Darryl D Holm et al 2004 Nonlinearity 17 2163 doi:10.1088/0951-7715/17/6/008
Darryl D Holm1,4, Vakhtang Putkaradze2 and Samuel N Stechmann1,3
Show affiliationsRecommended by E S Titi
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 H1 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.
02.30.Jr Partial differential equations
11.10.Ef Lagrangian and Hamiltonian approach
37D10 Invariant manifold theory
35Q51 Solitons (See also 37K40)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Issue 6 (November 2004)
Received 25 August 2003, in final form 5 July 2004
Published 16 August 2004
Darryl D Holm et al 2004 Nonlinearity 17 2163
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