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Center for Nonlinear Studies |
Models of water waves with long wavelength under the assumption of no diffusion and no viscosity have been of great interest in the literature for a long time. Both solitary and periodic traveling wave solutions were found with a flat bottom. However, many realistic situations, e.g. ocean floor, do not involve a flat bottom. As a simple realization of the non-flat bottom we first study how a periodic bottom can affect the wave solutions, especially the periodic solutions. In general, when the bottom profile is slowly varying, the relevant equation is a perturbation of the one with a flat bottom. First, we use the "multiply skill" technique to obtain an approximate solution. The leading order term has the form of the unperturbed solution. I will discuss some analytical results about how the bottom topography can influence the leading order term (and the first correction term) for a one-layer KdV equation for slowly varying bottom.