Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
zbMATH Open document ID: 0847.35130
DOI: 10.2307/2118656
author: Jürgen Pöschel
author: Sergei Kuksin
published in: Annals of Mathematics. Second Series
publication date: 1996-10-08T00:00:00Z
review text: This paper deals with the nonlinear Schrödinger equation \[ iu_t= u_{xx}- mu- f(|u|^2) u \] on \([0, T]\) with Dirichlet boundary conditions. After the introduction which contains the main results, the paper continues as follows. The Hamiltonian of the nonlinear equation is written in infinitely many coordinates, and its regularity is established. And then it is transformed into its Birkhoff normal form of order four. In the next section, another theorem is stated about the existence of invariant Cantor manifolds for Hamiltonians. This result is used to prove the basic theorem of the paper. The next result is reduced to a KAM-theorem regarding perturbations of families of linear Hamiltonians. Shortly, to summarize, the paper provides a number of small amplitude solutions which are quasi-periodic in time.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 854852
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: invariant Cantor manifolds
zbMATH Keywords: Hamiltonian
zbMATH Keywords: KAM-theorem
zbMATH Keywords: small amplitude solutions
Wikidata QID: Q56555288
reviewed by: Guy Jumarie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118656
OpenAlex ID: W2152673444
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description: scientific article; zbMATH DE number 854852