unknown in the Lorenz equation. Moreover, the new method has a good convergence rate. Noises are inevitable in observational data. The influence of observational noises on the performance of the presented method has been investigated. The results indicate that the strong noises, such as signal noise ratio (SNR) of 10 dB, have a larger influence on parameter estimation than the relatively weak noises. However, it is found that the precision of the parameter estimation remains acceptable for the relatively weak noises, e.g. SNR is 20 or 30 dB. It indicates that the presented method also has some anti-noise performance.
Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence
Hahm, T. S.; Wang, Lu; Madsen, J.
2008-08-01
An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E Χ B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρ_{i}<< ρ_{θ¡} ~ L_{E} ~ L_{p} << R (here ρ_{i} is the thermal ion Larmor radius and ρ_{θ¡} = B/B_{θ}] ρ_{i}), as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. We take κ perpendicular to ρ_{i} ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ /Τ_{i} ~ δΒ / Β up to the second order. Extending the electrostatic theory in the presence of high E Χ B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.
An almost symmetric Strang splitting scheme for nonlinear evolution equations.
Einkemmer, Lukas; Ostermann, Alexander
2014-07-01
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.
Implementation of nonreflecting boundary conditions for the nonlinear Euler equations
NASA Astrophysics Data System (ADS)
Atassi, Oliver V.; Galán, José M.
2008-01-01
Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier-Bessel modes. This leads to an 'exact' nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier-Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1995-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that we currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Karr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.
Nonlinear dirac and diffusion equations in 1+1 dimensions from stochastic considerations
Maharana
2000-08-01
We generalize the method of obtaining fundamental linear partial differential equations such as the diffusion and Schrodinger equation, the Dirac, and the telegrapher's equation from a simple stochastic consideration to arrive at a certain nonlinear form of these equations. A group classification through a one-parameter group of transformations for two of these equations is also carried out.
NASA Astrophysics Data System (ADS)
Reyes, M. A.; Gutiérrez-Ruiz, D.; Mancas, S. C.; Rosu, H. C.
2016-01-01
We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations when p = 2.
Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio
2014-01-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530
Estimation of Delays and Other Parameters in Nonlinear Functional Differential Equations.
1981-12-01
FSTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS by K. T. Banks and P. L. Daniel December 1981 LCDS Report #82...ESTIMATION OF DELAYS AND OTHER PARAMETERS IN NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS H. T. Banks and P. L. Daniel ABSTRACT We discuss a spline...based approximation scheme for nonlinear nonautonomous delay differential equations . Convergence results (using dissipative type estimates on the
Canonical equations of Hamilton for the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Liang, Guo; Guo, Qi; Ren, Zhanmei
2015-09-01
We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.
Nonequilibrium discrete nonlinear Schrödinger equation.
Iubini, Stefano; Lepri, Stefano; Politi, Antonio
2012-07-01
We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e., transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.
Rogue waves of a (3 + 1) -dimensional nonlinear evolution equation
NASA Astrophysics Data System (ADS)
Shi, Yu-bin; Zhang, Yi
2017-03-01
General high-order rogue waves of a (3 + 1) -dimensional Nonlinear Evolution Equation ((3+1)-d NEE) are obtained by the Hirota bilinear method, which are given in terms of determinants, whose matrix elements possess plain algebraic expressions. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. Two subclass of nonfundamental rogue waves are analyzed in details. By proper means of the regulations of free parameters, the dynamics of multi-rogue waves and high-order rogue waves have been illustrated in (x,t) plane and (y,z) plane by three dimensional figures.
Self-similar solutions for a nonlinear radiation diffusion equation
Garnier, Josselin; Malinie, Guy; Saillard, Yves; Cherfils-Clerouin, Catherine
2006-09-15
This paper considers the hydrodynamic equations with nonlinear conduction when the internal energy and the opacity have power-law dependences in the density and in the temperature. This system models the situation in which a dense solid is brought into contact with a thermal bath. It supports self-similar solutions that depend on the surface temperature. The self-similar solution can exhibit a shock wave followed by an ablation front if the surface temperature does not increase too fast in time, but it can exhibit a heat front followed by an isothermal shock otherwise. These flows are carefully studied in order to clarify the role of the initial solid density in the energy absorption and the ablation process. Comparisons with numerical simulations show excellent agreement.
A globalization procedure for solving nonlinear systems of equations
NASA Astrophysics Data System (ADS)
Shi, Yixun
1996-09-01
A new globalization procedure for solving a nonlinear system of equationsF(x)D0 is proposed based on the idea of combining Newton step and the steepest descent step WITHIN each iteration. Starting with an arbitrary initial point, the procedure converges either to a solution of the system or to a local minimizer off(x)D1/2F(x)TF(x). Each iteration is chosen to be as close to a Newton step as possible and could be the Newton step itself. Asymptotically the Newton step will be taken in each iteration and thus the convergence is quadratic. Numerical experiments yield positive results. Further generalizations of this procedure are also discussed in this paper.
Discrete nonlinear Schrödinger equation with defects.
Trombettoni, A; Smerzi, A; Bishop, A R
2003-01-01
We investigate the dynamical properties of the one-dimensional discrete nonlinear Schrödinger equation (DNLS) with periodic boundary conditions and with an arbitrary distribution of on-site defects. We study the propagation of a traveling plane wave with momentum k: the dynamics in Fourier space mainly involves two localized states with momenta +/-k (corresponding to a transmitted and a reflected wave). Within a two-mode ansatz in Fourier space, the dynamics of the system maps on a nonrigid pendulum Hamiltonian. The several analytically predicted (and numerically confirmed) regimes include states with a vanishing time average of the rotational states (implying complete reflections and refocusing of the incident wave), oscillations around fixed points (corresponding to quasi-stationary states), and, above a critical value of the nonlinearity, self-trapped states (with the wave traveling almost undisturbed through the impurity). We generalize this treatment to the case of several traveling waves and time-dependent defects. The validity of the two-mode ansatz and the continuum limit of the DNLS are discussed.
Hyperbolicity of the Nonlinear Models of Maxwell's Equations
NASA Astrophysics Data System (ADS)
Serre, Denis
. We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class Hs with s>1+d/2. The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex. Since the energy conservation law does not derive from the system of conservation laws itself (Faraday's and Ampère's laws), but also needs the compatibility relations divB=divD=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.
Controlled Nonlinear Stochastic Delay Equations: Part I: Modeling and Approximations
Kushner, Harold J.
2012-08-15
This two-part paper deals with 'foundational' issues that have not been previously considered in the modeling and numerical optimization of nonlinear stochastic delay systems. There are new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. There are two basic and interconnected themes for these models. The first, dealt with in this part, concerns the definition of admissible control. The classical definition of an admissible control as a nonanticipative relaxed control is inadequate for these models and needs to be extended. This is needed for the convergence proofs of numerical approximations for optimal controls as well as to have a well-defined model. It is shown that the new classes of admissible controls do not enlarge the range of the value functions, is closed (together with the associated paths) under weak convergence, and is approximatable by ordinary controls. The second theme, dealt with in Part II, concerns transportation equation representations, and their role in the development of numerical algorithms with much reduced memory and computational requirements.
Chaos control in the nonlinear Schrödinger equation with Kerr law nonlinearity
NASA Astrophysics Data System (ADS)
Yin, Jiu-Li; Zhao, Liu-Wei; Tian, Li-Xin
2014-02-01
The nonlinear Schrödinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.
Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrödinger Equations
NASA Astrophysics Data System (ADS)
Asad-uz-zaman, M.; Chachou Samet, H.; Khawaja, U. Al
2016-08-01
We derive the Lax pairs and integrability conditions of the nonlinear Schrödinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrödinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrödinger equation.
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.
Nonlinear Schrödinger equation with spatiotemporal perturbations.
Mertens, Franz G; Quintero, Niurka R; Bishop, A R
2010-01-01
We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation (NLSE) with the following perturbations: nonparametric spatiotemporal driving of the form f(x,t)=a exp[iK(t)x], damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations (ODEs) for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force f(x). The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of f(x). In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum P(t) and the soliton velocity V(t): This is a parameter representation of a curve P(V) which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
NASA Technical Reports Server (NTRS)
Tadmor, Eitan
1989-01-01
Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.
ON NONLINEAR EQUATIONS OF THE FORM F(x,\\, u,\\, Du,\\, \\Delta u) = 0
NASA Astrophysics Data System (ADS)
Soltanov, K. N.
1995-02-01
The Dirichlet problem for equations of the form F(x,\\, u,\\, Du,\\, \\Delta u) = 0 and also the initial boundary value problem for a parabolic equation with a nonlinearity are studied.Bibliography: 11 titles.
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.
Carasso, Alfred S
2013-01-01
Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930’s, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes. PMID:26401430
NASA Astrophysics Data System (ADS)
Parand, K.; Shahini, M.; Dehghan, Mehdi
2009-12-01
Lane-Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane-Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss-Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.
Analytic study of solutions for the Born-Infeld equation in nonlinear electrodynamics
NASA Astrophysics Data System (ADS)
Gao, Hui; Xu, Tianzhou; Fan, Tianyou; Wang, Gangwei
2017-03-01
The Born-Infeld equation is an important nonlinear partial differential equation in theoretical and mathematical physics. The Lie group method is used for simplifying the nonlinear partial differential equation, which is partly solved, in which there are some difficulties; to overcome the difficulties, we develop a power series method, and find the solutions in analytic form. In the mean time, a wave propagation (traveling wave) method is developed for solving the equation, and analytic solutions are also constructed.
Higher-order nonlinear Schrodinger equations for simulations of surface wavetrains
NASA Astrophysics Data System (ADS)
Slunyaev, Alexey
2016-04-01
Numerous recent results of numerical and laboratory simulations of waves on the water surface claim that solutions of the weakly nonlinear theory for weakly modulated waves in many cases allow a smooth generalization to the conditions of strong nonlinearity and dispersion, even when the 'envelope' is difficult to determine. The conditionally 'strongly nonlinear' high-order asymptotic equations still imply the smallness of the parameter employed in the asymptotic series. Thus at some (unknown a priori) level of nonlinearity and / or dispersion the asymptotic theory breaks down; then the higher-order corrections become useless and may even make the description worse. In this paper we use the higher-order nonlinear Schrodinger (NLS) equation, derived in [1] (the fifth-order NLS equation, or next-order beyond the classic Dysthe equation [2]), for simulations of modulated deep-water wave trains, which attain very large steepness (below or beyond the breaking limit) due to the Benjamin - Feir instability. The results are compared with fully nonlinear simulations of the potential Euler equations as well as with the weakly nonlinear theories represented by the nonlinear Schrodinger equation and the classic Dysthe equation with full linear dispersion [2]. We show that the next-order Dysthe equation can significantly improve the description of strongly nonlinear wave dynamics compared with the lower-order asymptotic models. [1] A.V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water. JETP 101, 926-941 (2005). [2] K. Trulsen, K.B. Dysthe, A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281-289 (1996).
New Traveling Wave Solutions for a Class of Nonlinear Evolution Equations
NASA Astrophysics Data System (ADS)
Bai, Cheng-Jie; Zhao, Hong; Xu, Heng-Ying; Zhang, Xia
The deformation mapping method is extended to solve a class of nonlinear evolution equations (NLEEs). Many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, and Jacobian elliptic function solutions, are obtained by a simple algebraic transformation relation between the solutions of the NLEEs and those of the cubic nonlinear Klein-Gordon (NKG) equation.
Soliton theory of two-dimensional lattices: the discrete nonlinear schrödinger equation.
Arévalo, Edward
2009-06-05
We theoretically investigate the motion of collective excitations in the two-dimensional nonlinear Schrödinger equation with cubic nonlinearity. The form of these excitations for a broad range of parameters is derived. Their evolution and interaction is numerically studied and the modulation instability is discussed. The case of saturable nonlinearity is revisited.
Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation.
Wang, L H; Porsezian, K; He, J S
2013-05-01
In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.
Bursting processes in plasmas and relevant nonlinear model equations
Basu, B.; Coppi, B.
1995-01-01
Important intrinsic plasma instabilities manifest themselves in the form of periodic bursts of fluctuations rather than as a state of stationary fluctuations, which a conventional application of quasilinear theory would lead to expect. A set of coupled nonlinear equations for the time evolution of the fluctuation amplitude and of the driving factor of the relevant instability is shown to have the features necessary to reproduce the variety of bursts that are observed experimentally. These are the periodicity, the duration, and the shape of the bursts, special consideration being given to the excitation of modes by high-energy particle populations in thermalized plasmas and to a model for the transition from a bursting state to one of stationary fluctuations. A model is introduced that is relevant to the case where the spatial dependence of the mode amplitude is important. The application of the given analysis to the bursty wave emissions observed in space is discussed. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.
Nonlinear grid error effects on numerical solution of partial differential equations
NASA Technical Reports Server (NTRS)
Dey, S. K.
1980-01-01
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
A new solution procedure for a nonlinear infinite beam equation of motion
NASA Astrophysics Data System (ADS)
Jang, T. S.
2016-10-01
Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.
Motsa, S S; Magagula, V M; Sibanda, P
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.
Motsa, S. S.; Magagula, V. M.; Sibanda, P.
2014-01-01
This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252
NASA Astrophysics Data System (ADS)
Yan, Zhenya
2003-04-01
In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.
Analytical solutions with the improved (G’/G)-expansion method for nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Bekir, Ahmet; Akbulut, Arzu
2016-10-01
To seek the exact solutions of nonlinear partial differential equations (NPDEs), the improved (G'/G)-expansion method is proposed in the present work. With the aid of symbolic computation, this effective method is applied to construct exact solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and (3+1)- dimensional Kudryashov-Sinelshchikov equation. As a result, new types of exact solutions are obtained.
Exact finite difference schemes for the non-linear unidirectional wave equation
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1985-01-01
Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.
Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions.
Guo, Boling; Ling, Liming; Liu, Q P
2012-02-01
In this paper, we construct a generalized Darboux transformation for the nonlinear Schrödinger equation. The associated N-fold Darboux transformation is given in terms of both a summation formula and determinants. As applications, we obtain compact representations for the Nth-order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third-order rogue wave is discussed and shown to exhibit interesting structures.
Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics
NASA Astrophysics Data System (ADS)
Mirzazadeh, Mohammad; Ekici, Mehmet; Sonmezoglu, Abdullah; Ortakaya, Sami; Eslami, Mostafa; Biswas, Anjan
2016-05-01
This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi's elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
NASA Astrophysics Data System (ADS)
Zhang, B.; Billings, S. A.
2015-08-01
Although a vast number of techniques for the identification of nonlinear discrete-time systems have been introduced, the identification of continuous-time nonlinear systems is still extremely difficult. In this paper, the Nonlinear Difference Equation with Moving Average noise (NDEMA) model which is a general representation of nonlinear systems and contains, as special cases, both continuous-time and discrete-time models, is first proposed. Then based on this new representation, a systematic framework for the identification of nonlinear continuous-time models is developed. The new approach can not only detect the model structure and estimate the model parameters, but also work for noisy nonlinear systems. Both simulation and experimental examples are provided to illustrate how the new approach can be applied in practice.
Perfectly matched layer absorbing boundary condition for nonlinear two-fluid plasma equations
NASA Astrophysics Data System (ADS)
Sun, X. F.; Jiang, Z. H.; Hu, X. W.; Zhuang, G.; Jiang, J. F.; Guo, W. X.
2015-04-01
Numerical instability occurs when coupled Maxwell equations and nonlinear two-fluid plasma equations are solved using finite difference method through parallel algorithm. Thus, a perfectly matched layer (PML) boundary condition is set to avoid the instability caused by velocity and density gradients between vacuum and plasma. A splitting method is used to first decompose governing equations to time-dependent nonlinear and linear equations. Then, a proper complex variable is used for the spatial derivative terms of the time-dependent nonlinear equation. Finally, with several auxiliary function equations, the governing equations of the absorbing boundary condition are derived by rewriting the frequency domain PML in the original physical space and time coordinates. Numerical examples in one- and two-dimensional domains show that the PML boundary condition is valid and effective. PML stability depends on the absorbing coefficient and thickness of absorbing layers.
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
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.
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.
NASA Technical Reports Server (NTRS)
Simon, M. K.
1980-01-01
A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated.
Integrable pair-transition-coupled nonlinear Schrödinger equations.
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.
Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave
NASA Astrophysics Data System (ADS)
Khachatryan, A. Kh.; Khachatryan, Kh. A.
2016-11-01
We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L 1[-r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.
Nonexistence of small, odd breathers for a class of nonlinear wave equations
NASA Astrophysics Data System (ADS)
Kowalczyk, Michał; Martel, Yvan; Muñoz, Claudio
2016-11-01
In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and φ ^4 and φ ^6 models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9-74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one.
Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)
1994-01-01
This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America
Slyusarchuk, V. E. E-mail: V.Ye.Slyusarchuk@NUWM.rv.ua
2014-06-01
The well-known theorems of Favard and Amerio on the existence of almost periodic solutions to linear and nonlinear almost periodic differential equations depend to a large extent on the H-classes and the requirement that the bounded solutions of these equations be separated. The present paper provides different conditions for the existence of almost periodic solutions. These conditions, which do not depend on the H-classes of the equations, are formulated in terms of a special functional on the set of bounded solutions of the equations under consideration. This functional is used, in particular, to test whether solutions are separated. Bibliography: 24 titles. (paper)
Ndzana, Fabien; Mohamadou, Alidou; Kofané, Timoléon C
2008-12-01
We study wave propagation in a nonlinear transmission line with dissipative elements. We show analytically that the telegraphers' equations of the electrical transmission line can be modeled by a pair of continuous coupled complex Ginzburg-Landau equations, coupled by purely nonlinear terms. Based on this system, we investigated both analytically and numerically the modulational instability (MI). We produce characteristics of the MI in the form of typical dependence of the instability growth rate on the wavenumbers and system parameters. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. We find that the initial modulated plane wave disintegrates into waves train. An apparently turbulent state takes place in the system during the propagation.
NASA Astrophysics Data System (ADS)
Rashidi, M. M.; Erfani, E.
2009-09-01
In this study, we present a numerical comparison between the differential transform method (DTM) and the homotopy analysis method (HAM) for solving Burgers' and nonlinear heat transfer problems. The first differential equation is the Burgers' equation serves as a useful model for many interesting problems in applied mathematics. The second one is the modeling equation of a straight fin with a temperature dependent thermal conductivity. In order to show the effectiveness of the DTM, the results obtained from the DTM is compared with available solutions obtained using the HAM [M.M. Rashidi, G. Domairry, S. Dinarvand, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708-717; G. Domairry, M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 489-499] and whit exact solutions. The method can easily be applied to many linear and nonlinear problems. It illustrates the validity and the great potential of the differential transform method in solving nonlinear partial differential equations. The obtained results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations and nonlinear ordinary differential equations that we are found to be in good agreement with the exact solutions.
Stationary states of extended nonlinear Schrödinger equation with a source
NASA Astrophysics Data System (ADS)
Borich, M. A.; Smagin, V. V.; Tankeev, A. P.
2007-02-01
Structure of nonlinear stationary states of the extended nonlinear Schrödinger equation (ENSE) with a source has been analyzed with allowance for both third-order and nonlinearity dispersion. A new class of particular solutions (solitary waves) of the ENSe has been obtained. The scenario of the destruction of these states under the effect of an external perturbation has been investigated analytically and numerically. The results obtained can be used to interpret experimental data on the weakly nonlinear dynamics of the magnetostatic envelope in heterophase ferromagnet-insulator-metal, metal-insulator-ferromagnet-insulator-metal, and other similar structures and upon the simulation of nonlinear processes in optical systems.
The (G'/G)-expansion method for the nonlinear time fractional differential equations
NASA Astrophysics Data System (ADS)
Unsal, Omer; Guner, Ozkan; Bekir, Ahmet; Cevikel, Adem C.
2017-01-01
In this paper, we obtain exact solutions of two time fractional differential equations using Jumarie's modified Riemann-Liouville derivative which is encountered in mathematical physics and applied mathematics; namely (3 + 1)-dimensional time fractional KdV-ZK equation and time fractional ADR equation by using fractional complex transform and (G/'G )-expansion method. It is shown that the considered transform and method are very useful in solving nonlinear fractional differential equations.
NASA Astrophysics Data System (ADS)
Gupta, A. K.; Ray, S. Saha
2014-09-01
In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
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.
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
Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2015-02-01
We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.
Analytical-numerical solution of a nonlinear integrodifferential equation in econometrics
NASA Astrophysics Data System (ADS)
Kakhktsyan, V. M.; Khachatryan, A. Kh.
2013-07-01
A mixed problem for a nonlinear integrodifferential equation arising in econometrics is considered. An analytical-numerical method is proposed for solving the problem. Some numerical results are presented.
Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations
NASA Astrophysics Data System (ADS)
Byun, Sun-Sig; Lee, Mikyoung; Palagachev, Dian K.
2016-03-01
We prove global regularity in weighted Lebesgue spaces for the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equations. As a consequence, regularity in Morrey spaces of the Hessian is derived as well.
NASA Astrophysics Data System (ADS)
Yang, Zhijian; Liu, Zhiming
2017-03-01
The paper investigates the well-posedness and the longtime dynamics of the quasilinear wave equations with structural damping and supercritical nonlinearities: {{u}tt}- Δ u+{{≤ft(- Δ \\right)}α}{{u}t}-\
Evolution of higher order nonlinear equation for the dust ion-acoustic waves in nonextensive plasma
Yasmin, S.; Asaduzzaman, M.; Mamun, A. A.
2012-10-15
There are three different types of nonlinear equations, namely, Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and mixed modified K-dV (mixed mK-dV) equations, for the nonlinear propagation of the dust ion-acoustic (DIA) waves. The effects of electron nonextensivity on DIA solitary waves propagating in a dusty plasma (containing negatively charged stationary dust, inertial ions, and nonextensive q distributed electrons) are examined by solving these nonlinear equations. The basic features of mixed mK-dV (higher order nonlinear equation) solitons are found to exist beyond the K-dV limit. The properties of mK-dV solitons are compared with those of mixed mK-dV solitons. It is found that both positive and negative solitons are obtained depending on the q (nonextensive parameter).
Analytic solutions for time-dependent Schrödinger equations with linear of nonlinear Hamiltonians
NASA Astrophysics Data System (ADS)
Adomian, G.; Efinger, H. J.
1994-10-01
The decomposition method is applied to the time-dependent Schrödinger equation for linear or nonlinear Hamiltonian operators, without linearization, perturbation, or numerical methods, to obtain a rapidly converging analytic solution
Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential
NASA Astrophysics Data System (ADS)
Alfimov, G. L.; Avramenko, A. I.
2013-07-01
We study nonlinear states for the NLS-type equation with additional periodic potential U(x), also called the Gross-Pitaevskii equation, GPE, in theory of Bose-Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N. These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.
Solving nonlinear or stiff differential equations by Laplace homotopy analysis method(LHAM)
NASA Astrophysics Data System (ADS)
Chong, Fook Seng; Lem, Kong Hoong; Wong, Hui Lin
2015-10-01
The initial value problems of nonlinear or stiff ordinary differential equation appear in many fields of engineering science, particularly in the studies of electrical circuits, chemical reactions, wave vibration and so on. In this research, the standard homotopy analysis method hybrids with Laplace transform method to solve nonlinear and stiff differential equations. Using this modification, the problems solved by LHAM successfully yield good solutions. Some examples are examined to highlight the convenience and effectiveness of LHAM.
On the structure of nonlinear constitutive equations for fiber reinforced composites
NASA Technical Reports Server (NTRS)
Jansson, Stefan
1992-01-01
The structure of constitutive equations for nonlinear multiaxial behavior of transversely isotropic fiber reinforced metal matrix composites subject to proportional loading was investigated. Results from an experimental program were combined with numerical simulations of the composite behavior for complex stress to reveal the full structure of the equations. It was found that the nonlinear response can be described by a quadratic flow-potential, based on the polynomial stress invariants, together with a hardening rule that is dominated by two different hardening mechanisms.
H[alpha]-stability of modified Runge-Kutta methods for nonlinear neutral pantograph equations
NASA Astrophysics Data System (ADS)
Ma, S. F.; Yang, Z. W.; Liu, M. Z.
2007-11-01
In this paper, we investigate H[alpha]-stability of algebraically stable Runge-Kutta methods with a variable stepsize for nonlinear neutral pantograph equations. As a result, the Radau IA, Radau IIA, Lobatto IIIC method, the odd-stage Gauss-Legendre methods and the one-leg [theta]-method with are H[alpha]-stable for nonlinear neutral pantograph equations. Some experiments are given.
Exact solutions of a generalized nonlinear Schrödinger equation.
Zhang, Shaowu; Yi, Lin
2008-08-01
Exact chirped soliton solutions of a generalized nonlinear Schrödinger equation with the cubic-quintic nonlinearities as well as the self-steeping were obtained using a variable parametric method. It was found that the formation of solutions is determined by the sign of a joint parameter solely. By performing numerical simulations, the chirped solutions are stable under perturbations.
Infinite hierarchy of nonlinear Schrödinger equations and their solutions.
Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.
Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades
NASA Technical Reports Server (NTRS)
Hodges, D. H.; Dowell, E. H.
1974-01-01
The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.
An Evolution Operator Solution for a Nonlinear Beam Equation
1990-12-01
uniqueness for the parabolic problem Ug + (-A) m u+ I I- u = f (14) on RN X (0, 1). Again, certain restrictions apply. The Schr ~ dinger equation , [68:pg 823...evolution equation because of the time dependence in the definition of the operator A. He identifies conditions for the existence of a unique solution. In...The arguments for the adjoint and dissipativity are not repeated. Because of the explicit time dependence , (71) is called an evolution equation . For
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks.
Smadbeck, Patrick; Kaznessis, Yiannis N
2014-08-01
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized.
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks
Smadbeck, Patrick; Kaznessis, Yiannis N.
2014-01-01
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized. PMID:25215268
NASA Astrophysics Data System (ADS)
Nakao, Mitsuhiro
We prove the existence of global decaying solutions to the exterior problem for the Klein-Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity. To derive the required estimates of solutions we employ a 'loan' method.
Differential geometry techniques for sets of nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Solitons in coupled nonlinear Schrödinger equations with variable coefficients
NASA Astrophysics Data System (ADS)
Han, Lijia; Huang, Yehui; Liu, Hui
2014-09-01
We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright-Dark and Bright-Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.
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.
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.
Galerkin/Runge-Kutta discretizations of nonlinear parabolic equations
NASA Astrophysics Data System (ADS)
Hansen, Eskil
2007-08-01
Global error bounds are derived for full Galerkin/Runge-Kutta discretizations of nonlinear parabolic problems, including the evolution governed by the p-Laplacian with p[greater-or-equal, slanted]2. The analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and an extended B-convergence theory. The global error is bounded in L2 by [Delta]xr/2+[Delta]tq, where r is the convergence order of the Galerkin method applied to the underlying stationary problem and q is the stiff order of the algebraically stable Runge-Kutta method.
The Poincaré-Bendixson Theorem and the non-linear Cauchy-Riemann equations
NASA Astrophysics Data System (ADS)
van den Berg, J. B.; Munaò, S.; Vandervorst, R. C. A. M.
2016-11-01
Fiedler and Mallet-Paret (1989) prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
Similarity solution to fractional nonlinear space-time diffusion-wave equation
NASA Astrophysics Data System (ADS)
Costa, F. Silva; Marão, J. A. P. F.; Soares, J. C. Alves; de Oliveira, E. Capelas
2015-03-01
In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.
State-Dependent Riccati Equation Regulation of Systems with State and Control Nonlinearities
NASA Technical Reports Server (NTRS)
Beeler, Scott C.; Cox, David E. (Technical Monitor)
2004-01-01
The state-dependent Riccati equations (SDRE) is the basis of a technique for suboptimal feedback control of a nonlinear quadratic regulator (NQR) problem. It is an extension of the Riccati equation used for feedback control of linear problems, with the addition of nonlinearities in the state dynamics of the system resulting in a state-dependent gain matrix as the solution of the equation. In this paper several variations on the SDRE-based method will be considered for the feedback control problem with control nonlinearities. The control nonlinearities may result in complications in the numerical implementation of the control, which the different versions of the SDRE method must try to overcome. The control methods will be applied to three test problems and their resulting performance analyzed.
A comparison between the propagators method and the decomposition method for nonlinear equations
Azmy, Y.Y.; Protopopescu, V. ); Cacuci, D.G. . Dept. of Chemical and Nuclear Engineering)
1990-01-01
Recently, a new formalism for solving nonlinear problems has been formulated. The formalism is based on the construction of advanced and retarded propagators that generalize the customary Green's functions in linear theory. One of the main advantages of this formalism is the possibility of transforming nonlinear differential equations into nonlinear integral equations that are usually easier to handle theoretically and computationally. The aim of this paper is to compare, on an example, the performances of the propagator method with other methods used for nonlinear equations, in particular, the decomposition method. The propagator method is stable, accurate, and efficient for all initial values and time intervals considered, while the decomposition method is unstable at large time intervals, even for very conveniently chosen initial conditions. 5 refs., 4 tabs.
Forward-backward equations for nonlinear propagation in axially invariant optical systems.
Ferrando, Albert; Zacarés, Mario; Fernández de Córdoba, Pedro; Binosi, Daniele; Montero, Alvaro
2005-01-01
We present a general framework to deal with forward and backward components of the electromagnetic field in axially invariant nonlinear optical systems, which include those having any type of linear or nonlinear transverse inhomogeneities. With a minimum amount of approximations, we obtain a system of two first-order equations for forward and backward components, explicitly showing the nonlinear couplings among them. The modal approach used allows for an effective reduction of the dimensionality of the original problem from 3 + 1 (three spatial dimensions plus one time dimension) to 1 + 1 (one spatial dimension plus one frequency dimension). The new equations can be written in a spinor Dirac-like form, out of which conserved quantities can be calculated in an elegant manner. Finally, these equations inherently incorporate spatiotemporal couplings, so that they can be easily particularized to deal with purely temporal or purely spatial effects. Nonlinear forward pulse propagation and nonparaxial evolution of spatial structures are analyzed as examples.
Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation
NASA Astrophysics Data System (ADS)
Mani Rajan, M. S.; Mahalingam, A.; Uthayakumar, A.
2014-07-01
We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.
Statistical mechanics of a discrete Schrödinger equation with saturable nonlinearity.
Samuelsen, Mogens R; Khare, Avinash; Saxena, Avadh; Rasmussen, Kim Ø
2013-04-01
We study the statistical mechanics of the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. 84, 3740 (2000)] regarding the statistical mechanics of the one-dimensional DNLS equation with a cubic nonlinearity. As in this earlier study, we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is nontrivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS, in fact, becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.
Proposal for detection of QED vacuum nonlinearities in Maxwell's equations by the use of waveguides.
Brodin, G; Marklund, M; Stenflo, L
2001-10-22
We present a novel method for detecting nonlinearities, due to quantum electrodynamics through photon-photon scattering, in Maxwell's equation. The photon-photon scattering gives rise to self-interaction terms which are similar to the nonlinearities due to the polarization in nonlinear optics. These self-interaction terms vanish in the limit of parallel propagating waves, but if, instead of parallel propagating waves, the modes generated in waveguides are used, there will be a nonzero total effect. Based on this idea, we calculate the nonlinear excitation of new modes and estimate the strength of this effect. Furthermore, we suggest a principal experimental setup.
Parametric autoresonant excitation of the nonlinear Schrödinger equation.
Friedland, L; Shagalov, A G
2016-10-01
Parametric excitation of autoresonant solutions of the nonlinear Schrodinger (NLS) equation by a chirped frequency traveling wave is discussed. Fully nonlinear theory of the process is developed based on Whitham's averaged variational principle and its predictions verified in numerical simulations. The weakly nonlinear limit of the theory is used to find the threshold on the amplitude of the driving wave for entering the autoresonant regime. It is shown that above the threshold, a flat (spatially independent) NLS solution can be fully converted into a traveling wave. A simplified, few spatial harmonics expansion approach is also developed for studying this nonlinear mode conversion process, allowing interpretation as autoresonant interaction within triads of spatial harmonics.
NASA Astrophysics Data System (ADS)
Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua
2016-12-01
Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205
Yan, Zhenya; Konotop, V V
2009-09-01
It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrödinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3) space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
Construction of the wave operator for non-linear dispersive equations
NASA Astrophysics Data System (ADS)
Tsuruta, Kai Erik
In this thesis, we will study non-linear dispersive equations. The primary focus will be on the construction of the positive-time wave operator for such equations. The positive-time wave operator problem arises in the study of the asymptotics of a partial differential equation. It is a map from a space of initial data X into itself, and is loosely defined as follows: Suppose that for a solution ψlin to the dispersive equation with no non-linearity and initial data ψ +, there exists a unique solution ψ to the non-linear equation with initial data ψ0 such that ψ behaves as ψ lin as t → infinity. Then the wave operator is the map W+ that takes ψ + to ψ0. By its definition, W+ is injective. An important additional question is whether or not the map is also surjective. If so, then every non-linear solution emanating from X behaves, in some sense, linearly as it evolves (this is known as asymptotic completeness). Thus, there is some justification for treating these solutions as their much simpler linear counterparts. The main results presented in this thesis revolve around the construction of the wave operator(s) at critical non-linearities. We will study the "semi-relativistic" Schrodinger equation as well as the Klein-Gordon-Schrodinger system on R2 . In both cases, we will impose fairly general quadratic non-linearities for which conservation laws cannot be relied upon. These non-linearities fall below the scaling required to employ such tools as the Strichartz estimates. We instead adapt the "first iteration method" of Jang, Li, and Zhang to our setting which depends crucially on the critical decay of the non-linear interaction of the linear evolution. To see the critical decay in our problem, careful analysis is needed to treat the regime where one has spatial and/or time resonance.
Nonlinear Instability of the Incoherent State for the Kuramoto-Sakaguchi-Fokker-Plank Equation
NASA Astrophysics Data System (ADS)
Ha, Seung-Yeal; Xiao, Qinghua
2015-07-01
We study the nonlinear instability of the incoherent solution to the Kuramoto-Sakaguchi-Fokker-Plank (KSFP) equation in a large coupling strength regime. For our instability analysis, we construct an approximate, exponentially growing perturbation mode using an elementary energy method. This method does not require spectral information from the linearized KSFP equation or an explicit growing solution for the corresponding linear equation. When the distribution function of oscillator's natural frequencies is either a Dirac measure or a bounded function with a compact support (in a small interval around the origin), the incoherent solution is nonlinearly unstable depending on the relative sizes of the coupling strength and diffusion coefficient.
Mártin, Daniel A; Hoyuelos, Miguel
2009-11-01
We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.
NASA Astrophysics Data System (ADS)
Mártin, Daniel A.; Hoyuelos, Miguel
2009-11-01
We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.
Continuous Dependence on Modeling in the Cauchy Problem for Nonlinear Elliptic Equations.
1987-04-01
parameter 4. AMON INTRODUCTION A problem in ordinary or partial differential equations is said to properly posed if it has a unique solution in the...problem for second-order nonlinear partial differential equations , Doctoral thesis, Cornell University, Ithaca, N.Y., 1986. [6] J. Conlan and G. N. Trytten...IModeling in the Cauchy Problem for Nonlinear Elliptic Equations by Allan Bennett DT1C A z1t17n m (It C ltd n Inttt " CENTER.FOR.NAVAL.ANALYSFS 4401
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.
Bright and dark soliton solutions for some nonlinear fractional differential equations
NASA Astrophysics Data System (ADS)
Ozkan, Guner; Ahmet, Bekir
2016-03-01
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona-Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann-Liouville sense.
NASA Astrophysics Data System (ADS)
Cui-Cui, Liao; Jin-Chao, Cui; Jiu-Zhen, Liang; Xiao-Hua, Ding
2016-01-01
In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space. Project supported by the National Natural Science Foundation of China (Grant No. 11401259) and the Fundamental Research Funds for the Central Universities, China (Grant No. JUSRR11407).
NASA Technical Reports Server (NTRS)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
Soliton solution and other solutions to a nonlinear fractional differential equation
NASA Astrophysics Data System (ADS)
Guner, Ozkan; Unsal, Omer; Bekir, Ahmet; Kadem, Abdelouahab
2017-01-01
In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the ansatz method and the functional variable method are used to construct exact solutions for (3+1)-dimensional time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation. This fractional equation is turned into another nonlinear ordinary differential equation by fractional complex transform then these methods are applied to solve it. As a result, some new exact solutions obtained.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions
Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.
2015-01-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256
Nonlinear Localized Dissipative Structures for Long-Time Solution of Wave Equation
2009-07-01
Fatemi, E., Engquist, B., and Osher, S., " Numerical Solution of the High Frequency Asymptotic Expansion for the Scalar Wave Equation ", Journal of...FINAL REPORT Grant Title: Nonlinear Localized Dissipative Structures for Long-Time Solution of Wave Equation By Dr. John Steinhoff Grant number... numerical method, "Wave Confinement" (WC), is developed to efficiently solve the linear wave equation . This is similar to the originally developed
Parallel Methods for Solving Nonlinear Block Bordered Systems of Equations
1989-12-31
pendix A. It is the 741 op-amp circuit (see e.g. Sedra and Smith [1982]), which was introduced in 1966 and is currently produced by almost every analog...Computing, edited by R. Wilhelmson, University of Illinois Press. A. Sedra , K. Smith [1982], Microelectronic Circuits, CBS College Publishing. J. Smith ...741 op-amp circuits (see e.g. Smith [1971], Valkenburg [1982]). This circuit leads to a 2-level block-bordered nonlinear system, as follows. The
Existence of solutions to nonlinear Hammerstein integral equations and applications
NASA Astrophysics Data System (ADS)
Li, Fuyi; Li, Yuhua; Liang, Zhanping
2006-11-01
In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L2(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2013-11-01
Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.
NASA Technical Reports Server (NTRS)
Nemeth, Michael P.
2010-01-01
A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.
The quadratically damped oscillator: A case study of a non-linear equation of motion
NASA Astrophysics Data System (ADS)
Smith, B. R.
2012-09-01
The equation of motion for a quadratically damped oscillator, where the damping is proportional to the square of the velocity, is a non-linear second-order differential equation. Non-linear equations of motion such as this are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions. Like all second-order ordinary differential equations, it has a corresponding first-order partial differential equation, whose independent solutions constitute the constants of the motion. These constants readily provide an approximate solution correct to first order in the damping constant. They also reveal that the quadratically damped oscillator is never critically damped or overdamped, and that to first order in the damping constant the oscillation frequency is identical to the natural frequency. The technique described has close ties to standard tools such as integral curves in phase space and phase portraits.
Approximation and Numerical Analysis of Nonlinear Equations of Evolution.
1980-01-31
les Espaces d’ Interpolation; Dualitg", Math. Scand., 9, 1961, pp. 147-177. 9. __ "Equations Diff~rentielles Op ~ rationnelles dan les Espaces de Hilbert...relaxation," Revue Francaise d’automatique, informatique, recherche operationnelle, R3, 1973, p. 5-32. Ill DOUGLAS, J. and GALLIE, T.MI. "On the Numerical
Existence of Forced Oscillations for Some Nonlinear Differential Equations.
1982-11-01
groups of level sets of the functional associated with the system are ", -t4 . I not trivial. Some more general results concerning systems of the type, f... general non autonomous systems of the type (1.3) 9 + v;(t,x) - 0 There is a vast literature devoted to the subject of nonlinear oscillations in systems...g(t,x) - 0 (x(t) .3) quite general results on the existence of periodic solutions have been obtained by Hartman 114] and Jacobovitz (151 (by using
Dynamics of cubic-quintic nonlinear Schrödinger equation with different parameters
NASA Astrophysics Data System (ADS)
Wei, Hua; Xue-Shen, Liu; Shi-Xing, Liu
2016-05-01
We study the dynamics of the cubic-quintic nonlinear Schrödinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter. Project supported by the National Natural Science Foundation of China (Grant Nos. 11301350, 11472124, and 11271158) and the Doctor Start-up Fund in Liaoning Province, China (Grant No. 20141050).
Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria
Frieman, E.A.; Chen, L.
1981-10-01
A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magentic field geometries. The specific case of axisymmetric tokamaks is then considered, and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating sceme, it is found that nonlinear ion Landau damping of kinetic shear-Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.
An ansatz for solving nonlinear partial differential equations in mathematical physics.
Akbar, M Ali; Ali, Norhashidah Hj Mohd
2016-01-01
In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.
Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2014-09-01
We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.
Initial value problem solution of nonlinear shallow water-wave equations.
Kânoğlu, Utku; Synolakis, Costas
2006-10-06
The initial value problem solution of the nonlinear shallow water-wave equations is developed under initial waveforms with and without velocity. We present a solution method based on a hodograph-type transformation to reduce the nonlinear shallow water-wave equations into a second-order linear partial differential equation and we solve its initial value problem. The proposed solution method overcomes earlier limitation of small waveheights when the initial velocity is nonzero, and the definition of the initial conditions in the physical and transform spaces is consistent. Our solution not only allows for evaluation of differences in predictions when specifying an exact initial velocity based on nonlinear theory and its linear approximation, which has been controversial in geophysical practice, but also helps clarify the differences in runup observed during the 2004 and 2005 Sumatran tsunamigenic earthquakes.
Freezing of nonlinear Bloch oscillations in the generalized discrete nonlinear Schrödinger equation.
Cao, F J
2004-09-01
The dynamics in a nonlinear Schrödinger chain in a homogeneous electric field is studied. We show that discrete translational invariant integrability-breaking terms can freeze the Bloch nonlinear oscillations and introduce new faster frequencies in their dynamics. These phenomena are studied by direct numerical integration and through an adiabatic approximation. The adiabatic approximation allows a description in terms of an effective potential that greatly clarifies the phenomena.
Long time behavior of some nonlinear dispersive equations
NASA Astrophysics Data System (ADS)
Deng, Yu
This thesis is divided into two parts. The first part consists of Chapters 2 and 3, in which we study the random data theory for the Benjamin-Ono equation on the periodic domain. In Chapter 2 we shall prove the invariance of the Gibbs measure associated to the Hamiltonian E1 of the equation, which was constructed in [49]. Despite the fact that the support of the Gibbs measure contains very rough functions that are not even in L2, we have successfully established the global dynamics by combining probabilistic arguments, Xs,b type estimates and the hidden structure of the equation. In Chapter 3, which is joint work with N. Tzvetkov and N. Visciglia, we extend this invariance result to the weighted Gaussian measures associated with the higher order conservation laws E2 and E3, thus completing the collection of invariant measures (except for the white noise), given the result of [51]. The second part concerns the global behavior of solutions to quasilinear dispersive systems in Rd with suitably small data. In Chapter 4 we shall prove global existence and scattering for small data solutions to systems of quasilinear Klein-Gordon equations with arbitrary speed and mass in 3 D, which extends the results in [20] and [32]. Moreover, the methods introduced here are quite general, and can be applied in a number of different situations. In Chapter 5, we briefly discuss how these methods, together with other techniques, are used in recent joint work with A. Ionescu and B. Pausader to study the 2D Euler-Maxwell system.
On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
NASA Astrophysics Data System (ADS)
Glatt-Holtz, Nathan; Mattingly, Jonathan C.; Richards, Geordie
2017-02-01
We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkina, Oxana; Rouvinskaya, Ekaterina; Talipova, Tatiana; Kurkin, Andrey; Pelinovsky, Efim
2016-10-01
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg-de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Transition behavior of the discrete nonlinear Schrödinger equation.
Rumpf, Benno
2008-03-01
Many nonlinear lattice systems exhibit high-amplitude localized structures, or discrete breathers. Such structures emerge in the discrete nonlinear Schrödinger equation when the energy is above a critical threshold. This paper studies the statistical mechanics at the transition and constructs the probability distribution in the regime where breathers emerge. The entropy as a function of the energy is nonanalytic at the transition. The entropy is independent of the energy in the regime of breathers above the transition.
Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Ma, Zheng-Yi; Ma, Song-Hua
2012-03-01
Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkin, Andrey; Talipova, Tatiana; Kurkina, Oxana; Rouvinskaya, Ekaterina; Pelinovsky, Efim
2016-04-01
Nonlinear disintegration of sine wave is studied in the framework of the Gardner equation (extended version of the Korteweg - de Vries equation with both quadratic and cubic nonlinear terms). Undular bores appear here as an intermediate stage of wave evolution. Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative solitary-like pulses. It is shown that nonlinear interaction of waves happens according to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k4/3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Conditional probability calculations for the nonlinear Schrödinger equation with additive noise.
Terekhov, I S; Vergeles, S S; Turitsyn, S K
2014-12-05
The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.
Optical rogue waves for the inhomogeneous generalized nonlinear Schrödinger equation.
Loomba, Shally; Kaur, Harleen
2013-12-01
We present optical rogue wave solutions for a generalized nonlinear Schrodinger equation by using similarity transformation. We have predicted the propagation of rogue waves through a nonlinear optical fiber for three cases: (i) dispersion increasing (decreasing) fiber, (ii) periodic dispersion parameter, and (iii) hyperbolic dispersion parameter. We found that the rogue waves and their interactions can be tuned by properly choosing the parameters. We expect that our results can be used to realize improved signal transmission through optical rogue waves.
The Painlevé test for nonlinear system of differential equations with complex chaotic behavior
NASA Astrophysics Data System (ADS)
Tsegel’nik, V.
2017-01-01
The Painlevé-analysis was performed for solutions of nonlinear third-order autonomous system of differential equations with quadratic nonlinearities on their right-hand sides. At certain values of two constant parameters incorporated into the system, the latter exhibits complex chaotic behavior. When the parameters attain the values corresponding to complex chaotic behavior, the system was found not to possess the Painlevé property.
NASA Technical Reports Server (NTRS)
Rosen, A.; Friedmann, P. P.
1978-01-01
A set of nonlinear equations of equilibrium for an elastic wind turbine or helicopter blades are presented. These equations are derived for the case of small strains and moderate rotations (slopes). The derivation includes several assumptions which are carefully stated. For the convenience of potential users the equations are developed with respect to two different systems of coordinates, the undeformed and the deformed coordinates of the blade. Furthermore, the loads acting on the blade are given in a general form so as to make them suitable for a variety of applications. The equations obtained in the study are compared with those obtained in previous studies.
1987-08-01
solution of the Korteweg-de Vries equation ( KdV ), working our way up to the derivation of the multi-soliton solution of the sine-Gordon equation (sG...SOLITARY WAVE SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS j DiS~~Uj~l. _’UDistribution/Willy Hereman AvaiiLi -itY Codes Technical Summary Report...Key Words: soliton theory, solitary waves, coupled KdV , evolution equations , direct methods, Harry Dym, sine-Gordon Mathematics Department, University
Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients.
Zhong, Wei-Ping; Belić, Milivoj R; Huang, Tingwen
2013-06-01
A similarity transformation is utilized to reduce the generalized nonlinear Schrödinger (NLS) equation with variable coefficients to the standard NLS equation with constant coefficients, whose rogue wave solutions are then transformed back into the solutions of the original equation. In this way, Ma breathers, the first- and second-order rogue wave solutions of the generalized equation, are constructed. Properties of a few specific solutions and controllability of their characteristics are discussed. The results obtained may raise the possibility of performing relevant experiments and achieving potential applications.
Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution.
Sicuro, Gabriele; Rapčan, Peter; Tsallis, Constantino
2016-12-01
We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.
1981-01-08
as it propagates over a small interval, and then to correct for absorption. Another nonlinear wave equation of great interest is the Korteweg - DeVries ...acoustics are described by the second-order-nonlinear wave equation , which is derived in this thesis and solved by numerical means. the validity of the...no approximations are made in the second-order-nonlinear acoustic wave equation as it is solved . This represents an advance on the prior art, in which
NASA Astrophysics Data System (ADS)
Haddad, L. H.; Carr, Lincoln D.
2015-09-01
We present the theoretical and mathematical foundations of stability analysis for a Bose-Einstein condensate (BEC) at Dirac points of a honeycomb optical lattice. The combination of s-wave scattering for bosons and lattice interaction places constraints on the mean-field description, and hence on vortex configurations in the Bloch-envelope function near the Dirac point. A full derivation of the relativistic linear stability equations (RLSE) is presented by two independent methods to ensure veracity of our results. Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex solutions of the nonlinear Dirac equation, which include Anderson-Toulouse skyrmions with lifetime ≈ 4 s. Beyond vortex stabilities the RLSE provide insight into the character of collective superfluid excitations, which we find to encode several established theories of physics. In particular, the RLSE reduce to the Andreev equations, in the nonrelativistic and semiclassical limits, the Majorana equation, inside vortex cores, and the Dirac-Bogoliubov-de Gennes equations, when nearest-neighbor interactions are included. Furthermore, by tuning a mass gap, relative strengths of various spinor couplings, for the small and large quasiparticle momentum regimes, we obtain weak-strong Bardeen-Cooper-Schrieffer superconductivity, as well as fundamental wave equations such as Schrödinger, Dirac, Klein-Gordon, and Bogoliubov-de Gennes equations. Our results apply equally to a strongly spin-orbit coupled BEC in which the Laplacian contribution can be neglected.
Chen, Xiang-Jun; Lam, Wa Kun
2004-06-01
An inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions is derived by introducing an affine parameter to avoid constructing Riemann sheets. A one-soliton solution simpler than that in the literature is obtained, which is a breather and degenerates to a bright or dark soliton as the discrete eigenvalue becomes purely imaginary. The solution is mapped to that of the modified nonlinear Schrödinger equation by a gaugelike transformation, predicting some sub-picosecond solitons in optical fibers.
NASA Astrophysics Data System (ADS)
Thakkar, Dipali; Ganguli, Ranjan
2003-10-01
Nonlinear equations of motion for elastic bending and torsion of isotropic rotor blades with surface bonded piezoceramic actuators are derived using Hamilton's principle. The equations are then solved using finite element discretization in the spatial and time domain. The effect of piezoceramic actuation is investigated for bending and torsion response of a rotating beam. It is found that the centrifugal stiffening effect reduces the tip transverse bending deflection and elastic twist as the rotation speed increases. However, the effect of rotation speed on the tip elastic twist is less pronounced. The importance of nonlinear terms for accurate prediction of torsion response is also shown.
Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation
NASA Astrophysics Data System (ADS)
Xiong, Chi; Good, Michael R. R.; Guo, Yulong; Liu, Xiaopei; Huang, Kerson
2014-12-01
We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the nonrelativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.
Dynamics of the optical solitons for a (2 + 1)-dimensional nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Zuo, Da-Wei; Jia, Hui-Xian; Shan, Dong-Ming
2017-01-01
In this paper, a nonlinear Schrödinger equation (NLS) has been studied, which can describe the propagation and interaction of optical solitons in a material with x-directional localized and y-directional nonlocal non-linearities. By the aid of variable separation and transformation, bilinear forms and multi-soliton solutions of the NLS equation are attained. Propagation and interaction of the solitons are discussed. As a special case of the optical solitons, Hermite-Gaussian vortex solitons are studied: the numbers of wave crests are increase with the order of the Hermite polynomial.
NASA Astrophysics Data System (ADS)
Zhong, Wei-Ping; Belić, Milivoj R.; Xia, Yuzhou
2011-03-01
Applying Hirota's binary operator approach to the (2+1)-dimensional nonlinear Schrödinger equation with the radially variable diffraction and nonlinearity coefficients, we derive a variety of exact solutions to the equation. Based on the solitary wave solutions derived, we obtain some special soliton structures, such as the embedded, conical, circular, breathing, dromion, ring, and hyperbolic soliton excitations. For some specific choices of diffraction and nonlinearity coefficients, we discuss features of the (2+1)-dimensional multisolitonic solutions.
Use of Picard and Newton iteration for solving nonlinear ground water flow equations
Mehl, S.
2006-01-01
This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems.
Modeling taper charge with a non-linear equation
NASA Technical Reports Server (NTRS)
Mcdermott, P. P.
1985-01-01
Work aimed at modeling the charge voltage and current characteristics of nickel-cadmium cells subject to taper charge is presented. Work reported at previous NASA Battery Workshops has shown that the voltage of cells subject to constant current charge and discharge can be modeled very accurately with the equation: voltage = A + (B/(C-X)) + De to the -Ex where A, B, D, and E are fit parameters and x is amp-hr of charge removed during discharge or returned during charge. In a constant current regime, x is also equivalent to time on charge or discharge.
Darboux transformation for the NLS equation
Aktosun, Tuncay; Mee, Cornelis van der
2010-03-08
We analyze a certain class of integral equations associated with Marchenko equations and Gel'fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schroedinger equation.
Lebedev, M E; Alfimov, G L; Malomed, Boris A
2016-07-01
We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.
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.
Nonlinear Schrödinger equation on graphs: recent results and open problems.
Noja, Diego
2014-01-28
In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of the nonlinear Schrödinger (NLS) equation. Special consideration is given to the existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning the NLS equation on a star graph: the standing waves of the NLS equation on a graph with a δ interaction at the vertex, and the scattering of fast solitons through a Y-junction in the cubic case. The emphasis is on a description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.
Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions
NASA Astrophysics Data System (ADS)
Feng, Lili; Yu, Fajun; Li, Li
2017-01-01
Starting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
Jerome L.V. Lewandowski
2005-01-25
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
Kinetic equations for a density matrix describing nonlinear effects in spectral line wings
Parkhomenko, A. I. Shalagin, A. M.
2011-11-15
Kinetic quantum equations are derived for a density matrix with collision integrals describing nonlinear effects in spectra line wings. These equations take into account the earlier established inequality of the spectral densities of Einstein coefficients for absorption and stimulated radiation emission by a two-level quantum system in the far wing of a spectral line in the case of frequent collisions. The relationship of the absorption and stimulated emission probabilities with the characteristics of radiation and an elementary scattering event is found.
A Nonlinear Volterra Integrodifferential Equation Describing the Stretching of Polymeric Liquids.
1981-05-01
AD-AICO 616 WISCONSIN UNIVNMADISON MATHEMATICS RESEARCH CENTER pis l NONLINEAR VOLTERRA INTEBRODIFFERENTIA4. EQUATION DESCRIBIMS TMIETC (U MAY 61 P ...INTEGRODIFFERENTIAL EQUATION DESCRIBING THE STRETCHIN/ OF POLYMERIC LIQUIDS, P ./Markowich-nd M. Renardy Mathematics Research Center University of Wisconsin...Army Research Office National Si once Foundat inn P . 0. Box 12211 Washinqton, D.C. Or",() Research Triangle Park North Carolina 27709 .- y -6 , I
Solution to the nonlinear field equations of ten dimensional supersymmetric Yang-Mills theory
NASA Astrophysics Data System (ADS)
Mafra, Carlos R.; Schlotterer, Oliver
2015-09-01
In this paper, we present a formal solution to the nonlinear field equations of ten-dimensional super Yang-Mills theory. It is assembled from products of linearized superfields which have been introduced as multiparticle superfields in the context of superstring perturbation theory. Their explicit form follows recursively from the conformal field theory description of the gluon multiplet in the pure spinor superstring. Furthermore, superfields of higher-mass dimensions are defined and their equations of motion are spelled out.
Integrability and structural stability of solutions to the Ginzburg-Landau equation
NASA Technical Reports Server (NTRS)
Keefe, Laurence R.
1986-01-01
The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).
On a method for constructing the Lax pairs for nonlinear integrable equations
NASA Astrophysics Data System (ADS)
Habibullin, I. T.; Khakimova, A. R.; Poptsova, M. N.
2016-01-01
We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.
The non-linear coupled spin 2-spin 3 Cotton equation in three dimensions
NASA Astrophysics Data System (ADS)
Linander, Hampus; Nilsson, Bengt E. W.
2016-07-01
In the context of three-dimensional conformal higher spin theory we derive, in the frame field formulation, the full non-linear spin 3 Cotton equation coupled to spin 2. This is done by solving the corresponding Chern-Simons gauge theory system of equations, that is, using F = 0 to eliminate all auxiliary fields and thus expressing the Cotton equation in terms of just the spin 3 frame field and spin 2 covariant derivatives and tensors (Schouten). In this derivation we neglect the spin 4 and higher spin sectors and approximate the star product commutator by a Poisson bracket. The resulting spin 3 Cotton equation is complicated but can be related to linearized versions in the metric formulation obtained previously by other authors. The expected symmetry (spin 3 "translation", "Lorentz" and "dilatation") properties are verified for Cotton and other relevant tensors but some perhaps unexpected features emerge in the process, in particular in relation to the non-linear equations. We discuss the structure of this non-linear spin 3 Cotton equation but its explicit form is only presented here, in an exact but not completely refined version, in appended files obtained by computer algebra methods. Both the frame field and metric formulations are provided.
Group theoretical approach to nonlinear evolution equations of lax type III. The Boussinesq equation
NASA Astrophysics Data System (ADS)
Levi, D.; Olshanetsky, M. A.; Perelomov, A. M.; Ragnisco, O.
1980-06-01
Within the group theoretical framework recently proposed by Berezin and Perelomov, we are able to derive an abstract (operator) generalization of the classical Boussinesq equation, which possesses an infinite sequence of conserved quantities.
The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links
NASA Astrophysics Data System (ADS)
Carillo, Sandra; Fuchssteiner, Benno
1989-07-01
Explicit computation for a Kawamoto-type equation shows that there is a rich associated symmetry structure for four separate hierarchies of nonlinear integrodifferential equations. Contrary to the general belief that symmetry groups for nonlinear evolution equations in 1+1 dimensions have to be Abelian, it is shown that, in this case, the symmetry group is noncommutative. Its semisimple part is isomorphic to the affine Lie algebra A(1)1 associated to sl(2,C). In two of the additional hierarchies that were found, an explicit dependence of the independent variable occurs. Surprisingly, the generic invariance for the Kawamoto-type equation obtained in Rogers and Carillo [Phys. Scr. 36, 865 (1987)] via a reciprocal link to the Möbius invariance of the singularity equation of the Kaup-Kupershmidt (KK) equation only holds for one of the additional hierarchies of symmetry groups. Thus the generic invariance is not a universal property for the complete symmetry group of equations obtained by reciprocal links. In addition to these results, the bi-Hamiltonian formulation of the hierarchy is given. A direct Bäcklund transformation between the (KK) hierarchy and the hierarchy of singularity equation for the Caudrey-Dodd-Gibbon-Sawada-Kotera equation is exhibited: This shows that the abundant symmetry structure found for the Kawamoto equation must exist for all fifth-order equations, which are known to be completely integrable since these equations are connected either by Bäcklund transformations or reciprocal links. It is shown that similar results must hold for all hierarchies emerging out of singularity hierarchies via reciprocal links. Furthermore, general aspects of the results are discussed.
Rogue waves for a system of coupled derivative nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Chan, Hiu Ning; Malomed, Boris; Chow, Kwok Wing
2015-11-01
Previous works in the literature on water waves have demonstrated that the fourth-order evolution of gravity waves in deep water will be governed by a higher order nonlinear Schrödinger equation. In the presence of two wave trains, the system is described by a higher order coupled nonlinear Schrödinger system. Through a gauge transformation, these evolution equations are reduced to a coupled derivative nonlinear Schrödinger system. The goal here is to study rogue waves, unexpectedly large displacements from an equilibrium position, through the Hirota bilinear transformation theoretically. The connections between the onset of rogue waves and modulation instability are investigated. The range of cubic nonlinearity allowing rogue wave formation is elucidated. Under a finite group velocity mismatch between the two components, the existence regime for rogue waves is extended as compared to the case with a single wave train. The amplification ratio of the amplitude can be higher than that of the single component nonlinear Schrödinger equation. Partial financial support has been provided by the Research Grants Council through contracts HKU711713E and HKU17200815.
NASA Astrophysics Data System (ADS)
Aricò, Costanza; Lo Re, Carlo
2016-12-01
We extend a recently proposed 2D depth-integrated Finite Volume solver for the nonlinear shallow water equations with non-hydrostatic pressure distribution. The proposed model is aimed at simulating both nonlinear and dispersive shallow water processes. We split the total pressure into its hydrostatic and dynamic components and solve a hydrostatic problem and a non-hydrostatic problem sequentially, in the framework of a fractional time step procedure. The dispersive properties are achieved by incorporating the non-hydrostatic pressure component in the governing equations. The governing equations are the depth-integrated continuity equation and the depth-integrated momentum equations along the x, y and z directions. Unlike the previous non-hydrostatic shallow water solver, in the z momentum equation, we retain both the vertical local and convective acceleration terms. In the former solver, we keep only the local vertical acceleration term. In this paper, we investigate the effects of these convective terms and the possible improvements of the computed solution when these terms are not neglected in the governing equations, especially in strongly nonlinear processes. The presence of the convective terms in the vertical momentum equation leads to a numerical solution procedure, which is quite different from the one of the previous solver, in both the hydrostatic and dynamic steps. We discretize the spatial domain using unstructured triangular meshes satisfying the Generalized Delaunay property. The numerical solver is shock capturing and easily addresses wetting/drying problems, without any additional equation to solve at wet/dry interfaces. We present several numerical applications for challenging flooding processes encountered in practical aspects over irregular topography, including a new set of experiments carried out at the Hydraulics Laboratory of the University of Palermo.
A new class of exact, nonlinear solutions to the Grad-Shafranov equation
NASA Technical Reports Server (NTRS)
Roumeliotis, George
1993-01-01
We have constructed a new class of exact, nonlinear solutions to the Grad-Shafranov equation, representing force-free magnetic fields with translational symmetry. These exact solutions are pertinent to the study of magnetic structures in the solar corona that are subjected to photospheric shearing motions.
Deep-Water Waves: on the Nonlinear Schrödinger Equation and its Solutions
NASA Astrophysics Data System (ADS)
Vitanov, Nikolay K.; Chabchoub, Amin; Hoffmann, Norbert
2013-06-01
We present a brief discussion on the nonlinear Schrödinger equation for modelling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions, that can be connected to the sudden formation of extreme waves, also known as rogue waves or freak waves.
Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.
Leszczynski, Henryk; Wrzosek, Monika
2017-02-01
We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.
ERIC Educational Resources Information Center
Lee, Sik-Yum; Song, Xin-Yuan; Cai, Jing-Heng
2010-01-01
Analysis of ordered binary and unordered binary data has received considerable attention in social and psychological research. This article introduces a Bayesian approach, which has several nice features in practical applications, for analyzing nonlinear structural equation models with dichotomous data. We demonstrate how to use the software…
Tensor-GMRES method for large sparse systems of nonlinear equations
NASA Technical Reports Server (NTRS)
Feng, Dan; Pulliam, Thomas H.
1994-01-01
This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.
Dynamics of a nonautonomous soliton in a generalized nonlinear Schrödinger equation.
Yang, Zhan-Ying; Zhao, Li-Chen; Zhang, Tao; Feng, Xiao-Qiang; Yue, Rui-Hong
2011-06-01
We solve a generalized nonautonomous nonlinear Schrödinger equation analytically by performing the Darboux transformation. The precise expressions of the soliton's width, peak, and the trajectory of its wave center are investigated analytically, which symbolize the dynamic behavior of a nonautonomous soliton. These expressions can be conveniently and effectively applied to the management of soliton in many fields.
Solution blow-up for a class of parabolic equations with double nonlinearity
Korpusov, Maxim O
2013-03-31
We consider a class of parabolic-type equations with double nonlinearity and derive sufficient conditions for finite time blow-up of its solutions in a bounded domain under the homogeneous Dirichlet condition. To prove the solution blow-up we use a modification of Levine's method. Bibliography: 13 titles.
Stochastic nonlinear wave equation with memory driven by compensated Poisson random measures
Liang, Fei; Gao, Hongjun
2014-03-15
In this paper, we study a class of stochastic nonlinear wave equation with memory driven by Lévy noise. We first show the existence and uniqueness of global mild solutions using a suitable energy function. Second, under some additional assumptions we prove the exponential stability of the solutions.
NASA Technical Reports Server (NTRS)
Banks, H. T.; Kunisch, K.
1982-01-01
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
NASA Technical Reports Server (NTRS)
Kilchinskaya, G. A.
1988-01-01
The fundamental concepts of the theory of simple thermomechanical materials are applied to a thermoelastic medium. Nonlinear coupled thermoelasticity equations are derived, with free energy and heat flux approximated by polynomials of deformation and temperature invariants. Details of the derivation procedure are presented.
NASA Astrophysics Data System (ADS)
Wang, Wei-Ting; Li, Ying-Ying; Yang, Shi-Jie
2013-06-01
We study the Bose-Einstein condensate trapped in a three-dimensional spherically symmetrical potential. Exact solutions to the stationary Gross-Pitaevskii equation are obtained for properly modulated radial nonlinearity. The solutions contain vortices with different winding numbers and exhibit the shell-soliton feature in the radial distributions.
ERIC Educational Resources Information Center
Butner, Jonathan; Amazeen, Polemnia G.; Mulvey, Genna M.
2005-01-01
The authors present a dynamical multilevel model that captures changes over time in the bidirectional, potentially asymmetric influence of 2 cyclical processes. S. M. Boker and J. Graham's (1998) differential structural equation modeling approach was expanded to the case of a nonlinear coupled oscillator that is common in bimanual coordination…
Maximum Likelihood Estimation of Nonlinear Structural Equation Models with Ignorable Missing Data
ERIC Educational Resources Information Center
Lee, Sik-Yum; Song, Xin-Yuan; Lee, John C. K.
2003-01-01
The existing maximum likelihood theory and its computer software in structural equation modeling are established on the basis of linear relationships among latent variables with fully observed data. However, in social and behavioral sciences, nonlinear relationships among the latent variables are important for establishing more meaningful models…
Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data
ERIC Educational Resources Information Center
Lee, Sik-Yum
2006-01-01
A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm is used to produce the joint Bayesian estimates of…
ERIC Educational Resources Information Center
Mooijaart, Ab; Satorra, Albert
2009-01-01
In this paper, we show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable to detect the presence of nonlinear terms in the model. As an example, we consider a regression model with latent variables and interactions terms. Not only the model test has zero power against that type of…
A novel approach to solve nonlinear Fredholm integral equations of the second kind.
Li, Hu; Huang, Jin
2016-01-01
In this paper, we present a novel approach to solve nonlinear Fredholm integral equations of the second kind. This algorithm is constructed by the integral mean value theorem and Newton iteration. Convergence and error analysis of the numerical solutions are given. Moreover, Numerical examples show the algorithm is very effective and simple.
Bounds on the Fourier coefficients for the periodic solutions of non-linear oscillator equations
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1988-01-01
The differential equations describing nonlinear oscillations (as seen in mechanical vibrations, electronic oscillators, chemical and biochemical reactions, acoustic systems, stellar pulsations, etc.) are investigated analytically. The boundedness of the Fourier coefficients for periodic solutions is demonstrated for two special cases, and the extrapolation of the results to higher-dimensionsal systems is briefly considered.
A quadrature based method of moments for nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin L.; Vedula, Prakash
2011-09-01
Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.
Study of Bunch Instabilities By the Nonlinear Vlasov-Fokker-Planck Equation
Warnock, Robert L.; /SLAC
2006-07-11
Instabilities of the bunch form in storage rings may be induced through the wake field arising from corrugations in the vacuum chamber, or from the wake and precursor fields due to coherent synchrotron radiation (CSR). For over forty years the linearized Vlasov equation has been applied to calculate the threshold in current for an instability, and the initial growth rate. Increasing interest in nonlinear aspects of the motion has led to numerical solutions of the nonlinear Vlasov equation, augmented with Fokker-Planck terms to describe incoherent synchrotron radiation in the case of electron storage rings. This opens the door to much deeper studies of coherent instabilities, revealing a rich variety of nonlinear phenomena. Recent work on this topic by the author and collaborators is reviewed.
Collapse, decay, and single-|k| turbulence from a generalized nonlinear Schrödinger equation.
Cui, Shaoyan; Yu, M Y; Zhao, Dian
2013-05-01
Turbulence governed by a generalized nonlinear Schrödinger equation (GNSE) including viscous heating and nonlinear damping is numerically investigated. It is found that a large localized pulse can suffer modulational instability and then collapse into the shortest-wavelength modes, as for the classical nonlinear Schrödinger equation. However, the total energy of the nonconservative GNSE can also become constant during the collapse via local balance of energy gain and loss in the phase space. After the collapse, instead of inverse cascading into a state of strong turbulence with broad spectrum, a single-step cascade, or condensation, into modes of one predominant wavelength can occur. In fact, after attaining total energy balance the turbulent system as a whole evolves like a closed adiabatic system.
Symmetries of the TDNLS equations for weakly nonlinear dispersive MHD waves
NASA Technical Reports Server (NTRS)
Webb, G. M.; Brio, M.; Zank, G. P.
1995-01-01
In this paper we consider the symmetries and conservation laws for the TDNLS equations derived by Hada (1993) and Brio, Hunter and Johnson, to describe the propagation of weakly nonlinear dispersive MHD waves in beta approximately 1 plasmas. The equations describe the interaction of the Alfven and magnetoacoustic modes near the triple umbilic, where the fast magnetosonic, slow magnetosonic and Alfven speeds coincide and a(g)(exp 2) = V(A)(exp 2) where a(g) is the gas sound speed and V(A) is the Alfven speed. We discuss Lagrangian and Hamiltonian formulations, and similarity solutions for the equations.
On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
NASA Astrophysics Data System (ADS)
Friedlander, Susan; Vicol, Vlad
2011-11-01
We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 283-301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.
NASA Astrophysics Data System (ADS)
Martin, I.; Tirado, F.; Vazquez, L.
We present a process to achieve the solution of the two dimensional nonlinear Schrödinger equation using a multigrid technique on a distributed memory machine. Some features about the multigrid technique as its good convergence and parallel properties are explained in this paper. This makes multigrid method the optimal one to solve the systems of equations arising at each time step from an implicit numerical scheme. We give some experimental results about the parallel numerical simulation of this equation on a message passing parallel machine.
A computerized implementation of a non-linear equation to predict barrier shielding requirements.
Chamberlain, A C; Strydom, W J
1997-04-01
A non-linear equation to predict barrier shielding thickness from the work function of x- and gamma-ray generators is presented. This equation is incorporated into a model that takes into account primary, scatter, and leakage radiation components to determine the amount of shielding necessary. The case of multiple wall materials is also considered. The equation accurately models the radiation attenuation curves given in NCRP 49 for concrete and lead, thus eliminating the necessity to use graphical or tabular methods to calculate shielding thickness, which can be inaccurate.
NASA Astrophysics Data System (ADS)
Kim, Bong-Sik
Three dimensional (3D) Navier-Stokes-alpha equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter f = 2O). The Navier-Stokes-alpha equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-alpha equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters f as alpha → 0. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear "2½-dimensional" limit resonant equations for f → 0. The existence and global regularity of solutions of limit resonant equations is established, uniformly in alpha. Bootstrapping from global regularity of the limit equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-alpha equations for large f for an infinite time is established. Then, the uniform convergence of a regular solution of the 3D rotating Navier-Stokes-alpha equations (alpha ≠ 0) to the one of the original 3D rotating NavierStokes equations (alpha = 0) for f large but fixed as alpha → 0 follows; this implies "shadowing" of trajectories of the limit dynamical systems by those of the perturbed alpha-dynamical systems. All the estimates are uniform in alpha, in contrast with previous estimates in the literature which blow up as alpha → 0. Finally, the existence of global attractors as well as exponential attractors is established for large f and the estimates are uniform in alpha.
Nature of transitions in augmented discrete nonlinear Schrödinger equations.
Amritkar, R E; Kenkre, V M
1999-06-01
We investigate the nature of the transitions between free and self-trapping states occurring in systems described by augmented forms of the discrete nonlinear Schrödinger equation. These arise from an interaction between a moving quasiparticle (such as an electron or an exciton) and lattice vibrations, when the effects of nonlinearities in interaction potential and restoring force are included. We derive analytic conditions for the stability of the free state and the crossover between first- and second-order transitions. We demonstrate our results for different types of nonlinearities in the interaction potential and restoring force. We find that, depending on the type of nonlinearity, it is possible to have both first- and second-order transitions. We discuss possible hysteresis effects.
NASA Astrophysics Data System (ADS)
Bich, Dao Huy; Xuan Nguyen, Nguyen
2012-12-01
In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge-Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency-amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.
Exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium.
Petrov, E Yu; Kudrin, A V
2010-05-14
The features of propagation of intense waves are of great interest for theory and experiment in electrodynamics and acoustics. The behavior of nonlinear waves in a bounded volume is of special importance and, at the same time, is an extremely complicated problem. It seems almost impossible to find a rigorous solution to such a problem even for any model of nonlinearity. We obtain the first exact solution of this type. We present a new method for deriving exact solutions of the Maxwell equations in a nonlinear medium without dispersion and give examples of the obtained solutions that describe propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium and free electromagnetic oscillations in a cylindrical cavity resonator filled with such a medium.
NASA Technical Reports Server (NTRS)
Mcdonald, B. Edward; Plante, Daniel R.
1989-01-01
The nonlinear progressive wave equation (NPE) model was developed by the Naval Ocean Research and Development Activity during 1982 to 1987 to study nonlinear effects in long range oceanic propagation of finite amplitude acoustic waves, including weak shocks. The NPE model was applied to propagation of a generic shock wave (initial condition provided by Sandia Division 1533) in a few illustrative environments. The following consequences of nonlinearity are seen by comparing linear and nonlinear NPE results: (1) a decrease in shock strength versus range (a well-known result of entropy increases at the shock front); (2) an increase in the convergence zone range; and (3) a vertical meandering of the energy path about the corresponding linear ray path. Items (2) and (3) are manifestations of self-refraction.
Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice.
Huang, Changming; Dong, Liangwei
2016-12-15
We predict the existence of gap solitons in the nonlinear fractional Schrödinger equation (NLFSE) with an imprinted optically harmonic lattice. Symmetric/antisymmetric nonlinear localized modes bifurcate from the lower/upper edge of the first/second band in defocusing/focusing Kerr media. A unique feature we revealed is that, in focusing Kerr media, stable solitons appear in the finite bandgaps with the decrease of the Lévy index, which is in sharp contrast to the standard NLSE with a focusing nonlinearity. Nonlinear bound states composed by in-phase and out-of-phase soliton units supported by the NLFSE are also uncovered. Our work may pave the way for the study of spatial lattice solitons in fractional dimensions.
Rogue waves for a system of coupled derivative nonlinear Schrödinger equations.
Chan, H N; Malomed, B A; Chow, K W; Ding, E
2016-01-01
Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.
NASA Astrophysics Data System (ADS)
Feng, Wenqiang; Salgado, Abner J.; Wang, Cheng; Wise, Steven M.
2017-04-01
We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems - including thin film epitaxy with slope selection and the square phase field crystal model - are carried out to verify the efficiency of the scheme.
Nonlinear diffusion-wave equation for a gas in a regenerator subject to temperature gradient
NASA Astrophysics Data System (ADS)
Sugimoto, N.
2015-10-01
This paper derives an approximate equation for propagation of nonlinear thermoacoustic waves in a gas-filled, circular pore subject to temperature gradient. The pore radius is assumed to be much smaller than a thickness of thermoviscous diffusion layer, and the narrow-tube approximation is used in the sense that a typical axial length associated with temperature gradient is much longer than the radius. Introducing three small parameters, one being the ratio of the pore radius to the thickness of thermoviscous diffusion layer, another the ratio of a typical speed of thermoacoustic waves to an adiabatic sound speed and the other the ratio of a typical magnitude of pressure disturbance to a uniform pressure in a quiescent state, a system of fluid dynamical equations for an ideal gas is reduced asymptotically to a nonlinear diffusion-wave equation by using boundary conditions on a pore wall. Discussion on a temporal mean of an excess pressure due to periodic oscillations is included.
NASA Technical Reports Server (NTRS)
Kaza, K. R. V.
1980-01-01
The second-degree nonlinear equations of motion for a flexible, twisted, nonuniform, horizontal axis wind turbine blade were developed using Hamilton's principle. A mathematical ordering scheme which was consistent with the assumption of a slender beam was used to discard some higher-order elastic and inertial terms in the second-degree nonlinear equations. The blade aerodynamic loading which was employed accounted for both wind shear and tower shadow and was obtained from strip theory based on a quasi-steady approximation of two-dimensional, incompressible, unsteady, airfoil theory. The resulting equations had periodic coefficients and were suitable for determining the aeroelastic stability and response of large horizontal-axis wind turbine blades.
NASA Astrophysics Data System (ADS)
Mohammadi, Reza
2014-03-01
In this study, the exponential spline scheme is implemented to find a numerical solution of the nonlinear Schrödinger equations with constant and variable coefficients. The method is based on the Crank-Nicolson formulation for time integration and exponential spline functions for space integration. The error analysis, existence, stability, uniqueness and convergence properties of the method are investigated using the energy method. We show that the method is unconditionally stable and accurate of orders O(k+kh+h2) and O(k+kh+h4). This method is tested on three examples by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and the Gross-Pitaevskii equation. The computed results are compared wherever possible with those already available in the literature. The results show that the derived method is easily implemented and approximate the exact solution very well.
Kaikina, Elena I.
2013-11-15
We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
NASA Astrophysics Data System (ADS)
Hesthaven, J. S.; Rasmussen, J. Juul; Bergé, L.; Wyller, J.
1997-12-01
Using a stable pseudospectral multi-domain method we investigate the dynamics of localized wavefields in the extended derivative nonlinear Schrödinger equation, with particular emphasis on the critical mass and structure of the initial conditions that promote wave collapse. The results are found to correspond well with theoretical observations based on a Lagrangian approach and through comparison with solutions of the critical nonlinear Schrödinger equation. Inclusion of high-order nonlinear dissipation due to the self-induced Raman effect, leading to the Raman-extended derivative nonlinear Schrödinger equation, is found to inhibit finite-time collapse in certain cases.
The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation
NASA Astrophysics Data System (ADS)
Dong, Huanhe; Zhang, Yong; Zhang, Xiaoen
2016-07-01
A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.
Family of almost periodic Schroedinger operators
Baker, G.A. Jr.; Bessis, D.; Moussa, P.
1983-01-01
Bellisard et al., introduced a one-dimensional, almost-periodic, discrete Schroedinger operator which is defined by a parameter lambda. We allow this parameter to become complex and develop a geometric formalism to control the operator. The support of the spectrum of this operator is the Julia set of the mapping x/sup 2/ - lambda ..-->.. x. We prove that the almost-periodicity holds over wide regions of the complex lambda-plane, even though Hermiticity fails. 14 references.
Atomic Schroedinger cat-like states
Enriquez-Flores, Marco; Rosas-Ortiz, Oscar
2010-10-11
After a short overview of the basic mathematical structure of quantum mechanics we analyze the Schroedinger's antinomic example of a living and dead cat mixed in equal parts. Superpositions of Glauber kets are shown to approximate such macroscopic states. Then, two-level atomic states are used to construct mesoscopic kittens as appropriate linear combinations of angular momentum eigenkets for j = 1/2. Some general comments close the present contribution.
NASA Astrophysics Data System (ADS)
Cardoso, W. B.; Zeng, J.; Avelar, A. T.; Bazeia, D.; Malomed, B. A.
2013-08-01
Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Muñoz-Mateo-Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at |x|→∞ faster than |x|. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation, for nodeless ground states and for excited modes with one, two, three and four nodes, in two versions of the model, with steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts.
Cardoso, W B; Zeng, J; Avelar, A T; Bazeia, D; Malomed, B A
2013-08-01
Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Muñoz-Mateo-Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at |x|→∞ faster than |x|. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation, for nodeless ground states and for excited modes with one, two, three and four nodes, in two versions of the model, with steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts.
Reflections on the nonlinear equation of state in rock based on experiment
TenCate, J.A.
1995-11-01
Measurements were made of the propagation of 1-D nonlinear waves (i.e., Young`s mode) in a bar of Berea sandstone 3.8 cm in diameter and 1.8 m long. Both waveforms (time domain) and spectra (frequency domain) were measured. The experimental results were then compared with waveforms calculated from a numerical scheme based on the simple wave solution for 1-D waves in rock using a classical nonlinear equation of state. The numerical solution is written in MATLAB and runs quickly on a small personal computer. Attenuation was added by propagating the waveform a small distance, transforming the waveform into the frequency domain, and applying the attenuation, and then transforming back into the time domain and propagating the new waveform. The same method was applied earlier for nonlinear propagation of a sound wave in a tube of air by Pestorius and Blackstock. The experiments and simulations clearly demonstrate that a classical nonlinear equation of state is incomplete or inappropriate for describing or modeling nonlinear propagation in sandstone. Results from another model (Van Den Abeele, paper 2aPA8) suggest the same conclusions. [Work supported by OBES/DOE through the University of California.
NASA Astrophysics Data System (ADS)
Cherniha, Roman; King, John R.; Kovalenko, Sergii
2016-07-01
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona-Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.
NASA Astrophysics Data System (ADS)
Ran, Maohua; Zhang, Chengjian
2016-12-01
This paper focuses on numerically solving the strongly coupled nonlinear space fractional Schrödinger equations. First, the laws of conservation of mass and energy are given. Then, an implicit difference scheme is proposed, under the assumption that the analytical solution decays to zero when the space variable x tends to infinity. We show that the scheme conserves the mass and energy and is unconditionally stable with respect to the initial values. Moreover, the solvability, boundedness and convergence in the maximum norm are established. To avoid solving nonlinear systems, a linear difference scheme with two identities is proposed. Several numerical experiments are provided to confirm the theoretical results.
Instability of standing waves for a class of nonlinear Schrodinger equations
NASA Astrophysics Data System (ADS)
Shu, Ji; Zhang, Jian
2007-03-01
This paper discusses a class of nonlinear Schrodinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.
A method for exponential propagation of large systems of stiff nonlinear differential equations
NASA Technical Reports Server (NTRS)
Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C.; Russo, Thomas V.
1989-01-01
A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.
Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation.
Conforti, M; Mussot, A; Kudlinski, A; Rota Nodari, S; Dujardin, G; De Biévre, S; Armaroli, A; Trillo, S
2016-07-01
We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.
Electromigration-induced step meandering on vicinal surfaces: Nonlinear evolution equation
NASA Astrophysics Data System (ADS)
Dufay, Matthieu; Debierre, Jean-Marc; Frisch, Thomas
2007-01-01
We study the effect of a constant electrical field applied on vicinal surfaces such as the Si(111) surface. An electrical field parallel to the steps induces a meandering instability with a nonzero phase shift. Using the Burton-Cabrera-Frank model, we extend the linear stability analysis performed by Liu, Weeks, and Kandel [Phys. Rev. Lett. 81, 2743 (1998)] to the nonlinear regime for which the meandering amplitude is large. We derive an amplitude equation for the step dynamics using a highly nonlinear expansion method. We investigate numerically two limiting regimes (small and large attachment lengths) which both reveal long-time coarsening dynamics.
Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation.
Zhao, Li-Chen; Liu, Jie
2013-01-01
We investigate rogue-wave solutions in a three-component coupled nonlinear Schrödinger equation. With certain requirements on the backgrounds of components, we construct a multi-rogue-wave solution that exhibits a structure like a four-petaled flower in temporal-spatial distribution, in contrast to the eye-shaped structure in one-component or two-component systems. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.
Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation
NASA Astrophysics Data System (ADS)
Conforti, M.; Mussot, A.; Kudlinski, A.; Rota Nodari, S.; Dujardin, G.; De Biévre, S.; Armaroli, A.; Trillo, S.
2016-07-01
We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.
A weak Kantorovich existence theorem for the solution of nonlinear equations
NASA Astrophysics Data System (ADS)
Uko, Livinus U.; Argyros, Ioannis K.
2008-06-01
The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we prove a weak Kantorovich-type theorem that gives the same conclusions as the previous two results under weaker conditions. Illustrative examples are provided in the paper.
Kim, Daewook; Kim, Dojin; Hong, Keum-Shik; Jung, Il Hyo
2014-01-01
The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form Ku'' + M(|A (1/2) u|(2))Au + g(u') = 0 under suitable assumptions on K, A, M(·), and g(·). Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation g. Lastly, numerical simulations in order to verify the analytical results are given.
Kim, Dojin; Hong, Keum-Shik; Jung, Il Hyo
2014-01-01
The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form Ku′′ + M(|A1/2u|2)Au + g(u′) = 0 under suitable assumptions on K, A, M(·), and g(·). Next, we derive decay estimates of the energy under some growth conditions on the nonlinear dissipation g. Lastly, numerical simulations in order to verify the analytical results are given. PMID:24977217
Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1 + 2) dimensions
Zarmi, Yair
2014-10-15
Despite the fact that it is not integrable, the (1 + 2)-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N ≥ 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in (1 + 2) dimensions, is constructed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to “annihilate” and “create” solitons – an effect that does not have an analog in perturbed classical nonlinear evolution equations.
Global series solutions of nonlinear differential equations with shocks using Walsh functions
NASA Astrophysics Data System (ADS)
Gnoffo, Peter A.
2014-02-01
An orthonormal basis set composed of Walsh functions is used for deriving global solutions (valid over the entire domain) to nonlinear differential equations that include discontinuities. Function gn(x) of the set, a scaled Walsh function in sequency order, is comprised of n piecewise constant values (square waves) across the domain xa⩽x⩽xb. Only two square wave lengths are allowed in any function and a new derivation of the basis functions applies a fractal-like algorithm (infinitely self-similar) focused on the distribution of wave lengths. This distribution is determined by a recursive folding algorithm that propagates fundamental symmetries to successive values of n. Functions, including those with discontinuities, may be represented on the domain as a series in gn(x) with no occurrence of a Gibbs phenomenon (ringing) across the discontinuity. A much more powerful, self-mapping characteristic of the series is closure under multiplication - the product of any two Walsh functions is also a Walsh function. This self-mapping characteristic transforms the solution of nonlinear differential equations to the solution of systems of polynomial equations if the original nonlinearities can be represented as products of the dependent variables and the convergence of the series for n→∞ can be demonstrated. Fundamental operations (reciprocal, integral, derivative) on Walsh function series representations of functions with discontinuities are defined. Examples are presented for solution of the time dependent Burger's equation and for quasi-one-dimensional nozzle flow including a shock.
NASA Technical Reports Server (NTRS)
Walker, K. P.; Freed, A. D.
1991-01-01
New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.
The stationary equations of a coupled nonlinear Schrödinger system
NASA Astrophysics Data System (ADS)
Wright, Otis C.
1999-02-01
The integrable coupled nonlinear Schrödinger (CNLS) equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases [M.G. Forest, D.W. McLaughlin, D. Muraki, O.C. Wright, Non-focussing instabilities in coupled, Integrable nonlinear Schrödinger PDEs, in preparation; D.J. Muraki, O.C. Wright, D.W. McLaughlin, Birefringent optical fibers: Modulational instability in a near-integrable system, Nonlinear Processes in Physics: Proceedings of III Postdam-V Kiev Workshop, 1991, pp. 242-245; O.C. Wright, Modulational stability in a defocussing coupled nonlinear Schrödinger system, Physica D 82 (1995) 1-10], whereas the scalar NLS equation is linearly unstable only in the focussing case [M.G. Forest, J.E. Lee, Geometry and modulation theory for the periodic Schrödinger equation, in: Dafermas et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, I.M.A. Math. Appl. 2 (1986) 35-70]. These instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced Duffing oscillator [Y. Li, D.W. McLaughlin, Morse and Melnikov functions for NLS pde’s, Commun. Math. Phys. 162 (1994) 175-214; D.W. McLaughlin, E.A. Overman, Whiskered tori for integrable Pde’s: Chaotic behaviour in near integrable Pde’s, in: Keller et al. (Eds.), Surveys in Applied Mathematics, vol. 1, Chapter 2, Plenum Press, New York, 1995]. The homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the NLS and the CNLS reside in the finite-dimensional phase space of certain stationary equations [S.P. Novikov, Funct. Anal. Prilozen, 8 (3) (1974) 54-66] of the infinite hierarchy of integrable commuting flows. The correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves [O.C. Wright, Near homoclinic orbits of the focussing nonlinear Schrödinger equation
Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory
NASA Astrophysics Data System (ADS)
Newhall, Katherine A.; Vanden-Eijnden, Eric
2017-01-01
This paper is concerned with the longtime dynamics of the nonlinear wave equation in one-space dimension, u_{tt} - κ^2 u_{xx} +V'(u) =0 quad xin [0,1] where κ >0 is a parameter and V(u) is a potential bounded from below and growing at least like u^2 as |u|→ ∞. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure, and when the potential is double-welled, for example when V(u) = 1/4 (1-u^2)^2 , there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here, we quantify this phenomenon by calculating exactly via transition state theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. We also investigate numerically how the mean TST frequency compares to the rate at which a typical solution crosses this dividing surface. These numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter κ in small enough. In this case, successive transitions between the two regions are roughly uncorrelated and their dynamics can be coarse-grained to jumps in a two-state Markov chain whose rate can be deduced from the mean TST frequency. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger κ, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency
Bouaricha, A.; Schnabel, R.B.
1996-12-31
This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns, in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F({ital x}) by a quadratic model, where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two- dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations.
NASA Astrophysics Data System (ADS)
Lee, C. T.; Lee, C. C.
2015-04-01
This paper introduces a systematic approach to investigate a higher order nonlinear dispersive wave equation for modeling different wave modes. We present both the conventional KdV-type soliton and anomaly type solitons for the equation. We also show the conservation laws and Hamiltonian structures for the equation. Our results suggest that the underlying equation has more interacting soliton phenomena than one would have known for the classical KdV and Boussinesq equation.
Random search algorithm for solving the nonlinear Fredholm integral equations of the second kind.
Hong, Zhimin; Yan, Zaizai; Yan, Jiao
2014-01-01
In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.
Breather management in the derivative nonlinear Schrödinger equation with variable coefficients
Zhong, Wei-Ping; Belić, Milivoj; Malomed, Boris A.; Huang, Tingwen
2015-04-15
We investigate breather solutions of the generalized derivative nonlinear Schrödinger (DNLS) equation with variable coefficients, which is used in the description of femtosecond optical pulses in inhomogeneous media. The solutions are constructed by means of the similarity transformation, which reduces a particular form of the generalized DNLS equation into the standard one, with constant coefficients. Examples of bright and dark breathers of different orders, that ride on finite backgrounds and may be related to rogue waves, are presented. - Highlights: • Exact solutions of a generalized derivative NLS equation are obtained. • The solutions are produced by means of a transformation to the usual integrable equation. • The validity of the solutions is verified by comparing them to numerical counterparts. • Stability of the solutions is checked by means of direct simulations. • The model applies to the propagation of ultrashort pulses in optical media.
Global Stability Analysis of Some Nonlinear Delay Differential Equations in Population Dynamics
NASA Astrophysics Data System (ADS)
Huang, Gang; Liu, Anping; Foryś, Urszula
2016-02-01
By using the direct Lyapunov method and constructing appropriate Lyapunov functionals, we investigate the global stability for the following scalar delay differential equation with nonlinear term y'(t)=f(1-y(t), y(t-τ ))-cy(t), where c is a positive constant and f: {R}^2 → R is of class C^1 and satisfies some additional requirements. This equation is a generalization of the SIS model proposed by Cooke (Rocky Mt J Math 7: 253-263, 1979). Criterions of global stability for the trivial and the positive equilibria of this delay equation are given. A special case of the function f depending only on the variable y(t-τ ) is also considered. Both general and special cases of this equation are often used in biomathematical modelling.
Analytical and numerical aspects in solving the controlled 3D Gross-Pitaevskii equation
Fedele, R.; Jovanovic, D.; De Nicola, S.; Eliasson, B.; Shukla, P. K.
2009-11-10
The results of recently developed investigations, that have been carried out within the framework of the controlling potential method (CPM), are reviewed. This method allows one to decompose a three dimensional (3D) Gross-Pitaevskii equation (GPE) into the pair of coupled Schroedinger-type equations. Under suitable mathematical conditions, the solutions of the 3D controlled GPE can be constructed from the solutions of a 2D linear Schroedinger equation (the transverse component of the GPE) coupled with a 1D nonlinear Schroedinger equation (the longitudinal component of the GPE). Such decomposition allows one to cast the solutions in the form of the product of the solutions of the transverse and the longitudinal components of the GPE. The coupling between these two equations is the functional of both the transverse and the longitudinal profiles. It is shown that the CPM can be used to obtain a new class of three-dimensional solitary waves solutions of the GPE, which governs the dynamics of Bose-Einstein condensates. By imposing an external controlling potential, the desired time-dependent shape of the localized BECs is obtained. The stability of the exact solutions was checked with direct simulations of the time -dependent, three-dimensional GPE. Our simulations show that the localized condensates are stable with respect to perturbed initial conditions.
NASA Astrophysics Data System (ADS)
Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang
2014-11-01
We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose-Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton.
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
NASA Astrophysics Data System (ADS)
Grahovski, Georgi G.; Mikhailov, Alexander V.
2013-12-01
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
Localization in physical systems described by discrete nonlinear Schrodinger-type equations.
Bishop, A R; Kalosakas, G; Rasmussen, K O; Kevrekidis, P G
2003-06-01
Following a short introduction on localized modes in a model system, namely the discrete nonlinear Schrodinger equation, we present explicit results pertaining to three different physical systems described by similar equations. The applications range from the Raman scattering spectra of a complex electronic material through intrinsic localized vibrational modes, to the manifestation of an abrupt and irreversible delocalizing transition of Bose-Einstein condensates trapped in two-dimensional optical lattices, and to the instabilities of localized modes in coupled arrays of optical waveguides.
Arnold, J.; Kosson, D.S.; Garrabrants, A.; Meeussen, J.C.L.; Sloot, H.A. van der
2013-02-15
A robust numerical solution of the nonlinear Poisson-Boltzmann equation for asymmetric polyelectrolyte solutions in discrete pore geometries is presented. Comparisons to the linearized approximation of the Poisson-Boltzmann equation reveal that the assumptions leading to linearization may not be appropriate for the electrochemical regime in many cementitious materials. Implications of the electric double layer on both partitioning of species and on diffusive release are discussed. The influence of the electric double layer on anion diffusion relative to cation diffusion is examined.
Daly, Edoardo; Porporato, Amilcare
2004-11-01
Similarity solutions of the shallow-water equation with a generalized resistance term are studied for open channel flows when both inertial and gravity forces are negligible. The resulting model encompasses various particular cases that appear, in addition to mathematical hydraulics, in diverse physical phenomena, such as gravity currents, creeping flows of Newtonian and non-Newtonian fluids, thin films, and nonlinear Fokker-Planck equations. Solutions of both source-type and dam-break problems are analyzed. Closed-form solutions are discussed, when possible, along with a qualitative study of two phase-plane formulations based on two different variable transformations.
Martin, P.; Castro, E.; Haines, M.G.
2005-10-01
Tokamak equilibrium has been analyzed with the magnetohydrodynamics nonlinear momentum equation in the low vorticity case. A large simplification in the analysis is obtained in this case compared with previous general treatments for rotating plasmas in tokamaks. Now pressure is not conserved around magnetic surfaces, however, other generalized functions have been found, which are conserved on each magnetic surface. A generalized Grad-Shafranov-type equation has been also derived for this case. How to determine the gradient of these new conserved functions on each magnetic surface from their value at one point of the corresponding surface is also shown.
Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong
2015-01-23
In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.
Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies
Chang, Jing; Gao, Yixian Li, Yong
2015-05-15
Consider the one dimensional nonlinear beam equation u{sub tt} + u{sub xxxx} + mu + u{sup 3} = 0 under Dirichlet boundary conditions. We show that for any m > 0 but a set of small Lebesgue measure, the above equation admits a family of small-amplitude quasi-periodic solutions with n-dimensional Diophantine frequencies. These Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proofs are based on an infinite dimensional Kolmogorov-Arnold-Moser iteration procedure and a partial Birkhoff normal form. .
Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.
Shah, Kamal; Khan, Rahmat Ali
2016-01-01
In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.
Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains
NASA Astrophysics Data System (ADS)
Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F.
2017-02-01
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.
Sqeezing generated by a nonlinear master equation and by amplifying-dissipative Hamiltonians
NASA Technical Reports Server (NTRS)
Dodonov, V. V.; Marchiolli, M. A.; Mizrahi, Solomon S.; Moussa, M. H. Y.
1994-01-01
In the first part of this contribution we show that the master equation derived from the generalized version of the nonlinear Doebner-Goldin equation leads to the squeezing of one of the quadratures. In the second part we consider two familiar Hamiltonians, the Bateman- Caldirola-Kanai and the optical parametric oscillator; going back to their classical Lagrangian form we introduce a stochastic force and a dissipative factor. From this new Lagrangian we obtain a modified Hamiltonian that treats adequately the simultaneous amplification and dissipation phenomena, presenting squeezing, too.
Exact Nonlinear Fourth-order Equation for Two Coupled Oscillators: Metamorphoses of Resonance Curves
NASA Astrophysics Data System (ADS)
Kyzioł, J.; Okniński, A.
We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov-Bogoliubov-Mitropolsky approach - we compute the amplitude profiles, which from mathematical point of view are algebraic curves. In the present paper we investigate metamorphoses of amplitude profiles induced by changes of control parameters near singular points of these curves. It follows that dynamics changes qualitatively in the neighbourhood of a singular point.
Alkhutov, Yu A; Zhikov, V V
2014-03-31
The paper is concerned with the solvability of the initial-boundary value problem for second-order parabolic equations with variable nonlinearity exponents. In the model case, this equation contains the p-Laplacian with a variable exponent p(x,t). The problem is shown to be uniquely solvable, provided the exponent p is bounded away from both 1 and ∞ and is log-Hölder continuous, and its solution satisfies the energy equality. Bibliography: 18 titles.
Approximated Lax pairs for the reduced order integration of nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Gerbeau, Jean-Frédéric; Lombardi, Damiano
2014-05-01
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations. It is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the basis on which the solution is searched for evolves in time according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive front or wave propagation. Another difference with other reduced-order methods is that it is not based on an off-line/on-line strategy. Numerical examples are shown for the linear advection, KdV and FKPP equations, in one and two dimensions.
NASA Astrophysics Data System (ADS)
Pelinovsky, Dmitry; Penati, Tiziano; Paleari, Simone
2016-08-01
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrödinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss the applications of the discrete nonlinear Schrödinger equation in the context of existence and stability of breathers of the Klein-Gordon lattice.
Stationary dark localized modes: discrete nonlinear Schrödinger equations.
Konotop, V V; Takeno, S
1999-07-01
Various kinds of stationary dark localized modes in discrete nonlinear Schrödinger equations are considered. A criterion for the existence of such excitations is introduced and an estimation of a localization region is provided. The results are illustrated in examples of the deformable discrete nonlinear Schrödinger equation, of the model of Frenkel excitons in a chain of two-level atoms, and of the model of a one-dimensional Heisenberg ferromagnetic in the stationary phase approximation. The three models display essentially different properties. It is shown that at an arbitrary amplitude of the background it is impossible to reach strong localization of dark modes. In the meantime, in the model of Frenkel excitons, exact dark compacton solutions are found.
Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths
NASA Astrophysics Data System (ADS)
Chu, Jixun; Coron, Jean-Michel; Shang, Peipei
2015-10-01
We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on the finite interval (0, 2 kπ) where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear Korteweg-de Vries stationary equation is of dimension 1. This is for example the case if k = 1.
Analytical solutions for nonlinear cable equations with calcium dynamics. I: Derivations.
Iannella, Nicolangelo; Tanaka, Shigeru
2006-06-01
The interaction between membrane potential and internal calcium concentration plays many important roles in regulating synaptic integration and neuronal firing. In order to gain a better theoretical understanding between the voltage-calcium interaction, a nonlinear cable equation with calcium dynamics is solved analytically. This general reaction-diffusion system represents a model of a cylindrical dendritic segment in which calcium diffuses internally in the presence of buffers, pumps and exchangers, and where ion channels are sparsely distributed over the membrane,in the form of hotspots, acting as point current sources along the dendritic membrane. In order to proceed, the reaction-diffusion system is recast into a system of coupled nonlinear integral equations, with which a perturbative expansion in dimensionless voltage and calcium concentration are used to find analytical solutions to this general system. The resulting solutions can be used to investigate, the interaction between the membrane potential and underlying calcium dynamics in a natural (non-discretized) setting.
Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations
NASA Astrophysics Data System (ADS)
Feola, Roberto; Procesi, Michela
2015-10-01
In this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo-differential operators. This procedure automatically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.
Nonlinear self-adjointness and conservation laws of Klein-Gordon-Fock equation with central symmetry
NASA Astrophysics Data System (ADS)
Abdulwahhab, Muhammad Alim
2015-05-01
The concept of nonlinear self-adjointness, introduced by Ibragimov, has significantly extends approaches to constructing conservation laws associated with symmetries since it incorporates the strict self-adjointness, the quasi self-adjointness as well as the usual linear self-adjointness. Using this concept, the nonlinear self-adjointness condition for the Klein-Gordon-Fock equation was established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. The Noether's theorem was further applied to the Klein-Gordon-Fock equation to explore more distinct first integrals, result shows that conservation laws constructed through this approach are exactly the same as those obtained under strict self-adjointness of Ibragimov's method.
A HAM-based wavelet approach for nonlinear ordinary differential equations
NASA Astrophysics Data System (ADS)
Yang, Zhaochen; Liao, Shijun
2017-07-01
Based on the homotopy analysis method (HAM) and the generalized Coiflet-type orthogonal wavelet, a new analytic approximation approach for solving nonlinear boundary value problems (governed by nonlinear ordinary differential equations), namely the wavelet homotopy analysis method (wHAM), is proposed. The basic ideas of the wHAM are described using the one-dimensional Bratu's equation as an example. This method not only keeps the main advantages of the normal HAM, but also possesses some new properties and advantages. First of all, the wHAM possesses high computational efficiency. Besides, based on multi-resolution analysis, it provides us a convenient way to balance the accuracy and efficiency by simply adjusting the resolution level. Furthermore, different from the normal HAM, the wHAM provides us much larger freedom to choose the auxiliary linear operator. In addition, just like the normal HAM, iteration can greatly accelerate the computational efficiency of the wHAM without loss of accuracy.
NASA Astrophysics Data System (ADS)
Hafez, M. G.; Talukder, M. R.; Ali, M. Hossain
2017-03-01
The Burgers equation is obtained to study the characteristics of nonlinear propagation of ion-acoustic shock, singular kink, and periodic waves in weakly relativistic plasmas containing relativistic thermal ions, nonextensive distributed electrons, Boltzmann distributed positrons, and kinematic viscosity of ions using the well-known reductive perturbation technique. This equation is solved by employing the (G'/G)-expansion method taking unperturbed positron-to-electron concentration ratio, electron-to-positron temperature ratio, strength of electrons nonextensivity, ion kinematic viscosity, and weakly relativistic streaming factor. The influences of plasma parameters on nonlinear propagation of ion-acoustic shock, periodic, and singular kink waves are displayed graphically and the relevant physical explanations are described. It is found that these parameters extensively modify the shock structures excitation. The obtained results may be useful in understanding the features of small but finite amplitude localized relativistic ion-acoustic shock waves in an unmagnetized plasma system for some astrophysical compact objects and space plasmas.
A Haar wavelet collocation method for coupled nonlinear Schrödinger-KdV equations
NASA Astrophysics Data System (ADS)
Oruç, Ömer; Esen, Alaattin; Bulut, Fatih
2016-04-01
In this paper, to obtain accurate numerical solutions of coupled nonlinear Schrödinger-Korteweg-de Vries (KdV) equations a Haar wavelet collocation method is proposed. An explicit time stepping scheme is used for discretization of time derivatives and nonlinear terms that appeared in the equations are linearized by a linearization technique and space derivatives are discretized by Haar wavelets. In order to test the accuracy and reliability of the proposed method L2, L∞ error norms and conserved quantities are used. Also obtained results are compared with previous ones obtained by finite element method, Crank-Nicolson method and radial basis function meshless methods. Error analysis of Haar wavelets is also given.
A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Gong, Yuezheng; Wang, Qi; Wang, Yushun; Cai, Jiaxiang
2017-01-01
A Fourier pseudo-spectral method that conserves mass and energy is developed for a two-dimensional nonlinear Schrödinger equation. By establishing the equivalence between the semi-norm in the Fourier pseudo-spectral method and that in the finite difference method, we are able to extend the result in Ref. [56] to prove that the optimal rate of convergence of the new method is in the order of O (N-r +τ2) in the discrete L2 norm without any restrictions on the grid ratio, where N is the number of modes used in the spectral method and τ is the time step size. A fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method. Numerical examples are presented to show the efficiency and accuracy of the new method.
Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Meza, L. E. Arroyo; Dutra, A. de Souza; Hott, M. B.; Roy, P.
2015-01-01
By using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with cubic and quintic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) coefficients. We obtain a class of wide localized exact solutions of NLSE in the presence of a number of non-Hermitian parity-time (PT )-symmetric external potentials, which are constituted by a mixing of external potentials and source or drain terms. The exact solutions found here can be applied to theoretical studies of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. We show that, even in the presence of gain or loss terms, stable solutions can be found and that the PT symmetry is an important feature to guarantee the conservation of the average energy of the system.
Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations.
Chen, Shihua; Baronio, Fabio; Soto-Crespo, Jose M; Liu, Yi; Grelu, Philippe
2016-06-01
We shed light on the fundamental form of the Peregrine soliton as well as on its frequency chirping property by virtue of a pertinent cubic-quintic nonlinear Schrödinger equation. An exact generic Peregrine soliton solution is obtained via a simple gauge transformation, which unifies the recently-most-studied fundamental rogue-wave species. We discover that this type of Peregrine soliton, viable for both the focusing and defocusing Kerr nonlinearities, could exhibit an extra doubly localized chirp while keeping the characteristic intensity features of the original Peregrine soliton, hence the term chirped Peregrine soliton. The existence of chirped Peregrine solitons in a self-defocusing nonlinear medium may be attributed to the presence of self-steepening effect when the latter is not balanced out by the third-order dispersion. We numerically confirm the robustness of such chirped Peregrine solitons in spite of the onset of modulation instability.
Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Chen, Shihua; Baronio, Fabio; Soto-Crespo, Jose M.; Liu, Yi; Grelu, Philippe
2016-06-01
We shed light on the fundamental form of the Peregrine soliton as well as on its frequency chirping property by virtue of a pertinent cubic-quintic nonlinear Schrödinger equation. An exact generic Peregrine soliton solution is obtained via a simple gauge transformation, which unifies the recently-most-studied fundamental rogue-wave species. We discover that this type of Peregrine soliton, viable for both the focusing and defocusing Kerr nonlinearities, could exhibit an extra doubly localized chirp while keeping the characteristic intensity features of the original Peregrine soliton, hence the term chirped Peregrine soliton. The existence of chirped Peregrine solitons in a self-defocusing nonlinear medium may be attributed to the presence of self-steepening effect when the latter is not balanced out by the third-order dispersion. We numerically confirm the robustness of such chirped Peregrine solitons in spite of the onset of modulation instability.
NASA Astrophysics Data System (ADS)
Lou, Sen-yue
1998-05-01
To study a nonlinear partial differential equation (PDE), the Painleve expansion developed by Weiss, Tabor and Carnevale (WTC) is one of the most powerful methods. In this paper, using any singular manifold, the expansion series in the usual Painleve analysis is shown to be resummable in some different ways. A simple nonstandard truncated expansion with a quite universal reduction function is used for many nonlinear integrable and nonintegrable PDEs such as the Burgers, Korteweg de-Vries (KdV), Kadomtsev-Petviashvli (KP), Caudrey-Dodd-Gibbon-Sawada-Kortera (CDGSK), Nonlinear Schrödinger (NLS), Davey-Stewartson (DS), Broer-Kaup (BK), KdV-Burgers (KdVB), λf4 , sine-Gordon (sG) etc.
Collapse for the higher-order nonlinear Schrödinger equation
Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; ...
2016-02-01
We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data,more » are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.« less
Collapse for the higher-order nonlinear Schrödinger equation
Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; Horikis, T. P.; Karachalios, N. I.; Kevrekidis, P. G.
2016-02-01
We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.
A Family of Ellipse Methods for Solving Non-Linear Equations
ERIC Educational Resources Information Center
Gupta, K. C.; Kanwar, V.; Kumar, Sanjeev
2009-01-01
This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the…