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

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.

Automated summary

This paper takes a closer look at the relationship between one-gap solutions of the Korteweg-de Vries equation and the Riemann-Hilbert problem. By reformulating it as a scalar Riemann-Hilbert problem on the torus, the authors deduce the vector-valued and singular matrix-valued solutions using Jacobi theta functions. The results are compared with those in recent literature.