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On the central quadric ansatz: integrable models and Painleve reductions

181   0   0.0 ( 0 )
 Added by E. V. Ferapontov
 Publication date 2012
  fields Physics
and research's language is English




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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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We compare the results of our two papers with the results of the paper Aratyn H., Gomes J.F., Zimerman A.H., Higher order Painleve equations and their symmetries via reductions of a class of integrable models, J. Phys. A: Math. Theor., V. 44} (2011), Art. No. 235202.
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