Comparison of model equations for small-amplitude long waves

Consider 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 (0.1) and the RLW-equation u(t) + u(x) + uu(x) - u(xxl) = 0 (0.2) 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 (0.2) from (0.1) in small-amplitude, long-wavelength regimes also produces a third equation, namely u(t) + u(x) + uu(x) + u(xtt) = 0. Kruskal, in a wide-ranging discussion of modelling considerations, pointed to this equation as an example that might not accurately describe water waves. Its status has remained unresolved. It is our purpose here to show that the initial-value problem for the latter equation is indeed well posed Moreover, we show that for small-amplitude, long waves, solutions of this model also agree to the neglected order with solutions of either (0.1) or (0.2) provided the initial data is properly imposed.