Existence of chaos for nonlinear Schrödinger equation under singular perturbations

The work \cite{Li99} is generalized to the singularly perturbednonlinear Schrödinger (NLS) equation of which the regularlyperturbed NLS studied in \cite{Li99} is a mollification. Specifically,the existence of Smale horseshoes and Bernoulli shift dynamics isestablished in a neighborhood of a symmetric pair of Silnikov homoclinicorbits under certain generic conditions, and the existence of thesymmetric pair of Silnikov homoclinic orbits has been proved in\cite{Li01}. The main difficulty in the current horseshoe constructionis introduced by the singular perturbation $\e \pa_x^2$ which turnsthe unperturbed reversible system into an irreversible system. It turnsout that the equivariant smooth linearization can still be achieved,and the Conley-Moser conditions can still be realized.

Keywords

homoclinic orbits, chaos, Samle horseshoes, equivariant smooth linearization, Conley-Moser conditions

2010 Mathematics Subject Classification

Primary 35Q30, 35Q55. Secondary 37L10, 37L50.

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Published 1 January 2004