On nonlinear Schr\"odinger equations with almost periodic initial data

TL;DR

This paper establishes local well-posedness of nonlinear Schrödinger equations with almost periodic initial data and demonstrates finite time blowup for certain nonlinearities, marking the first such blowup result in this setting.

Contribution

It proves local well-posedness in an algebra of almost periodic functions and shows finite time blowup for NLS with specific nonlinearities, a novel result for generic almost periodic data.

Findings

NLS is locally well-posed in the algebra of almost periodic functions.

Finite time blowup occurs for NLS with nonlinearities |u|^p, p even.

First example of finite time blowup with generic almost periodic initial data.

Abstract

We consider the Cauchy problem of nonlinear Schr\"odinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmb{\omega} =\{\omega_j\}_{j = 1}^\infty$, NLS is local well-posed in the algebra $\mathcal{A}_{\pmb{\omega}}(\mathbb R)$ of almost periodic functions with absolutely convergent Fourier series. Then, we prove a finite time blowup result for NLS with a nonlinearity $|u|^p$, $p \in 2\mathbb{N}$. This elementary argument presents the first instance of finite time blowup solutions to NLS with generic almost periodic initial data.