Knocking out teeth in one-dimensional periodic NLS

TL;DR

This paper proves the existence and uniqueness of weak solutions for the one-dimensional cubic nonlinear Schrödinger equation with mixed initial data on the real line and torus, using a normal form reduction technique.

Contribution

It establishes the existence of weak solutions in extended function spaces and proves uniqueness under certain regularity conditions, employing a novel normal form reduction method.

Findings

Existence of weak solutions in mixed Sobolev spaces.

Uniqueness of solutions for specific regularity ranges.

Application of a differentiation by parts technique for normal form reduction.

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T)$ where $\epsilon>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.