Integrable vs Nonintegrable Geodesic Soliton Behavior

Abstract

We study confined solutions of certain evolutionary partial differential equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler-Poincare equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L2 norm of the velocity. These pde possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form singularities.