Dynamics of Partial Differential Equations
Volume 3 (2006)
Arnold diffusion of the discrete nonlinear Schrödinger equation
Pages: 235 – 258
In this article, we prove the existence of Arnold diffusion for an interesting specific system — discrete nonlinear Schrödinger 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.
Arnold diffusion, Darboux transformation, isospectral theory, Melnikov-Arnold integrals, λ-lemma, transition chain
2010 Mathematics Subject Classification
34-xx, 35-xx, 37-xx, 76-xx, 78-xx