KAM tori for quintic nonlinear schrödinger equations with given potential

Abstract

This paper is concerned with the 1-dimensional quintic nonlinear Schrodinger equations with real valued \begin{document}$ C^{\infty} $\end{document} -smooth given potential \begin{document}$ \sqrt{-1}u_{t} = u_{xx}-V(x)u-|u|^4u $\end{document} 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.