Dynamics of Partial Differential Equations

Volume 3 (2006)

Number 3

Arnold diffusion of the discrete nonlinear Schrödinger equation

Pages: 235 – 258

DOI: http://dx.doi.org/10.4310/DPDE.2006.v3.n3.a4


Y. Charles Li (Department of Mathematics, University of Missouri, Columbia, Missouri, U.S.A.)


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

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