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

In this paper, we study the following nonlinear Schr"odinger equation \begin{eqnarray}\label{maineq0} $\mathbf{\text{i}}$u_{t}-u_{xx}+Vu+\epsilon f(x)|u|^4u=0,\ x\in$\mathbb{T}$=$\mathbb{R}$/2\pi$\mathbb{Z}$, \end{eqnarray} where $V\ast$ 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 $0\le \mathrm{\mid }ϵ\mathrm{\mid }\ll 1$, 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.