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arXiv · 1201.5061

On the central quadric ansatz: integrable models and Painleve reductions

E. V. Ferapontov·B. Huard·A. Zhang
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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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E. V. Ferapontov, B. Huard, A. Zhang. 2012-01-24. On the central quadric ansatz: integrable models and Painleve reductions. https://doi.org/10.1088/1751-8113%2F45%2F19%2F195204

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