On the central quadric ansatz: integrable models and Painlevé reductions

Eugene Ferapontov, Benoit Huard, Aobo Zhang

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)
10 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)195204
JournalJournal of Physics A: Mathematical and Theoretical
Issue number19
Publication statusPublished - 2012


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