Camassa–Holm, Korteweg–de Vries and related models for water waves

Abstract

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, ε, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, ε → 0, δ → 0, independently. Retaining terms O(ε), O(δ2) and O(εδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

Variable depth KdV equations and generalizations to more nonlinear regimes

“…the BBM equations (1.8) should be replaced by the following family (see [8,18]): [12,15,19,20,26,29,30,32,33,36]. We justify in this paper the derivation of the generalized KdV equation and also we show that the correct generalization of equation (1.10) under the scaling (1.9) and with the following conditions on the topographical variations:…”

“…In 2008 Constantin and Lannes [8] rigorously justified the relevance of more nonlinear generalization of the KdV equations (linked to the CamassaHolm equation [7] and the Degasperis-Procesi equations [11]) as models for the propagation of shallow water waves. They proved that these equations can be used to furnish approximations to the governing equations for water waves, and in their investigation they put earlier (formal) asymptotic procedures due to Johnson [18] on a firm and mathematically rigorous basis. However, all these results hold for flat bottoms only.…”

A local discontinuous Galerkin method for the Burgers–Poisson equation

“…Author's personal copy been suggested to capture one aspect or another of the classical water-wave problem, see e.g., [4,5,14,15,17,21,22,28,29].…”

Algebro-geometric solutions for the two-component Camassa–Holm Dym hierarchy

“…Similarly, let ω (3) P∞ − ,ν 2 (x) (P ) be another normalized differential of the third kind holomorphic on K n \ {P ∞ − ,ν 2 (x)} with simple poles at P ∞ − andν 2 (x) and residues 1 and −1, respectively, 22) where ζ in (4.21) denotes the local coordinate…”