On the Radius of Analyticity of Solutions to the Cubic Szeg\"o Equation

TL;DR

This paper investigates the spatial analyticity of solutions to the cubic Szegő equation on the torus, establishing that solutions remain analytic for all time and providing a lower bound for their radius of analyticity.

Contribution

It proves the global preservation of analyticity for solutions and introduces a method to estimate the lower bound of the radius of analyticity using energy-like estimates in the Wiener algebra.

Findings

Solutions remain spatially analytic for all time.

A lower bound for the radius of analyticity is established.

The method involves energy estimates in a Gevrey class based on the Wiener algebra.

Abstract

This paper is concerned with the cubic Szeg\H{o} equation $$ i\partial_t u=\Pi(|u|^2 u), $$ defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb T$, where $\Pi: L^2(\mathbb T)\rightarrow L^2_+(\mathbb T)$ is the Szeg\H{o} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time $t\in (-\infty,\infty)$. In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the $\ell^1$ norm of Fourier transforms (the Wiener algebra).