On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

Abstract

In this paper, we study the following nonlinear Schrödinger equation \begin{document}$ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $\end{document} where $ V* $ 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\leq|\epsilon|\ll1 $, there is some $ (V_n)_{n\in\mathbb{Z}} $ such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] 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.