On nonlinear Schrödinger equations with 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.