• DOI:10.1088/1751-8113/45/19/195204
• Corpus ID: 54864921

On the central quadric ansatz: integrable models and Painlevé reductions

@article{Ferapontov2012OnTC,
  title={On the central quadric ansatz: integrable models and Painlev{\'e} reductions},
  author={Eugene V. Ferapontov and Benoit Huard and Alex Zhongyi Zhang},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2012},
  volume={45}
}

• E. Ferapontov, B. Huard, A. Zhang
• Published 24 January 2012
• Mathematics
• Journal of Physics A: Mathematical and Theoretical

It was observed by Tod (1995 Class. Quantum Grav.12 1535–47) and later by Dunajski and Tod (2002 Phys. Lett. A 303 253–64) 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 Painlevé equations PIII and PII, respectively. The aim of our paper is threefold: (1) 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). (2) Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painlevé equations PI–PVI, with PVI corresponding to the generic case of our classification. (3) We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.