Singular Solutions of Euler–Poincaré Equations on Manifolds with Symmetry

The Euler–Poincare equation EPDiff 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. For a strong enough norm, EPDiff admits 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 itself is a momentum map. Consequently, the diffeons evolve according to canonical Hamiltonian equations.This paper examines diffeon solutions on Einstein spaces that are “mostly” symmetric, i.e., whose quotient by a subgroup of the isometry group is one-dimensional. An example is the two-sphere, whose isometry group SO3 contains S 1. In this situation, the singular diffeons are supported on latitudes of the sphere. For this S 1 symmetry of the two-sphere, the canonical Hamiltonian dynamics for diffeons reduces from integral partial differential equations to a dynamical system of ordinary differential equations for their co-latitudes. Explicit examples are computed numerically for the motion and interaction of the Puckons on the sphere with respect to the H 1 norm. We analyze this case and several other two-dimensional examples. From consideration of these two-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.