Integrable vs. nonintegrable geodesic soliton behavior

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–Poincaré 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. The results for head-on antisymmetric collisions of pulsons tend to be singularity formation. Numerical simulations of these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the same type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture this 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 initial conditions of finite extent.