The singular manifold method revisited

Abstract

For many completely integrable partial differential equations (PDEs) the singular manifold method of Weiss allows the recovery of the Lax pair and Darboux transformation (DT), and so also the Bäcklund transformation, from a truncated Painlevé expansion. Recently the so-called ‘‘two-singular manifold method’’ has been proposed in order to handle PDEs such as the modified Korteweg–de Vries (MKdV) equation. Here we present a more natural extension of the Weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such PDEs. In this approach we allow the possibility that the DT might in fact correspond to 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’’ (SME); this can give SMEs different from those usually presented. The summation of infinite Painlevé expansions is effected by seeking a truncation in a new Riccati variable Z. The use of this variable greatly simplifies the recovery of Lax pairs from Painlevé analysis. Practical and theoretical aspects of our approach are illustrated using MKdV as an example. The results of this analysis are confirmed by the consideration of fifth-order MKdV. We then make a further extension of this method which allows it to be applied to a PDE in 2+1 dimensions, and so simultaneously to reductions of the latter to PDEs in 1+1 dimensions. A corollary of our analysis is a direct proof of the convergence of infinite WTC expansions for a certain choice of the arbitrary coefficients therein. In addition, the approach developed here allows us to place within the context of Painlevé analysis a larger class of exact solutions than was possible hitherto. Again, our analysis greatly simplifies the recovery of such solutions.