Existence of chaos for nonlinear Schrödinger equation under singular perturbations

Abstract

The work [1] is generalized to the singularly perturbed nonlinear Schrödinger (NLS) equation of which the regularly perturbed NLS studied in [1] 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 [2]. The main difficulty in the current horseshoe construction is introduced by the singular perturbation ǫ∂ 2x 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. Contents 1. Introduction 225 2. Equivariant Smooth Linearization 227 3. The Poincaré Map and Its Representation 228 4. The Fixed Points of the Poincaré Map P 230 5. Existence of Chaos 231 6. Numerical Evidence for the Generic Assumptions 236 References 237

Arnold diffusion of the discrete nonlinear Schrödinger equation

“…There is a good description on the locations of these critical points z c j in the NLS setting [19]. These F j 's can be used to build a complete set of MelnikovArnold integrals for the Arnold diffusion purpose.…”

“…Denote by {F u,s ( q) : q ∈ A} the C 1 families of C 2 one dimensional unstable and stable Fenichel fibers with base points in A [19] such that for any q * ∈ F u ( q) or q * ∈ F s ( q), ( q ∈ A),…”

“…For more details on the topic of this subsection, see [19]. Expressions of unstable manifolds of tori can be generated via Darboux transformations.…”

“…Together they provide a complete set of Melnikov-Arnold integrals with elegant universal formulae. This is the case for both DNLS and NLS [17] [18] [19] [20] [21]. On the other hand, specific calculation of these integrals is the challenge.…”

“…On the other hand, specific calculation of these integrals is the challenge. For the purpose of proving the existence of chaos, often one Melnikov integral is enough and easily computable [19] [21] [22] [25] [26]. The reason is that one can utilize locally invariant center manifolds instead of KAM tori.…”