The Ostrovsky-Vakhnenko equation on the half-line: a Riemann-Hilbert approach

TL;DR

This paper develops a Riemann-Hilbert approach to solve the initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line, linking it to the Degasperis-Procesi equation and providing a method to recover solutions from initial and boundary data.

Contribution

It introduces a novel Riemann-Hilbert formulation for the Ostrovsky-Vakhnenko equation on the half-line, enabling solution reconstruction from spectral data.

Findings

Solution u(x,t) can be recovered from initial and boundary values.

Formulation involves a 3x3 vector Riemann-Hilbert problem.

Connects the Ostrovsky-Vakhnenko equation to the Degasperis-Procesi equation.

Abstract

We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a 3\times 3 vector Riemann-Hilbert problem formulated in the complex plane of a spectral parameter z.