On billiard solutions of nonlinear PDEs

This Letter presents some special features of a class of integrable PDEs ad mitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions ar e weak solutions that are piecewise smooth and have first derivative discon tinuities at peaks in their profiles. A connection is established between t he peak locations and finite dimensional billiard systems moving inside n-d imensional quadrics under the field of Hooke potentials. Points of reflecti on are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynam ics of the peaks is described in the context of the algebraic-geometric app roach to integrable systems. (C) 1999 Published by Elsevier Science B.V. Al l rights reserved.