Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings

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Abstract

A Hamiltonian difference scheme associated with the integrable nonlinear Schrödinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution. The chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the solutions are not subject to even constraints, the traditional mechanism of crossings of the unperturbed homoclinic orbits/manifolds is not observed.

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Ablowitz, M. J., Herbst, B. M., & Schober, C. M. (1996). Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings. Physica A: Statistical Mechanics and Its Applications, 228(1–4), 212–235. https://doi.org/10.1016/0378-4371(95)00434-3

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