  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.
