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The Camassa--Holm equation for water waves moving over a shear flow. (English) Zbl 1032.76519
Summary: 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 (${\varepsilon}$, amplitude; ${\delta}$, long-wave) are introduced. We seek a solution of these equations in the form of a double asymptotic expansion, for $\varepsilon \to 0$, $\delta \to 0$, retaining terms O(1), O($\varepsilon$), O($\delta^2$) and O($\varepsilon \delta^2$) 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. It is shown that a CH equation can exist (at this order) for a simple nonlinear function of the horizontal velocity component of the perturbed flow field, at a certain depth. The results agree with those previously found for the no-shear case, and then the CH equation is precisely for this velocity component. Even so, the presence of the underlying flow has some important and new consequences. Although the wave which is propagating downstream always has an associated CH equation at a depth below the surface (and above the bed), this is not true for upstream propagation. As the difference between the top and bottom speeds of the underlying flow increases (i.e. the constant vorticity increases), so the depth at which the CH equation is valid for this wave moves downwards. At a critical value of this difference, the CH equation is valid on the bottom and thereafter it moves outside the physical flow field. (The CH equation itself, the equation for the surface wave, and the transformation between them, still exist even when the flow is not physically realisable.) The variation of the depth, for each direction of propagation, as the linear shear is varied, is presented, as is the form of the resulting CH equation.

76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25Solitary waves (inviscid fluids)
35Q51Soliton-like equations
35Q58Other completely integrable PDE (MSC2000)
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