On the central quadric ansatz: integrable models and Painlevé reductions

Ferapontov, Eugene, Huard, Benoit and Zhang, Aobo (2012) On the central quadric ansatz: integrable models and Painlevé reductions. Journal of Physics A: Mathematical and Theoretical, 45 (19). p. 195204. ISSN 1751-8113

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Abstract

It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painleve equations PIII and PII, respectively. The aim of our paper is threefold:
-- Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP).
-- Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painleve equations PI - PVI, with PVI corresponding to the generic case of our classification.
-- We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.

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