The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data

In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schr\"odinger equation in higher dimensions will asymptotically approach the associated linear solution within a specific time scale. The proof is based on a combinatorial analysis method. Our results and methods work for {\em arbitrary} space dimensions and focusing/defocusing {\em arbitrary} power-law nonlinearities.