# On a Schwarzian PDE associated with the KdV Hierarchy

Nijhoff, Frank W., Hone, Andrew N.W., Joshi, Nalini (2000) On a Schwarzian PDE associated with the KdV Hierarchy. Physics Letters A, 267 (2-3). pp. 147-156. ISSN 0375-9601. (doi:10.1016/S0375-9601(00)00063-3)

## Abstract

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under M\"obius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlev\'e VI equation, i.e. with four arbitary parameters.

Item Type: Article 10.1016/S0375-9601(00)00063-3 Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equationsQ Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Andrew N W Hone 21 Jun 2014 23:12 UTC 29 May 2019 12:42 UTC https://kar.kent.ac.uk/id/eprint/41499 (The current URI for this page, for reference purposes)