The Weakly Nonlinear Schr\"odinger 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 present through diagrams. Our results and methods apply to {\em arbitrary} space dimensions and general power-law nonlinearities of the form $\pm|u|^{2p}u$, where $1\leq p\in\mathbb N$.