On the existence of full dimensional KAM tori for 1D periodic nonlinear Schrödinger equation

In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schrödinger equation \begin{eqnarray}\label{maineq0} $\mathbf{i}$u_{t}-u_{xx}+V*u+εf(x)|u|^{4}u=0,\ x\in$\mathbb{T}$=$\mathbb{R}$/2π$\mathbb{Z}$, \end{eqnarray} with boundary conditions, where $V\ast$ is the Fourier multiplier, and $f(x)$ is Gevrey smooth. Here the radius of the invariant tori satisfies a slower decay, i.e. $$I_n\sim e^{-2{\ln }^{\sigma }\mathrm{\mid }n\mathrm{\mid }},asn\to \mathrm{\infty },$$ 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 $x$.