On a Schwarzian PDE associated with the KdV Hierarchy

Frank Nijhoff; Andrew Hone; Nalini Joshi

doi:10.1016/s0375-9601(00)00063-3

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On a Schwarzian PDE associated with the KdV Hierarchy

Frank Nijhoff·Andrew Hone·Nalini Joshi

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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.

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Frank Nijhoff, Andrew Hone, Nalini Joshi. 1999-09-24. On a Schwarzian PDE associated with the KdV Hierarchy. https://doi.org/10.1016/s0375-9601(00)00063-3

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