Arnold Diffusion of the Discrete Nonlinear Schrödinger Equation

Details Arnold Diffusion of the Discrete Nonlinear Schrödinger Equation Arnold Diffusion of the Discrete Nonlinear Schrödinger Equation

2006-07-18

arxiv.org/abs/math/0607443v1

Dynamics of PDE, Vol.3, no.3, (2006), 235-258

Mathematics

Dynamical Systems

24pp

Scientific paper

In this article, we prove the existence of Arnold diffusion for an interesting specific system -- discrete nonlinear Schr\"odinger equation. The proof is for the 5-dimensional case with or without resonance. In higher dimensions, the problem is open. Progresses are made by establishing a complete set of Melnikov-Arnold integrals in higher and infinite dimensions. The openness lies at the concrete computation of these Melnikov-Arnold integrals. New machineries introduced here into the topic of Arnold diffusion are the Darboux transformation and isospectral theory of integrable systems.

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