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.
Numerical study of a multiscale expansion of the Korteweg–de Vries equation and Painlevé-II equation
