On the radius of analyticity of solutions to the cubic Szegő equation

Abstract

This paper is concerned with the cubic Szegő equation

defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb{T}$, where $\Pi :L^2(\mathbb{T})\to L_{+}^2(\mathbb{T})$ is the Szegő 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).