Integrable vs. nonintegrable geodesic soliton behavior

We study confined solutions of certain evolutionary partial differential eq uations (PDE) in 1 + 1 space-time. The PDE we study are Lie-Poisson Hamilto nian systems for quadratic Hamiltonians defined on the dual of the Lie alge bra of vector fields on the real line. These systems are also Euler-Poincar e equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric i s different from the L-2 norm of the velocity. These PDE possess a finite-d imensional invariant manifold of particle-like (measure-valued) solutions w e call "pulsons". We solve the particle dynamics of the two-pulson interact ion analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pu lson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. The results for head-on antisymmetric coll isions of pulsons tend to be singularity formation. Numerical simulations o f these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the s ame type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture th is behavior occurs because the integrable two-pulson interactions dominate the dynamics on the invariant pulson manifold, and this dynamics dominates the PDE initial value problem for most choices of confined pulses and initi al conditions of finite extent. (C) 2001 Published by Elsevier Science B.V.