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.