The Singular Manifold Method

John Weiss
49 Grandview Road
Arlington, MA 02174

Abstract:

For systems with the Painlevé Property, Bäcklund transformations can be defined. These appear as specializations (truncations) of certain expansions of the solution about its' singular manifold. With reference to the Lax pair for a system, the Bäcklund transformations are equivalent to transformations of linear systems developed by Darboux (Bäcklund-Darboux transformations).

For specific systems the Bäcklund-Darboux transformations lead to a reformulation of these systems in terms of the Schwarzian derivative. We find the Bäcklund transformations of these system and study their periodic fixed points.

The periodic fixed points of the Bäcklund transformations determine a finite dimensional invariant manifold for the flow of the system. The resulting (ordinary) differential equations have a hamiltonian structure and the flow of the (partial) differential system is represented by commuting flows on the finite dimensional manifold.

The numerical resolution of the KdV periodic fixed points in the neighborhood of steady states and reductions is presented. These are analogous to orbits in the neighborhood of heteroclinic points.