The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations
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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 incopressible fluid, the equation of mass conservation and the simplest bottom and surface conditions) contains so may different and important equations. We will briefly show how such equations are generated, by carefully describing the asymptotic procedures that we adopt. We will then present a number of examples, some of which will be the familiar integrable equations (Korteweg-de Vries (KdV), Nonlinear Schrödinger (NLS)) and also variants of these which are, in a sense, nearlintegrable. These will include, for example, a variable coefficient KdV equation, a nearly concentric KdV equation and higher-order NLS equation. Some of these results have significant consequences for water-wave propagation; for example, the existence of a KdV equation for arbitrary velocity profiles below the surface helps to explain why solitary waves are observed in nature. We conclude with a discussion of some of the more recent, exciting work on the r^ole of the CamassaHolm (CH) equation in water waves. We will also outline how, for this type of problem, a number of interesting and relevant variants of the CH equation can arise.
- © 2006, the Authors. Published by Atlantis Press.
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- This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
TY - JOUR AU - Robin S. Johnson PY - 2003 DA - 2003/08/01 TI - The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations JO - Journal of Nonlinear Mathematical Physics SP - 72 EP - 92 VL - 10 IS - Supplement 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2003.10.s1.6 DO - 10.2991/jnmp.2003.10.s1.6 ID - Johnson2003 ER -