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

Abstract

In this paper, we study the following nonlinear Schrödinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+εf(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2π\mathbb{Z}, \end{eqnarray} where $V\ast$ is the Fourier multiplier defined by $\widehat{(V\ast u}{)}_n=V_n{\hat{u}}_n,V_n\in [-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\le |\epsilon |\ll 1$, there is some $(V_n{)}_{n\in \mathbb{Z}}$ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \cite{BJFA2005} and Cong-Liu-Shi-Yuan \cite{CLSY} 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.