The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

Abstract

We present in this paper the singular manifold method (SMM) derived from Painlevé analysis, as a helpful tool to obtain much of the characteristic features of nonlinear partial differential equations. As is well known, it provides in an algorithmic way the Lax pair and the Bäcklund transformation for the PDE under scrutiny.

Moreover, the use of singular manifold equations under homographic invariance consideration leads us to point out the connection between the SMM and so–called nonclassical symmetries as well as those obtained from direct methods. It is illustrated here by means of some examples.

We introduce at the same time a new procedure that is able to determine the Darboux transformations. In this way, we obtain as a bonus the one and two soliton solutions at the same step of the iterative process to evaluate solutions.