Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces

Abstract

In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrödinger equation with a general nonlinearityunder periodic boundary conditions is studied, where M ξ is a real Fourier multiplier and ε is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω 1 , ω 2 . . . , ω m ), and h(|u| 2 ) is a real analytic function near u = 0 with h(0) = 0. It is shown that, under suitable hypothesis on φ(t), there are many quasi-periodic solutions for the above equation via KAM theory.