The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

2003/01/01 by R. S. Johnson · 1 citation
Physics and Astronomy · Earth and Planetary Sciences · Mathematics · #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing #Nonlinear Photonic Systems #Integrable system #Mathematics #Dispersionless equation #Mathematical physics #Camassa–Holm equation #Mathematical analysis #Pure mathematics #Partial differential equation #Kadomtsev–Petviashvili equation

paper · pdf · doi:10.2991/jnmp.2003.10.s1.6

openalex publication_date 2003/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations (the Euler equation for an incompressible fluid, the equation of mass conservation and the simplest bottom and surface conditions) contains so may different -and important -equations.

Citations

• A New Integrable Shallow Water Equation
• The Camassa–Holm equation for water waves moving over a shear flow
• XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• On the scattering problem for the Camassa-Holm equation
• On three-dimensional packets of surface waves
• On the Korteweg—de Vries equation for a gradually varying channel
• The disintegration of wave trains on deep water Part 1. Theory
• On solutions of the Camassa-Holm equation

Cited by

• Camassa–Holm equation [wikipedia]