On the central quadric ansatz: integrable models and Painleve reductions

TL;DR

This paper classifies integrable models with solutions characterized by central quadrics, linking them to Painleve equations, and shows these solutions form a special subclass of two-phase solutions.

Contribution

It provides a classification of integrable models with the central quadric ansatz and connects these solutions to all Painleve equations P I - P VI.

Findings

Five canonical forms of integrable models identified.

All Painleve equations P I - P VI derived from the ansatz.

Solutions form a subclass of two-phase solutions.

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.