Lower dimensional invariant tori with prescribed frequency for the nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut−uxx+mu+|u|2u+f(|u|2)u=0, subject to Dirichlet boundary conditions is considered, where the nonlinearity f is a real analytic function near u=0 with f(0)=f′(0)=0. It is proved that for each given constant potential m and each pres...

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Bibliographic Details
Published inNonlinear analysis Vol. 92; pp. 30 - 46
Main Authors Ren, Xiufang, Geng, Jiansheng
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.11.2013
Subjects
Analytic functions
Boundary conditions
Dirichlet problem
Invariants
KAM tori
Mathematical analysis
Nonlinearity
Normal form
Quasi-periodic solutions
Schroedinger equation
Schrödinger equation
Vectors (mathematics)
KAM tori
Quasi-periodic solutions
Schrödinger equation
Normal form
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ISSN0362-546X
1873-5215
DOI10.1016/j.na.2013.07.001

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Summary:In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut−uxx+mu+|u|2u+f(|u|2)u=0, subject to Dirichlet boundary conditions is considered, where the nonlinearity f is a real analytic function near u=0 with f(0)=f′(0)=0. It is proved that for each given constant potential m and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude time quasi-periodic solutions, whose b-dimensional frequencies are just small dilation of a prescribed Diophantine vector. Accordingly, we obtain the existence of lower dimensional invariant KAM tori with tangential frequencies constrained to a given Diophantine direction in an infinite-dimensional phase space setting.
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ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2013.07.001