On numerical inverse scattering for the Korteweg-de Vries equation with discontinuous step-like data

TL;DR

This paper develops a numerical method based on inverse scattering and Riemann-Hilbert problems to compute dispersive shock wave solutions of the Korteweg-de Vries equation with step-like initial data, including discontinuities.

Contribution

It introduces two Riemann-Hilbert problem formulations for step-like potentials and applies the Deift-Zhou method for numerical solution, extending applicability to discontinuous initial data.

Findings

Successfully computes solutions for small times and asymptotic regions.

Handles discontinuous step-like initial data effectively.

Ensures unique solvability of the inverse scattering problems.

Abstract

We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with the inverse scattering transform for the classical Schr\"odinger operator with possibly discontinuous, step-like potentials and develop relevant theory to ensure unique solvability of these problems. We then numerically implement the Deift-Zhou method of nonlinear steepest descent to compute the solution of the Cauchy problem for small times and in two asymptotic regions. Our method applies to continuous and discontinuous data.