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

In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrodinger equation with a general nonlinearity iu(t) - Delta u + M(xi)u + epsilon phi(t)(u + h(vertical bar u vertical bar(2))u) = 0, x is an element of T-d, t is an element of R under periodic boundary conditions is studied, where M-xi, is a real Fourier multiplier and epsilon is a small positive parameter, phi(t) is a real analytic quasi -periodic function in t with frequency vector omega = (omega(1), omega(2) ... , omega(,)), and h(vertical bar u vertical bar(2)) is a real analytic function near u = 0 with h(0) = 0. It is shown that, under suitable hypothesis on phi(t), there are many quasi -periodic solutions for the above equation via KAM theory. (C) 2017 All rights reserved.