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April  2020, 40(4): 2421-2439. doi: 10.3934/dcds.2020120

## KAM tori for quintic nonlinear schrödinger equations with given potential

 1 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China 2 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

* Corresponding author: Dongfeng Yan

Received  July 2019 Revised  November 2019 Published  January 2020

Fund Project: The first author is supported by NSFC grant 11371132, and the second author is supported by NSFC grant 11601487.

This paper is concerned with the 1-dimensional quintic nonlinear Schrödinger equations with real valued
 $C^{\infty}$
-smooth given potential
 $\sqrt{-1}u_{t} = u_{xx}-V(x)u-|u|^4u$
subject to Dirichlet boundary conditions. By means of normal form theory and an infinite-dimensional Kolmogorov-Arnold-Moser (KAM, for short) theorem, it is proved that the above equation admits a family of elliptic tori where lies small amplitude quasi-periodic solutions with two frequencies of high modes.
Citation: Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120
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