Geodesic solitons in nonlinear internal-wave interactions and in computational anatomy
Darryl D. Holm
Los Alamos National Laboratory
The depth-averaged three-dimensional Euler equations for shallow water flow are well approximated by two-dimensional equations of geodesic motion for a certain Sobolev norm. The initial value problem for these equations produces filamentary, soliton-like solutions that are measure-valued; that is, they are supported on curves that evolve in the plane. Numerical simulations show that these filamentary solutions emerge and dominate the initial value problem for any confined initial velocity distribution.
These solutions possess three notable properties: they superpose, they form an invariant manifold, and their nonlinear interactions allow them to reconnect with each other in two dimensions. This reconnection phenomenon is also observed in oceanic internal waves through synthetic aperture radar measurements taken from the space shuttle. Thus, these filamentary solutions of the depth-averaged three-dimensional Euler equations provide a simplified framework for studying evolving two-dimensional arrays of interacting internal waves.
Remarkably, the same family of geodesic equations also arises in image processing within the template matching framework of computational anatomy. In this context, a measure-valued solution at two time points corresponds to the “cartoons” of an image and its target image under the template-matching map.
The existence of these measure-valued solutions of geodesic flow is guaranteed—for any Sobolev norm and in any number of spatial dimensions—because the solution ansatz is a momentum map for the action of diffeomorphisms on the measure-valued support set of the solutions.
