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The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach

Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky, Lech Zielinski

TL;DR

The paper extends the Fokas unified transform to the periodic Camassa–Holm equation by formulating a master RH problem whose data are determined by the initial spectral data. A change of variables to $(y,t)$ clarifies the RH construction and enables a direct reconstruction of the solution $u(x,t)$ (via $m=u-u_{xx}$) from the RH data, while the global relations ensure consistency between initial and boundary values and enforce periodicity. The approach yields a rigorous, invariant RH representation for periodic CH on the interval, including a detailed residue and branch-cut structure that accommodates possible discrete spectra, and it provides a pathway to finite-band (algebro-geometric) CH solutions. Overall, the work demonstrates that periodic CH IBVPs admit a complete spectral characterization and a compatible RH formulation, with potential implications for both analytic understanding and computational implementations of CH dynamics under periodic boundary conditions.

Abstract

This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly, we present a representation of the solution to this problem in terms of the solution of the associated Riemann-Hilbert problem, the data for which are determined by the initial data for the problem in terms of the associated spectral functions.

The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach

TL;DR

The paper extends the Fokas unified transform to the periodic Camassa–Holm equation by formulating a master RH problem whose data are determined by the initial spectral data. A change of variables to clarifies the RH construction and enables a direct reconstruction of the solution (via ) from the RH data, while the global relations ensure consistency between initial and boundary values and enforce periodicity. The approach yields a rigorous, invariant RH representation for periodic CH on the interval, including a detailed residue and branch-cut structure that accommodates possible discrete spectra, and it provides a pathway to finite-band (algebro-geometric) CH solutions. Overall, the work demonstrates that periodic CH IBVPs admit a complete spectral characterization and a compatible RH formulation, with potential implications for both analytic understanding and computational implementations of CH dynamics under periodic boundary conditions.

Abstract

This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly, we present a representation of the solution to this problem in terms of the solution of the associated Riemann-Hilbert problem, the data for which are determined by the initial data for the problem in terms of the associated spectral functions.

Paper Structure

This paper contains 25 sections, 24 theorems, 207 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Assumptions on initial data
  3. RH problem formalism for the CH equation
  4. The Lax pair
  5. RH problem formalism for problem on the line
  6. Jost solutions and scattering relation
  7. Pre-Riemann-Hilbert problem in $x,t$ variables
  8. RH problem formalism for problems on an interval
  9. Dedicated solutions of the Lax pair equations
  10. Global relations
  11. Pre-Riemann-Hilbert problem for the problem on an interval: jump conditions
  12. Pre-Riemann-Hilbert problem for the problem on an interval: symmetries
  13. Pre-Riemann-Hilbert problem for the problem on an interval: behavior at $k=\infty$
  14. Pre-Riemann-Hilbert problem for the problem on an interval: residue conditions
  15. Pre-Riemann-Hilbert problem for the problem on an interval: behavior at $k=0$
  16. ...and 10 more sections

Key Result

Proposition 3.2

The $2\times 2$ function $M(x,t,k)$ constructed from the Jost solutions of the Lax pair equations generated by a solution of the CH equation $u(x,t)$ can be characterized as that satisfying the following conditions: Here $\alpha(x,t)$, $q(x,t)$, and $f(x,t)$ are not specified whereas $a(k)$, $b(k)$, $\{\nu_j\}_1^N$, and $\{c_j\}_1^N$ are determined by $u_0(x)=u(x,0)$ through the solutions of the

Figures (4)

  • Figure 1: Paths of integration for $\Phi_j$
  • Figure 2: Domains and the separating contour $\Sigma$
  • Figure 3: Contour $\hat{\Sigma}$
  • Figure 4: Contour for $M^{(t)}$ and $\hat{M}^{(t)}$

Theorems & Definitions (51)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 41 more