Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry
| dc.creator | Holm, Darryl D. | |
| dc.creator | Munn, Jonathan | |
| dc.creator | Stechmann, Samuel N. | |
| dc.date | 2004-02-24 | |
| dc.date | 2004-09-27 | |
| dc.date.accessioned | 2026-07-07T05:35:21Z | |
| dc.date.available | 2026-07-07T05:35:21Z | |
| dc.description | 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. | |
| dc.description | 46 pages, 5 figures, numerical results added | |
| dc.identifier | https://arxiv.org/abs/nlin/0402044 | |
| dc.identifier | http://arxiv.org/abs/nlin/0402044 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/80673 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.title | Reduced Singular Solutions of EPDiff Equations on Manifolds with Symmetry | |
| dc.type | text |