On Billiard Solutions of Nonlinear PDE's

Alber, M. S., R. Camassa, Y. N. Fedorov, D. D. Holm, and J. E. Marsden

Physics Letters A 264, 171-178

Abstract:

This letter presents some special features of a class of integrable PDE's admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and nite dimensional billiard systems moving inside n-dimensional quadrics under the eld of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDE's weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.