A scalar Riemann-Hilbert problem on the torus: Applications to the KdV equation

TL;DR

This paper reformulates the Riemann-Hilbert problem for one-gap KdV solutions as a scalar problem on the torus, deriving explicit solutions using Jacobi theta functions and comparing with recent results.

Contribution

It introduces a scalar Riemann-Hilbert formulation on the torus for one-gap KdV solutions, providing explicit theta function solutions and insights into the problem's structure.

Findings

Derived explicit solutions in terms of Jacobi theta functions

Reformulated the problem as a scalar Riemann-Hilbert problem on the torus

Compared new results with existing literature

Abstract

We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature.