Quasi-periodic solutions for derivative nonlinear  Schrödinger equation

Meina Gao; Jianjun Liu

doi:10.3934/dcds.2012.32.2101

Article Contents

2012, Volume 32, Issue 6: 2101-2123. Doi: 10.3934/dcds.2012.32.2101

This issue Previous Article A direct proof of the Tonelli's partial regularity result Next Article Generalized Stokes system in Orlicz spaces

Quasi-periodic solutions for derivative nonlinear Schrödinger equation

Meina Gao^1, and
Jianjun Liu^2,

1.
School of Science, Shanghai Second Polytechnic University, Shanghai 201209
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: February 2011

Revised: June 2011

Published: June 2012

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Abstract

In this paper, we discuss the existence of time quasi-periodic solutions for the derivative nonlinear Schrödinger equation $$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0 $$ subject to Dirichlet boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and Birkhoff normal form, we will prove that there exist a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.

Keywords:

KAM theory,
Derivative nonlinear Schrödinger equation,
Normal form.

Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q55.

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