The Degasperis-Procesi equation on the half-line

TL;DR

This paper studies the Degasperis-Procesi equation on the half-line, demonstrating how solutions can be reconstructed from initial and boundary data using a Riemann-Hilbert problem framework.

Contribution

It introduces a method to recover solutions of the Degasperis-Procesi equation on the half-line via a Riemann-Hilbert problem based on spectral analysis.

Findings

Solution can be reconstructed from initial and boundary data

Formulation of a Riemann-Hilbert problem for the equation

Framework applicable to initial-boundary value problems

Abstract

We analyze a class of initial-boundary value problems for the Degasperis-Procesi equation on the half-line. Assuming that the solution $u(x,t)$ exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $k$.