Title: Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation Show Chinese title

Title: 三次非聚焦非线性薛定谔方程的几乎周期解

Abstract: This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schr\"odinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions). Show Chinese abstract

Abstract: 本文研究了三次非聚焦非线性薛定谔方程(NLS)的柯西问题，初始数据为几乎周期性的。我们证明了对于满足迪奥法恩特频率条件的小的解析准周期初始数据，柯西问题存在一个在空间和时间上都是几乎周期性的解，并且在某种合适的意义下局部有界的解中，这个解是唯一的。分析结合了直接和逆谱理论。在逆谱理论部分，我们证明了当初始数据相关的狄拉克算子具有纯绝对连续谱且谱不那么稀疏时，解的存在性、几乎周期性和唯一性。这解决了NLS层次结构带来的新挑战，例如额外的自由度和额外的对易流。在直接谱理论部分，对于满足迪奥法恩特频率条件的小的解析准周期势的狄拉克算子，我们证明了纯绝对连续谱、指数衰减的谱间隙以及谱厚度条件（齐次性和Craig型条件）。