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

TL;DR

This work analyzes the standard nonlinear Schrödinger equation with spatially quasi-periodic initial data, formulating the problem for the Fourier coefficients and proving local existence, uniqueness, and decay of quasi-periodic solutions that retain the same frequency vector. The authors develop an infinite-dimensional ODE framework, introducing a novel combinatorial tree expansion and a power-of-conjugation labeling to organize the Picard iteration, complemented by Feynman diagrams to visualize interactions. They establish a local well-posedness theory with polynomial decay and demonstrate asymptotic convergence to the linear solution in both $L^{\infty}$ and $H^s$ norms for small nonlinearities on an appropriate time scale, highlighting the influence of decay rates and frequency non-resonance. The results lay groundwork for a second paper addressing derivative NLS and provide techniques (infinite ODEs, tree expansions, and diagrammatic representations) that enhance understanding of quasi-periodic dispersive dynamics with non-decaying data.

Abstract

This is the first part of a two-paper series studying nonlinear Schrödinger equations with quasi-periodic initial data. In this paper, we consider the standard nonlinear Schrödinger 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 Schrödinger 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.

Figures (2)

• Figure 1: $\Gamma^{(1)}$-family.
• Figure 2: $\Gamma^{(2)}$-family.

Theorems & Definitions (52)

• Conjecture 1
• Remark 1.1
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7