Inverse Scattering Transform for the Degasperis-Procesi Equation

TL;DR

This paper develops an inverse scattering transform method for solving the Degasperis-Procesi equation, involving spectral analysis and Riemann-Hilbert problem formulation, advancing analytical techniques for this nonlinear PDE.

Contribution

It introduces an IST framework for the Degasperis-Procesi equation based on an $ ext{sl}(3)$ spectral problem with boundary conditions and reduction group considerations.

Findings

Constructed fundamental analytic solutions

Formulated a Riemann-Hilbert problem

Implemented the dressing method

Abstract

We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method are presented.