On Nonlinear Schrödinger Equations with Almost Periodic Initial Data

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We consider the Cauchy problem of nonlinear Schrodinger 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 provides the first instance of finite time blowup solutions to NLS with generic almost periodic initial data.