Existence, Uniqueness and Asymptotic Dynamics of Nonlinear Schr\"odinger Equations With Quasi-Periodic Initial Data: I. The Standard NLS

TL;DR

This paper proves local existence, uniqueness, and asymptotic convergence of solutions to the standard nonlinear Schr"odinger equation with quasi-periodic initial data, using Fourier analysis and combinatorial methods.

Contribution

It establishes the first rigorous results on existence, uniqueness, and asymptotic behavior for NLS with quasi-periodic initial data, extending the understanding of such equations.

Findings

Local existence of quasi-periodic solutions with Fourier decay

Uniqueness of solutions within the quasi-periodic class

Asymptotic convergence to linear solutions as nonlinearity diminishes

Abstract

This is the first part of a two-paper series studying nonlinear Schr\"odinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schr\"odinger equation. Under the assumption that the Fourier coefficients of the initial data obey a power-law upper bound, we establish local existence of a solution that retains quasi-periodicity in space with a slightly weaker Fourier decay. Moreover, the solution is shown to be unique within this class of quasi-periodic functions. In addition, for the nonlinear Schr\"odinger equation with small nonlinearity, within the time scale, as the small parameter of nonlinearity tends to zero, we prove that the nonlinear solution converges asymptotically to the linear solution with respect to both the sup-norm $\|\cdot\|_{L_x^\infty(\mathbb R)}$ and the Sobolev-norm $\|\cdot\|_{H^s_x(\mathbb R)}$. The proof proceeds via a consideration of an associated infinite system of coupled ordinary differential equations for the Fourier coefficients and a combinatorial analysis of the resulting tree expansion of the coefficients. For this purpose, we introduce a Feynman diagram for the Picard iteration and $\ast^{[\cdot]}$ to denote the complex conjugate label.