Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

Abstract

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, PII . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive autoBäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P34 equations, and a little known Bäcklund transformation for PII itself. We then use our results on Bäcklund transformations to obtain, for each member of the PII hierarchy, a sequence of special integrals.

Cite this paper

@inproceedings{Clarkson1999BacklundTF, title={Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach}, author={Peter A Clarkson and Nalini Joshi and Andrew Pickering}, year={1999} }