Global well-posedness of the short-pulse and sine-Gordon equations in energy space

TL;DR

This paper establishes the global well-posedness of the short-pulse and sine-Gordon equations in energy space, leveraging conserved quantities and coordinate transformations, for small initial data.

Contribution

It proves the global well-posedness of both equations in energy space, connecting the short-pulse and sine-Gordon equations through characteristic coordinates.

Findings

Global well-posedness of the short-pulse equation for small initial data.

Global well-posedness of the sine-Gordon equation in an appropriate space.

Utilization of conserved quantities to establish results.

Abstract

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space $H^2$. Our analysis relies on local well-posedness results of Sch\"afer & Wayne, the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine-Gordon equation in an appropriate function space.