I will discuss the problem of nonlinear wave motion of the free surface of a body of fluid with a varying bottom. The ob ject is to describe the character of wave propagation in a long-wave asymptotic regime. The case in which the bottom topography is periodic is shown to homogenize completely, and we compute how the bottom irregularities affects the effective Boussinesq equations and in the appropriate unidirectional limit, the Korteweg de Vries (KdV) equation. We also consider the case of a bottom described by a random, stationary ergodic process with sufficiently strong mixing conditions. We show that in the long wave limit, the random effects are governed by a canonical limit process which is equivalent to a white noise through Donsker’s invariance principle, with one free parameter the variance. The coherent wave motions of the KdV limit are shown to be preserved, while at the same time the random effects are described, and the degree of scattering due to the variable bottom is quantified. Our analysis is performed from the point of view of perturbation theory for Hamilto- nian PDEs with a small parameter.

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