Modulation of the Camassa-Holm equation and reciprocal transformations

2005/01/01 by Simonetta Abenda, Тамара Грава · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Nonlinear Photonic Systems #Korteweg–de Vries equation #Camassa–Holm equation #Hamiltonian (control theory) #Poisson bracket #Lax pair #Mathematics #Reciprocal #Partial differential equation #Mathematical physics #Mathematical analysis #Integrable system #Physics #Pure mathematics #Nonlinear system #Quantum mechanics

paper · pdf · doi:10.5802/aif.2142

openalex publication_date 2005/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit a bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations are quite different: indeed the KdV averaged bi- Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.

Citations

• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
• On the scattering problem for the Camassa-Holm equation
• The Liouville correspondence between the Korteweg—de Vries and the Camassa‐Holm hierarchies
• Group velocity and nonlinear dispersive wave propagation
• Multipeakons and the Classical Moment Problem
• The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type
• Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory
• Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation
• Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians

Cited by

• Camassa–Holm equation [wikipedia]