Generalised Painlevé truncation: expansion in Riccati pseudopotentials

Summary

Abstract

In the search for the Lax pair and Darboux transformation of a (completely integrable) partial differential equation, a natural generalisation of the constant-level truncation in Painlevé analysis is to seek an expansion as a polynomial in the components of some (to be determined) Riccati pseudopotential. In this direct approach the relationship with Painlevé analysis becomes a secondary consideration. Here we give two generalisations of this method. The first is to finding Lax pairs of higher order, which is relatively straightforward. The second concerns the assumption of the form of the Darboux transformation within the context of a Lax pair having some gauge freedom: this allows us to deal easily with a modification of the Boussinesq equation.

Introduction

Given a partial differential equation (PDE) suspected of being completely integrable, for example because it passes the Weiss-Tabor-Carnevale (WTC) Painlevé test, there are several techniques that might be used to find its Lax pair. One could use Walquhist-Estabrook prolongation, or one could put the PDE into Hirota bilinear form and then try to find the Bäcklund transformation (BT), or one could use the information provided by truncating the WTC Painlevé expansion. Overviews of these approaches can be found in. Here we will be concerned with an alternative method, which involves seeking a solution to the PDE as an expansion in some Riccati pseudopotential. As we will see, this provides a natural extension of the constant-level truncation in Painlevé analysis.

From the truncated Painlevé expansion there are several ways to proceed in order to find the Lax pair. Perhaps the method most commonly referred to is the “singular manifold method” of Weiss.