On Billiard Solutions of Nonlinear PDE's

This Letter presents some special features of a class of integrable PDEs 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 finite dimensional billiard systems moving inside n-dimensional quadrics under the field 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 PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.