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A dressing method for soliton solutions of the Camassa-Holm equation

Rossen Ivanov, Tony Lyons, Nigel Orr

Abstract

The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.

A dressing method for soliton solutions of the Camassa-Holm equation

Abstract

The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.

Paper Structure

This paper contains 12 sections, 1 theorem, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Spectral Problem for the Camassa-Holm Equation
  3. From the scalar to the matrix Lax pair
  4. The Gauge Transformed SL(2) Spectral Problem
  5. Diagonlaisation
  6. Symmetry reductions of the Spectral Problem
  7. The Soliton Solutions
  8. The Dressing Method
  9. Dressing factor with a simple pole
  10. The one-soliton solution
  11. The two-soliton solution
  12. Discussion

Key Result

Proposition 3.1

The dressing factor $g(y,t,\lambda)$ is assumed to be of the form and $A$ is a matrix-valued residue of rank 1.

Figures (2)

  • Figure 1: The one-soliton solution at $t=0$ where $\nu_1=1$, $\nu_2=2$, $u_0=1$ and $\omega=0.25$.
  • Figure 2: Snapshots of the two soliton solution of the Camassa-Holm equation (\ref{['eq:ch']}), for three values of $t\in\{-10, 0, 10\}$. The other parameters are $u_0=1$, $\omega_1=0.35$ and $\omega_2=0.25$. The constants of integration were chosen as $\mu_1=1,\ \mu_2=2,\ \nu_1=2,\ \nu_2=3$.

Theorems & Definitions (1)

  • Proposition 3.1