Long-time asymptotic behavior for the complex short pulse equation

TL;DR

This paper analyzes the long-time behavior of solutions to the complex short pulse equation, deriving explicit asymptotics for non-soliton solutions and expressing solutions via Riemann-Hilbert problems.

Contribution

It introduces a Riemann-Hilbert problem framework for the complex short pulse equation and derives explicit long-time asymptotics using the Deift-Zhou method.

Findings

Explicit long-time asymptotics for non-soliton solutions

Parametric expression of solutions via Riemann-Hilbert problem

Implicit one-soliton solution on discrete spectrum

Abstract

In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the solution of $2\times 2$-matrix Riemann-Hilbert problem, from which an implicit one-soliton solution is obtained on the discrete spectrum. While on the continuous spectrum we further establish the explicit long-time asymptotic behavior of the non-soliton solution by using Deift-Zhou nonlinear steepest descent method.