Quasi-periodic solutions of Schr¨odinger equations with quasi-periodic forcing in higher dimensional spaces

In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schr¨odinger equation with a general nonlinearity iut − ∆u + Mξu + εφ(t)(u + h(|u|²)u) = 0, x ∈ T^d , t ∈ R under periodic boundary conditions is studied, where Mξ 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 = (ω1, ω2 . . . , ωm), 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 tion via KAM theory.