# On the existence of full dimensional KAM torus for nonlinear Schr\"odinger equation

## TL;DR

This paper proves the existence of full dimensional KAM tori for a nonlinear Schrödinger equation with space-dependent nonlinearities, extending previous results to more general cases with explicit space dependence.

## Contribution

It demonstrates the existence of time almost periodic solutions for a nonlinear Schrödinger equation with space-dependent nonlinearities, extending prior work to cases lacking zero momentum.

## Key findings

- Existence of full dimensional KAM tori in Gevrey space.
- Extension of Bourgain's and Cong-Liu-Shi-Yuan's results.
- Handling the absence of zero momentum in the analysis.

## Abstract

In this paper, we study the following nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, \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 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.