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Riemann-Hilbert approach to the Algebro-Geometric solution of the modified Camassa-Holm equation with linear dispersion term

Engui Fan, Gaozhan Li, Yiling Yang

TL;DR

This work develops an exact algebro-geometric (finite-gap) solution to the modified Camassa-Holm equation with linear dispersion by formulating and solving a Riemann-Hilbert problem tied to a high-genus hyperelliptic curve. It employs a Baker–Akhiezer function and a g-function mechanism to produce theta-function representations on a genus $4(p+q)-1$ curve, yielding explicit $u^{(AG)}(y,t;\mathbf{P}_1,\mathbf{P}_2,\mathbf{A},\mathbf{B})$ and $x^{(AG)}(y,t;\mathbf{P}_1,\mathbf{P}_2,\mathbf{A},\mathbf{B})$ via reconstruction formulas. The paper provides two solvable RH formulations, including an alternative $\tilde{g}$-based approach, both leading to the same algebro-geometric solutions up to additive constants, thereby enabling exact finite-gap solutions and setting groundwork for long-time asymptotics and soliton-gas analyses in the mCH hierarchy. Extending prior ω=0 results to the ω>0 case, this work highlights the central role of hyperelliptic curves and theta functions in the algebro-geometric structure of the mCH equation and its Lax pair.

Abstract

This paper aims at providing an exact algebro-geometric solution of the modified Camassa-Holm (mCH) equation derived from hyperelliptic curves in $4(p+q)-1$ genus. To achieve this goal, we construct the Riemann-Hilbert problems cosponsoring to the mCH equation, which can be solved exactly by the Baker-Akhiezer function. Then the precise expression of the algebro-geometric solution of the mCH equation can be obtained through reconstructed formula.

Riemann-Hilbert approach to the Algebro-Geometric solution of the modified Camassa-Holm equation with linear dispersion term

TL;DR

This work develops an exact algebro-geometric (finite-gap) solution to the modified Camassa-Holm equation with linear dispersion by formulating and solving a Riemann-Hilbert problem tied to a high-genus hyperelliptic curve. It employs a Baker–Akhiezer function and a g-function mechanism to produce theta-function representations on a genus curve, yielding explicit and via reconstruction formulas. The paper provides two solvable RH formulations, including an alternative -based approach, both leading to the same algebro-geometric solutions up to additive constants, thereby enabling exact finite-gap solutions and setting groundwork for long-time asymptotics and soliton-gas analyses in the mCH hierarchy. Extending prior ω=0 results to the ω>0 case, this work highlights the central role of hyperelliptic curves and theta functions in the algebro-geometric structure of the mCH equation and its Lax pair.

Abstract

This paper aims at providing an exact algebro-geometric solution of the modified Camassa-Holm (mCH) equation derived from hyperelliptic curves in genus. To achieve this goal, we construct the Riemann-Hilbert problems cosponsoring to the mCH equation, which can be solved exactly by the Baker-Akhiezer function. Then the precise expression of the algebro-geometric solution of the mCH equation can be obtained through reconstructed formula.

Paper Structure

This paper contains 6 sections, 2 theorems, 64 equations, 6 figures.

Table of Contents

  1. Introduction
  2. RH problem framework
  3. High Genus Algebro-Geometric solution
  4. $g-$function mechanism
  5. Explicitly solvable RH problem
  6. Alternative form of the solvable RH problem

Key Result

Theorem 1

For $M(\lambda)$ satisfies an RH problem RHP0, a real and non-singular solution $u(y,t)$ of mCH equation y-mCH is given by the following reconstruction formulae where

Figures (6)

  • Figure 2: Jump curve of the RH problem \ref{['RHP0 g']}, $\mathcal{R}$ and basis of its first homology group for $p=q=1$ case.
  • Figure 3: Jump curve $\Gamma=\bigcup_{j=0}^{4p+4q-1}\Gamma_j$ of the RH problem \ref{['RHP0 g']}, $\mathcal{R}$ and basis of its first homology group for $p=2$, $q=1$ case.
  • Figure 4: Jump curve $\tilde{\Gamma}$ of the RH problem \ref{['RHP g=7 alt']}.
  • Figure : (a)
  • Figure : (a)
  • ...and 1 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Corollary 2