Quasi-periodic solutions of two dimensional Schrödinger equations with Quasi-periodic forcing

This work focuses on two-dimensional (2D)(2D) quasi-periodically forced nonlinear Schrödinger equations. This means studying iut−Δu+μu+εϕ(t)(u+|u|2u)=0,μ≥0,x∈T2,t∈R with periodic boundary conditions, where εε is a small positive parameter, ϕ(t)ϕ(t) is a real analytic quasi-periodic function in tt with frequency vector ω=(ω1,ω2…,ωm)ω=(ω1,ω2…,ωm). It is shown that, under suitable hypothesis on ϕ(t)ϕ(t), there are many quasi-periodic solutions for the above equation via KAM theory.