Arnold diffusion of the discrete nonlinear Schrödinger equation

Abstract

Abstract. 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.

Chaos in Partial Differential Equations, Navier-Stokes Equations and Turbulence

“…Arnold diffusion can be proved for discrete nonlinear Schrödinger equation under certain Hamiltonian perturbation [29].…”