On a Schwarzian PDE associated with the KdV Hierarchy

Details On a Schwarzian PDE associated with the KdV Hierarchy On a Schwarzian PDE associated with the KdV Hierarchy

1999-09-24

arxiv.org/abs/solv-int/9909026v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

11 pages

Scientific paper

10.1016/S0375-9601(00)00063-3

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.

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