Four lectures on KAM for the non-linear Schrödinger equation

L. H. Eliasson, S. B. Kuksin

Research output: Contribution to journalArticle

Abstract

We discuss the KAM-theory for lower-dimensional tori for the non-linear Schrödinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d= 2. We discuss this difficulty, and we show that a block decomposition and a Töplitz- Lipschitz-property, present for non-linear Schrödinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06]. © 2008 Springer Science + Business Media B.V.

Original languageEnglish
Pages (from-to)179-212
Number of pages34
JournalNATO Science for Peace and Security Series B: Physics and Biophysics
DOIs
Publication statusPublished - 2008

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Nonlinear Equations
Small Divisors
Lipschitz Property
KAM Theory
Periodic Boundary Conditions
Convolution
Torus
Decompose
Business

Cite this

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abstract = "We discuss the KAM-theory for lower-dimensional tori for the non-linear Schr{\"o}dinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d= 2. We discuss this difficulty, and we show that a block decomposition and a T{\"o}plitz- Lipschitz-property, present for non-linear Schr{\"o}dinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06]. {\circledC} 2008 Springer Science + Business Media B.V.",
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Four lectures on KAM for the non-linear Schrödinger equation. / Eliasson, L. H.; Kuksin, S. B.

In: NATO Science for Peace and Security Series B: Physics and Biophysics, 2008, p. 179-212.

Research output: Contribution to journalArticle

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