Long-time asymptotics for the Degasperis-Procesi equation on the half-line

TL;DR

This paper derives explicit long-time asymptotic formulas for solutions of the Degasperis-Procesi equation on the half-line using nonlinear steepest descent applied to a matrix Riemann-Hilbert problem.

Contribution

It introduces a method to compute the leading order asymptotics of the Degasperis-Procesi equation on the half-line in the similarity region.

Findings

Explicit formula for the solution's asymptotics in the similarity region

Application of nonlinear steepest descent to a 3x3 Riemann-Hilbert problem

Connection of asymptotics with initial and boundary data

Abstract

We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.