A model for the periodic water wave problem and its long wave amplitude equations

Abstract

We are interested in the validity of the KdV and of the long wave NLS approximation for the water wave problem over a periodic bottom. Approximation estimates are non-trivial, since solutions of order O(ε^2 ), resp. O(ε), have to be controlled on an O(1/ε^3 ), resp. O(1/ε^2 ), time scale. In contrast to the spatially homogeneous case, in the periodic case new quadratic resonances occur and make a more involved analysis necessary. For a phenomenological model we present some results and explain the underlying ideas. The focus is on results which are robust in the sense that they hold under very weak non-resonance conditions without a detailed discussion of the resonances. This robustness is achieved by working in spaces of analytic functions. We explain that, if analyticity is dropped, the KdV and the long wave NLS approximation make wrong predictions in case of unstable resonances and suitably chosen periodic boundary conditions. Finally we outline, how, we think, the presented ideas can be transferred to the water wave problem