The short pulse equation by a Riemann-Hilbert approach

TL;DR

This paper introduces a Riemann-Hilbert approach to solve the short pulse equation, enabling explicit solution representation, long-time behavior analysis, and conditions for wave breaking.

Contribution

It develops a novel Riemann-Hilbert framework directly applied to the Lax pair of the short pulse equation, enhancing solution analysis and wave breaking criteria.

Findings

Explicit parametric solution representation for the Cauchy problem

Analysis of long-time asymptotics of solutions

Spectral condition for wave breaking

Abstract

We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation $u_{xt}=u+\frac{1}{6}(u^3)_{xx}$ with zero boundary conditions (as $|x|\to\infty$). This approach is directly applied to the Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the long-time behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the long-time behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.