On a Schwarzian PDE associated with the KdV Hierarchy

Abstract

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius 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} Painlevé VI equation, i.e. with four arbitary parameters.

11 pages

References in corpus (2)

• The KP Hierarchy in Miwa Coordinates
• On the discrete and continuous Miura Chain associated with the Sixth Painlevé Equation

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