Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology,
Heriot-Watt University, Edinburgh, July 95

The singular manifold method: a fresh approach

A. Pickering

Dienst Theoretische Natuurkunde,
Vrije Universiteit Brussel,
B-1050 Brussels, Belgium


The singular manifold method of Weiss allows for many completely integrable partial differential equations the recovery of the Lax pair and Darboux transformation (DT), and so also the Bäcklund transformation, from a truncated Painlevé expansion. Here we present a natural extension of this singular manifold method whereby the DT is now identified with an infinite Painlevé expansion for a certain choice of the arbitrary coefficients: this then leads us to a new and more consistent definition of ``singular manifold equation.'' This summation of an infinite Painlevé expansion, which involves of course only one singular manifold, is achieved by seeking a truncated expansion in a new Riccati variable Z. This new Riccati variable also allows us to place within the context of Painlevé analysis a larger class of exact solutions than was possible hitherto. Full details of this work are presented elsewhere.

Conference paper.


Last modified Mon Apr 8 16:25:00 GB-Eire 1996 (DBD)