Existence of Chaos for a Singularly Perturbed NLS Equation

Details Existence of Chaos for a Singularly Perturbed NLS Equation Existence of Chaos for a Singularly Perturbed NLS Equation

2002-06-25

arxiv.org/abs/math/0206270v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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