Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry

Details Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry

2004-02-24

arxiv.org/abs/nlin/0402044v2

Nonlinear Sciences

Pattern Formation and Solitons

46 pages, 5 figures, numerical results added

Scientific paper

The EPDiff equation governs geodesic flow on the diffeomorphisms with respect to a chosen metric, which is typically a Sobolev norm on the tangent space of vector fields. EPDiff admits a remarkable ansatz for its singular solutions, called ``diffeons,'' whose momenta are supported on embedded subspaces of the ambient space. Diffeons are true solitons for some choices of the norm. The diffeon solution ansatz is a momentum map. Consequently. the diffeons evolve according to canonical Hamiltonian equations. We examine diffeon solutions on Einstein spaces that are "mostly" symmetric, i.e., whose quotient by a subgroup of the isometry group is 1-dimensional. An example is the two-sphere, whose isometry group $\SO{3}$ contains $S^1$. In this situation, the singular diffeons (called ``Puckons'') are supported on latitudes (``girdles'') of the sphere. For this $S^1$ symmetry of the two-sphere, the canonical Hamiltonian dynamics for Puckons reduces from integral partial differential equations to a dynamical system of ordinary differential equations for their colatitudes. Explicit examples are computed numerically for the motion and interaction of the Puckons on the sphere with respect to the $H^1$ norm. We analyse this case and several other 2-dimensional examples. From consideration of these 2-dimensional spaces, we outline the theory for reduction of diffeons on a general manifold possessing a metric equivalent to the warped product of the line with the bi-invariant metric of a Lie group.

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