Quasi-periodic solutions for a Schrödinger equation with a quintic nonlinear term depending on the time and space variables

Abstract

This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.