Complex Billiard Hamiltonian Systems and Nonlinear Waves

There is a deep connection between solutions of nonlinear equations and both geodesic flows and billiards on Riemannian manifolds. For example, in Alber, Camassa, Holm and Marsden [1995], a link between umbilic geodesics and billiards on n-dimensional quadrics and new soliton-like solutions of nonlinear equations in the Dym hierarchy was investigated. The geodesic flows provide the spatial x-flow, or the instantaneous profile of the solution of a partial differential equation. When combined with a prescription for advancing the solution in time, a t-flow, one is able to determine a class of solutions for the partial differential equation under study (see, for example, Alber and Alber [1985, 1987] and Alber and Marsden [1994]). New classes of solutions can be obtained using deformations of finite dimensional level sets in the phase space. In particular, to obtain soliton, billiard and peakon solutions of nonlinear equations, one applies limiting procedures to the system of differential equations on the Riemann surfaces describing quasiperiodic solutions. To carry this out, one can use the method of asymptotic reduction—for details see Alber and Marsden [1992, 1994] and Alber, Camassa, Holm and Marsden [1994, 1995].