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 eqnarraymaineq0 iuₓ-uₗₗ+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. \ Iₙ e^-2^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.