Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry

• Published in: arXiv.org 2004-09
• Main Authors: Holm, Darryl D, Munn, Jonathan, Stechmann, Samuel N
• Format: Article
• Language: English
• Subjects: Fields (mathematics); Girdles; Isomorphism; Lie groups; Ordinary differential equations; Partial differential equations; Solitary waves; Subgroups; Subspaces; Symmetry
• ISSN: 2331-8422

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.