A Riemann-Hilbert approach to the Harry-Dym equation on the line

TL;DR

This paper applies the Riemann-Hilbert method to solve the Harry-Dym equation on the real line with decaying initial data, providing a new analytical framework for this nonlinear integrable system.

Contribution

It introduces a Riemann-Hilbert problem formulation for the Harry-Dym equation using Fokas' unified method, enabling explicit solution construction.

Findings

Solution expressed via a 2x2 matrix Riemann-Hilbert problem

Explicit one-soliton solution derived from the Riemann-Hilbert problem

Framework applicable to decaying initial conditions

Abstract

In this paper, we consider the Harry-Dym equation on the line with decaying initial value. The Fokas unified method is used to construct the solution of the Harry-Dym equation via a $2 \times 2$ matrix Riemann Hilbert problem in the complex plane. Further, one-cups soltion solution is expressed in terms of solutions of the Riemann Hilbert problem.