Global analytic solutions of a pseudospherical Novikov equation

TL;DR

This paper proves that solutions to a Novikov equation describing pseudospherical surfaces are globally analytic in space and time, using Sobolev and Gevrey space techniques, and explores the geometric implications of this analyticity.

Contribution

It extends regularity results for the Novikov equation by establishing global analyticity of solutions in both variables, linking PDE analysis with geometric surface theory.

Findings

Solutions are globally analytic in space and time.

A lower bound for the radius of spatial analyticity is established.

The solution induces a global analytic metric on pseudospherical surfaces.

Abstract

In this paper we consider a Novikov equation, recently shown to describe pseudospherical surfaces, to extend some recent results of regularity of its solutions. By making use of the global well-posedness in Sobolev spaces, for analytic initial data in Gevrey spaces we prove some new estimates for the solution in order to use the Kato-Masuda Theorem and obtain a lower bound for the radius of spatial analyticity. After that, we use embeddings between spaces to then conclude that the unique solution is, in fact, globally analytic in both variables. Finally, the global analyticity of the solution is used to prove that it endows a certain strip with a global analytic metric associated to pseudospherical surfaces obtained in previous results in the literature.