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 {eqnarray}{maineq0} iuₜ-uₓₓ+V*u+εf(x)|u|⁴u=0,\ x=R/2πZ, {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. \[ I_n~ e-2ln^σ|n|, as\ n→∞, \] for any $ σ> 2, $ which extends results of Bourgain {BJFA2005} and Cong {cong2024} to the case that the nonlinear perturbation depends explicitly on the space variable x.