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

Mateusz Piorkowski1, Gerald Teschl2

  • 1Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium.

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This study analyzes the Riemann-Hilbert problem for Korteweg-de Vries equation solutions. It reformulates the problem on a torus to derive new solutions using Jacobi theta functions.

Keywords:
Jacobi theta functionsKdV equationRiemann–Hilbert problem

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics

Background:

  • The Korteweg-de Vries (KdV) equation models shallow water waves and other phenomena.
  • One-gap solutions of the KdV equation are often studied using the Riemann-Hilbert problem.

Purpose of the Study:

  • To investigate the Riemann-Hilbert problem associated with one-gap solutions of the KdV equation.
  • To reformulate the problem on a torus for deeper insight.
  • To derive novel vector-valued and matrix-valued solutions.

Main Methods:

  • Reformulation of the Riemann-Hilbert problem as a scalar problem on a torus.
  • Deductive derivation of solutions.
  • Utilizing Jacobi theta functions for solution representation.

Main Results:

  • Successfully reformulated the Riemann-Hilbert problem on a torus.
  • Derived model vector-valued solutions.
  • Derived singular matrix-valued solutions expressed using Jacobi theta functions.

Conclusions:

  • The torus reformulation provides a new perspective on KdV solutions.
  • The derived solutions offer a comprehensive understanding of one-gap KdV solutions.
  • Results are consistent with and extend existing literature findings.