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The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach

Iryna Karpenko, Dmitry Shepelsky

TL;DR

This work develops a comprehensive Riemann–Hilbert framework for the initial-boundary value problem of the modified Camassa–Holm equation on the half-line. By carefully analyzing the Lax pair, defining direct and inverse spectral mappings, and formulating a master RH problem parametrized by y and t, the authors reconstruct the solution from spectral data that couple the initial profile and boundary measurements via global relations. The approach provides a rigorous, unique representation of the solution through RH problems, and clarifies the role of background, positivity of m, and the moving-boundary formulation when translating between (x,t) and (y,t) variables. Overall, the paper extends the CH RH machinery to the mCH setting, offering a robust route to asymptotics and boundary-value control for this integrable system on the half-line.

Abstract

We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation $ \tilde m_t+\left((\tilde u^2-\tilde u_x^2+2\tilde u)\tilde m\right)_x = 0$, $\tilde m:=\tilde u-\tilde u_{xx}+1$ on the half line $x \ge 0$. We provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann--Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated with the initial and boundary values of the solution, whose compatibility is characterized in spectral terms.

The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach

TL;DR

This work develops a comprehensive Riemann–Hilbert framework for the initial-boundary value problem of the modified Camassa–Holm equation on the half-line. By carefully analyzing the Lax pair, defining direct and inverse spectral mappings, and formulating a master RH problem parametrized by y and t, the authors reconstruct the solution from spectral data that couple the initial profile and boundary measurements via global relations. The approach provides a rigorous, unique representation of the solution through RH problems, and clarifies the role of background, positivity of m, and the moving-boundary formulation when translating between (x,t) and (y,t) variables. Overall, the paper extends the CH RH machinery to the mCH setting, offering a robust route to asymptotics and boundary-value control for this integrable system on the half-line.

Abstract

We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation , on the half line . We provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann--Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated with the initial and boundary values of the solution, whose compatibility is characterized in spectral terms.

Paper Structure

This paper contains 24 sections, 16 theorems, 122 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Eigenfunctions and Spectral Functions
  3. Eigenfunctions near $\lambda=\infty$
  4. Eigenfunctions near $\lambda=0$
  5. Spectral functions
  6. Compatibility of initial and boundary values
  7. Spectral Mappings: Direct Problems
  8. The direct $x$-spectral problem m_0(x) -> (a(mu), b(mu))
  9. The direct $t$-spectral problem vj(x) -> (A(mu), B(mu))
  10. Inverse Spectral Mappings
  11. The Inverse $x$-Spectral Mapping
  12. The Inverse $t$-Spectral Mapping
  13. The main Riemann--Hilbert problem
  14. The pre-Riemann--Hilbert problem
  15. The main Riemann--Hilbert problem parametrized by $y$ and $t$
  16. ...and 9 more sections

Key Result

Proposition 4.1

Figures (4)

  • Figure 1: Paths of integration for $\Phi_{01}, \Phi_{02},$ and $\Phi_{03}$ ($\Phi_{\infty1}, \Phi_{\infty2},$ and $\Phi_{\infty3}$).
  • Figure 2: Signature tables for $p(x,t,\mu)$
  • Figure 3: The oriented contour $\check \Sigma$ for the RH problem RH$^{(t)}$
  • Figure 4: The oriented contour $\Sigma$ for the RH problem RH$^{(xt)}$

Theorems & Definitions (29)

  • Remark 2.1
  • Remark 2.2
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Proposition 5.3
  • Proposition 5.4
  • proof
  • ...and 19 more