On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

Hongzi Cong; Lufang Mi; Yunfeng Shi; Yuan Wu

doi:10.3934/dcds.2019287

Article Contents

2019, Volume 39, Issue 11: 6599-6630. Doi: 10.3934/dcds.2019287

This issue Previous Article Almost surely invariance principle for non-stationary and random intermittent dynamical systems Next Article Relative entropy dimension of topological dynamical systems

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.
College of Science, The Institute of Aeronautical Engineering and Technology, Binzhou University, Binzhou 256600, China
3.
School of Mathematical Sciences, Peking University, Beijing 100871, China
4.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^* Corresponding author: Yuan Wu
^* Corresponding author: Yuan Wu

Received: January 31, 2019

Published: August 2019

H.C. is supported by the NNSFC (No. 11671066). L.M. is supported by the NNSFC (No. 11401041) and SPNSF (ZR2019MA062). Y.S. is supported by China Postdoctoral Science Foundation (No. 2018M641050). Y.W. is supported by NNSFC (No. 11790272 and No. 11421061)

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Abstract

In this paper, we study the following nonlinear Schrödinger equation

$ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $

where $ V* $ is the Fourier multiplier defined by $ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $ and $ f(x) $ is Gevrey smooth. It is shown that for $ 0\leq|\epsilon|\ll1 $, there is some $ (V_n)_{n\in\mathbb{Z}} $ such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable $ x $. The main difficulty here is the absence of zero momentum of the equation.

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Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q56, 35K55.

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