On solutions of the Camassa-Holm equation

2003/07/08 by R. S. Johnson · 6 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Algebraic structures and combinatorial models #Peakon #Camassa–Holm equation #Limit (mathematics) #Maple #Simple (philosophy) #Inverse scattering transform #Soliton #Parametric statistics #Applied mathematics #Mathematics #Parametric equation #Inverse scattering problem #Mathematical analysis #Inverse problem #Physics #Integrable system #Nonlinear system #Geometry

paper · doi:10.1098/rspa.2002.1078

openalex publication_date 2003/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The techniques that have been developed for the application of the inverse-scattering transform method to the solution of the Camassa–Holm equation, principally by Constantin, are implemented. We use this approach, first, to represent the known solitary-wave solution in a simple parametric form, and then, second, to obtain the general two- and three-soliton solutions. (These latter two solutions require rather extensive use of mathematical packages, Mathematica and Maple, in order to complete the construction of solutions.) A number of examples are presented; the phase shifts, evident after an interaction, are found and the special limit that recovers the peakon solutions is discussed.

Citations

• A New Integrable Shallow Water Equation
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod
• On the scattering problem for the Camassa-Holm equation
• The Camassa-Holm Equation: A Loop Group Approach

Cited by

• On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions
• On the Camassa-Holm equation and a direct method of solution I. Bilinear form and solitary waves
• On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions
• Evolution of the Scattering Coefficients of the Camassa–Holm Equation, for General Initial Data
• Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity
• The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations
• Camassa–Holm equation [wikipedia]