KdV-like solitary waves in two-dimensional FPU-lattices

We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles.Comment: revised version with several changes in the presentation of the technical details; 25 pages, 15 figure

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

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

Justification of the log-KdV equation in granular chains: the case of precompression

For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg-de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of small anharmonicity. We prove control of the approximation error for the travelling solutions satisfying differential advance-delay equations, as well as control of the approximation error for time-dependent solutions to the lattice equations on long but finite time intervals. We also show nonlinear stability of the travelling waves on long but finite time intervals.Comment: 29 page