Measure-valued solutions (strings and branes) in compressible geodesic motion

Darryl Holm, Los Alamos National Laboratory, Mathematical Modeling and Analysis Group

[ For full abstract see: http://www.cds.caltech.edu/~mleok/dholm.pdf]

The Lie-Poisson Hamiltonian equation for compressible geodesic motion in n dimensions is a vector evolutionary equation with nonlinear convection and stretching [1].

The Hamiltonian is the kinetic energy and the velocity is a convolution of (isotropic) G with the momentum. Besides its geometrical meaning, this equation arises as the ``template matching equation'' in 3D image processing [2,3], and in 2D as an equation for vertically averaged shallow water motion [4].

After discussing 1D variants of this type of motion, such as the ``peakon'' solutions of a shallow water equation, we shall report the following superposeable, measure-valued momentum solutions of the vector geodesic equation in n dimensions, and 2N parameters

These measure-valued solutions for momentum are defined on N surfaces, (curves, for k=1). The 2N parameters satisfy canonical Hamiltonian equations for geodesic motion on the space of k-surfaces taking n-vector values with co-metric G. These solutions are found numerically to emerge from smooth initial distributions of momentum and show quasi-one-dimensional interaction behavior (including reconnection) for (n,k)=(2,1) [5].

[1] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler--Poincar\'e equations and semidirect products with applications to continuum theories. Adv. in Math. 137, 1-81 (1998).

[2] D. Mumford, Pattern theory and vision. In Questions Matheematiques En Traitement Du Signal et de L'Image, Chapter 3, pp. 7-13. Paris: Institut Henri Poincare (1998).

[3] M. I. Miller, A. Trouve and L. Younes, On the metrics and Euler-Lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375-405 (2002).

[4] H. P. Kruse, J. Schreule and W. Du, A two-dimensional version of the CH equation. In Symmetry and Perturbation Theory: SPT 2001 Edited by D. Bambusi, G. Gaeta and M. Cadoni. World Scientific: New York, pp 120-127 (2001).

[5] D. D. Holm and M. F. Staley, Wave structures and nonlinear balances in a family of evolutionary PDEs. SIADS, to appear (2003).