Comparison of model equations for small-amplitude long waves.

*(English)*Zbl 0948.35108Consider a body of water of finite depth under the influence of gravity, bounded below by a flat, impermeable surface. If viscous and surface tension effects are ignored, and assuming that the flow is incompressible and irrotational, the fluid motion is governed by the Euler equations together with suitable boundary conditions on the rigid surfaces and on the air-water interface. In special regimes, the Euler equations admit of simpler, approximate models that describe pretty well the fluid response to a disturbance. In situations where the wavelength is long and the amplitude is small relative to the undisturbed depth, and if the Stokes number is of order one, then various model equations have been derived. Two of the most standard are the KdV-equation
\[
u_t+u_x+ uu_x+u_{xxx} =0\tag{1}
\]
and the RLW-equation
\[
u_t+ u_x+uu_x- u_{xxt}=0. \tag{2}
\]
Bona, Pritchard and Scott showed that solutions of these two evolution equations agree to the neglected order of approximation over a long time scale, if the initial disturbance in question is genuinely of small-amplitude and long-wavelength. The same formal argument that allows one to infer (2) from (1) in small-amplitude, long-wavelength regimes also produces a third equation, namely
\[
u_t+ux+ uu_x+u_{xtt} =0.\tag{3}
\]
These models (1)–(3) are compared in the paper.

Reviewer: Eryk Infeld (Warszawa)