Existence of Chaos for a Singularly Perturbed NLS Equation

TL;DR

This paper proves the existence of chaotic dynamics, specifically Smale horseshoes and Bernoulli shifts, in a singularly perturbed nonlinear Schrödinger equation, extending previous results to a more complex, irreversible system.

Contribution

It generalizes the existence of chaos from regular to singular perturbations in the NLS equation, overcoming difficulties caused by the singular perturbation term.

Findings

Existence of Smale horseshoes near symmetric Silnikov homoclinic orbits.

Chaotic dynamics persist despite the singular perturbation.

Equivariant smooth linearization is achievable in the perturbed system.

Abstract

The work [Li,99] is generalized to the singularly perturbed nonlinear Schr\"odinger (NLS) equation of which the regularly perturbed NLS studied in [Li,99] is a mollification. Specifically, the existence of Smale horseshoes and Bernoulli shift dynamics is established in a neighborhood of a symmetric pair of Silnikov homoclinic orbits under certain generic conditions, and the existence of the symmetric pair of Silnikov homoclinic orbits has been proved in [Li,01]. The main difficulty in the current horseshoe construction is introduced by the singular perturbation $\e \pa_x^2$ which turns the unperturbed reversible system into an irreversible system. It turns out that the equivariant smooth linearization can still be achieved, and the Conley-Moser conditions can still be realized.