Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

By Darryl D. Holm, Jerrold E. Marsden. 2003

162 citations recorded by the source corpus.

This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincare equations associated with the diffeomorphism group (of ℝn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincare (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/61048ec87afd5ff76adc7966ac83ff51da687eef

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Citations in the corpus

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions (2009, methodology): “In this case the reduction to multi-peakon solutions can be elegantly understood in terms of momentum maps using the ideas of [HM].”
The geometry of the universal Teichmuller space and the Euler-Weil-Petersson equation (2015, methodology): “As in the case of the peakons of the Camassa–Holm equations, see [18], the momentum m = Qopu associated to the Teichons solutions is given by momentum mapping.”