The Weakly Nonlinear Schr\"odinger Equation in Higher Dimensions with Quasi-periodic Initial Data

TL;DR

This paper proves that solutions to the weakly nonlinear Schrödinger equation in higher dimensions with quasi-periodic initial data asymptotically behave like linear solutions, using combinatorial diagram analysis under decay conditions.

Contribution

It extends asymptotic analysis of nonlinear Schrödinger equations to higher dimensions with general nonlinearities using a novel combinatorial approach.

Findings

Solutions approach linear behavior over time

Applicable to arbitrary dimensions and nonlinearities

Uses diagram-based combinatorial analysis

Abstract

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 $1\leq p\in\mathbb N$.