On the existence of full dimensional KAM tori for 1D periodic nonlinear Schr\"odinger equation

Abstract

In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \mathbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^{4}u=0,\ x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, \end{eqnarray} with boundary conditions, where $V*$ is the Fourier multiplier, and $f(x)$ is Gevrey smooth. Here the radius of the invariant tori satisfies a slower decay, i.e. \[ I_n\sim e^{-2\ln^{\sigma}|n|}, \mbox{as}\ n\rightarrow\infty, \] for any $ \sigma> 2, $ which extends results of Bourgain \cite{BJFA2005} and Cong \cite{cong2024} to the case that the nonlinear perturbation depends explicitly on the space variable $x$.