On the central quadric ansatz: integrable models and Painleve reductions
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.
Keywords
Cite
@article{arxiv.1201.5061,
title = {On the central quadric ansatz: integrable models and Painleve reductions},
author = {E. V. Ferapontov and B. Huard and A. Zhang},
journal= {arXiv preprint arXiv:1201.5061},
year = {2015}
}
Comments
12 pages