Four lectures on KAM for the non-linear Schrödinger equation

L. H. Eliasson; S. B. Kuksin

doi:10.1007/978-1-4020-6964-2-10

Four lectures on KAM for the non-linear Schrödinger equation

L. H. Eliasson, S. B. Kuksin

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

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 language	English
Pages (from-to)	179-212
Number of pages	34
Journal	NATO Science for Peace and Security Series B: Physics and Biophysics
DOIs	https://doi.org/10.1007/978-1-4020-6964-2-10
Publication status	Published - 2008

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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]. {\textcopyright} 2008 Springer Science + Business Media B.V.",

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AB - 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.

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JO - NATO Science for Peace and Security Series B: Physics and Biophysics

JF - NATO Science for Peace and Security Series B: Physics and Biophysics

ER -

Four lectures on KAM for the non-linear Schrödinger equation

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