Rotating concentric circular peakons

We study circularly symmetric solution behaviour of invariant manifolds of singular solutions of the partial differential equation EPDiff for geodesic flow of a pressureless fluid whose kinetic energy is the H 1 norm of the fluid velocity. These singular solutions describe interaction dynamics on lower-dimensional support sets, for example, curves, or filaments, of momentum in the plane. The 2 + 1 solutions we study are planar generalizations of the 1 + 1 peakon solutions of Camassa and Holm (1993 Phys. Rev. Lett. 71 1661–4) for shallow water solitons. As an example, we study the canonical Hamiltonian interaction dynamics of N rotating concentric circles of peakons whose solution manifold is 2N -dimensional. Thus, the problem is reduced from infinite dimensions to a finite-dimensional, canonical, invariant manifold. We show both analytical and numerical results. Just as occurs in soliton dynamics, these solutions are found to exhibit elastic collision behaviour. That is, their interactions exchange momentum and angular momentum but do not excite any internal degrees of freedom. One expects the same type of elastic collision behaviour to occur in other, more geometrically complicated cases. PACS numbers: 47.35, 11.10.Ef, 45.20.Jj (Some figures in this article are in colour only in the electronic version) 4 Also at: Mathematics Department, Imperial College London, London SW7 2AZ, UK. 0951-7715/04/062163+24$30.00 © 2004 IOP Publishing Ltd and London Mathematical Society Printed in the UK 2163