Unexpected behavior of nonlinear Schrödinger solitons in an external potential (original) (raw)

Colliding Solitons for the Nonlinear Schrödinger Equation

Communications in Mathematical Physics, 2009

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L 2-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.

0 A pr 2 00 8 COLLIDING SOLITONS FOR THE NONLINEAR SCHRÖDINGER EQUATION

2021

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a spatially adiabatic external potential. For a high initial relative speed ‖v‖ of the solitons, we show that, up to times of order log ‖v‖ after the collision, the solitons preserve their shape (in L-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.

Soliton-defect collisions in the nonlinear Schrödinger equation

Physics Letters A, 1995

We report results of systematic numerical simulations of collisions between a soliton and an attractive defect described by the perturbing term in the nonlinear Schr6dinger equation proportional to the &function. This model has applications in nonlinear optics and other physical systems. In the parametric range where the defect's strength is of the same order of magnitude as the soliton's amplitude, i.e., the interaction is essentially nonlinear, only two outcomes of the collision are possible: transmission and capture. We have found a border between the corresponding parametric regions. The border is represented in a universal form which does not depend on any free parameter. In this essentially nonlinear regime of the interaction, the soliton keeps its integrity, being transmitted or captured as a whole, practically without emission of radiation. At larger values of the defect's strength, the interaction becomes nearly linear. In this case, the soliton is split into reflected, transmitted, and trapped wave packets. Using the corresponding linear Schr/Sdinger equation, we calculate analytically the shares of the reflected and transmitted energy in the limiting case when the defect's strength and the soliton's velocity are essentially larger than its amplitude.

On soliton dynamics in nonlinear schrödinger equations

GAFA Geometric And Functional Analysis, 2006

In this paper we announce the result of asymptotic dynamics of solitons of nonlinear Schrödinger equations with external potentials. To each local minima of the potential there is a soliton centered around it. Under some conditions on the nonlinearity, the potential and the datum, we prove that the solution can be decomposed into two parts: the soliton and the term dissipating to infinity. * This paper is part of the first author's Ph.D thesis. † Supported by NSERC under Grant NA7901 and NSF under Grant DMS-0400526. 1 In this case Equation (1) is called the Gross-Pitaevskii equation.

Scattering of The Vector Soliton in Coupled Nonlinear Schrödinger Equation with Gaussian Potential

Malaysian Journal of Fundamental and Applied Sciences, 2020

Nonlinear Schrodinger equation (NLSE) is the fundamental equation which describes the wave field envelope dynamics in a nonlinear and dispersive medium. However, if the fields have many components, one should consider the Coupled Nonlinear Schrodinger equation (CNLSE). We considered the interactions of orthogonally polarized and equal-amplitude vector solitons with two polarization directions. In this paper, we focused on the effect of Gaussian potential on the scattering of the vector soliton in CNLSE. The scattering process was investigated by the variational approximation method and direct numerical solution of CNLSE. Analytically, we analyzed the dynamics of the width and center of mass position of a soliton by the variational approximation method. Soliton may be reflected from each other or transmitted through or trapped. Initially, uncoupled solitons may form the coupled state if the kinetic energy of solitons less than the potential of attractive interaction between solitons ...

Soliton dynamics and scattering

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006, 2007

A survey of results and problems of soliton dynamics in dispersive and hyperbolic nonlinear PDE's and the related spectral and scattering theory. I focus on the problem of large time behavior of the nonlinear Schrödinger equation, with both solitary and radiative waves appearing in the solution. The equations are nonintegrable in general and in arbitrary dimension. I will formulate the main conjectures relevant to soliton dynamics. Mathematics Subject Classification (2000). Primary 35Qxx.

A P ] 2 2 Se p 20 06 Scattering of Solitons for Schrödinger Equation Coupled to a Particle

2006

We establish soliton-like asymptotics for finite energy solutions to the Schrödinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold in the Hilbert phase space. On leave Department Mechanics and Mathematics of Moscow State University. Supported partly by Max-Planck Institute of Mathematics in the Sciences (Leipzig), and Wolfgang Pauli Institute of Vienna University. 2

Interaction of vector solitons with a nonlinear interface

Optics Communications, 2003

We develop the analytical method of field momenta for analyzing the dynamics of optical vector solitons in photorefractive nonlinear media. First, we derive the effective evolution equations for the parameters of multi-component solitons composed of incoherently coupled beams and investigate the soliton internal oscillations associated with the relative motion of the soliton components. Then, we apply this method for analyzing the vector soliton scattering by a nonlinear interface. In particular, we show that a vector soliton can be reflected, transmitted, captured, or split into separate components, depending on the initial energy of its internal degree of freedom. The results are verified by direct numerical simulations of spatial optical solitons in photorefractive nonlinear media.

Resonant scattering of solitons

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2003

We study the scattering of solitons in the nonlinear Schrödinger equation on local inhomogeneities which may give rise to resonant transmission and reflection. In both cases, we derive resonance conditions for the soliton's velocity. The analytical predictions are tested numerically in regimes characterized by various time scales. Special attention is paid to intermode interactions and their effect on coherence, decoherence, and dephasing of plane-wave modes which build up the soliton.

Quantum scattering of solitons

Nuclear Physics B, 1976

The quantum formalism of soliton scattering is presented using the path integral collective coordinate method. Constructing the asymptotic states we develop a general method leading to a perturbation expansion about the ground state of an effective multi-soliton sector Hamiltonian. This is discussed explicitly on the two-soliton sector example of sine-Gordon theory where exact multisoliton classical solutions are known. As the first approximation in this perturbation expansion we obtain the classical phase shifts found earlier by other authors using different methods. Next, we calculate the first quantum corrections to these classical phase shifts. Our result is found to be in precise agreement with the conjectured exact soliton-antisoliton scattering amplitude of Faddeev, Kulish and Korepin. It also gives further support to the statement of Dashen, Hasslacher and Neveu that for sine-Gordon theory, the semi-classical WKB method gives exact results.