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

@inproceedings{Piorkowski2021ASR, title={A scalar Riemann-Hilbert problem on the torus: Applications to the KdV equation}, author={Mateusz Piorkowski and Gerald Teschl}, year={2021} }

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 matrixvalued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature.

## References

SHOWING 1-10 OF 16 REFERENCES

A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation

- Mathematics
- 1993

In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior…

Algebro-geometric approach to nonlinear integrable equations

- Mathematics, Physics
- 1994

A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving…

Riemann Surfaces

- Mathematics
- 2007

This course serves as a follow-up of Math 506. The students are assumed to know complex analysis and have some basic knowledge in algebra and topology. The courses are divided into two parts: The…

On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation

- Mathematics
- 2016

We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the…

Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent

- Mathematics
- 2009

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries equation for decaying initial data in the soliton and similarity region. This paper can…

Theta functions and non-linear equations

- Mathematics
- 1981

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori §…

Rigorous Asymptotics of a KdV Soliton Gas

- Mathematics
- 2018

We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are…

Matrix Riemann–Hilbert problems with jumps across Carleson contours

- MathematicsMonatshefte fur Mathematik
- 2018

A notion of Lp-Riemann–Hilbert problem is introduced and basic uniqueness results and Fredholm properties are established and the implications of Fredholmness for the unique solvability are investigated.