Arnold Diffusion of the Discrete Nonlinear Schr\"odinger Equation
Dynamical Systems
2007-05-23 v1 Mathematical Physics
Analysis of PDEs
math.MP
Chaotic Dynamics
Abstract
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.
Cite
@article{arxiv.math/0607443,
title = {Arnold Diffusion of the Discrete Nonlinear Schr\"odinger Equation},
author = {Y. Charles Li},
journal= {arXiv preprint arXiv:math/0607443},
year = {2007}
}
Comments
24pp