The Camassa–Holm equation for water waves moving over a shear flow

The rôle of the Camassa–Holm (CH) equation within the classical water-wave problem, which incorporates an ambient underlying flow, is described. The governing equations for gravity waves over a flow with non-zero vorticity are presented, and the two familiar parameters (, amplitude; δ, long-wave) are introduced. We seek a solution of these equations in the form of a double asymptotic expansion, for → 0, δ → 0, retaining terms O(1), O(), O(δ2) and O(δ2) only. The development initially allows for an arbitrary underlying 'shear' flow and some of the terms in the asymptotic expansions are presented for this general case. However, significant complications soon become evident, to the extent that a complete description for arbitrary flows—an obvious aim—would be a considerable undertaking (and it is doubtful if any useful general conclusions would be possible). Thus, the calculation is completed for the case of a linear shear, i.e. the underlying flow has constant vorticity.

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