Efficient computation of soliton gas primitive potentials

TL;DR

This paper introduces an efficient numerical method for computing soliton gas primitive potentials for the Korteweg--de Vries equation, based on solving a specialized Riemann--Hilbert problem with square-root behavior.

Contribution

It develops a novel numerical approach using $g$-function deformation and endpoint singularity incorporation to accurately solve the Riemann--Hilbert problem for soliton gas potentials.

Findings

The method effectively computes potentials from the Riemann--Hilbert problem.

It handles square-root jump matrices with high accuracy.

The approach improves computational efficiency for soliton gas analysis.

Abstract

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $-\infty$. This accumulation results in an associated Riemann--Hilbert problem on a number of disjoint intervals. In the case where the jump matrices have specific square-root behavior, we describe an efficient and accurate numerical method to solve this Riemann--Hilbert problem and extract the potential. The keys to the method are, first, the deformation of the Riemann--Hilbert problem, making numerical use of the so-called $g$-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation.