On the radius of spatial analyticity for defocusing nonlinear Schr\"odinger equations

TL;DR

This paper investigates how the spatial analyticity radius of solutions to defocusing nonlinear Schrödinger equations evolves over time, showing it cannot decay faster than 1/|t|, extending previous results to higher odd powers.

Contribution

It extends the understanding of analyticity radius decay from the cubic case to all odd integer powers greater than 3 in defocusing NLS equations.

Findings

Analytic radius decays no faster than 1/|t| over time

Extension of previous results to higher odd powers p>3

Provides bounds on the spatial analyticity for long-time solutions

Abstract

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schr\"odinger equations $iu_t + \Delta u = |u|^{p-1}u$, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ cannot decay faster than $1/|t|$ as $|t|\rightarrow\infty$. This extends the previous work of Tesfahun for the cubic case $p=3$ to the cases where $p$ is any odd integer greater than $3$.