The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces

TL;DR

This paper explores multi-dimensional generalisations of the dispersionless KP equation, characterising solutions with a quadric ansatz and linking to Einstein-Weyl structures, revealing new geometric spaces and integrability properties.

Contribution

It introduces new multi-dimensional generalisations of the dKP equation, characterises solutions with a quadric ansatz, and constructs explicit Einstein-Weyl spaces with specific geometric properties.

Findings

Quadric ansatz leads to Painleve I or II in 2D but not in higher dimensions.

A new family of Einstein-Weyl spaces depending on one arbitrary function is constructed.

The second generalisation yields Einstein-Weyl structures with a weighted parallel vector field.

Abstract

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev-Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painleve I or II for the dKP equation, but fails to pass the Painlev\'e test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein-Weyl structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of Einstein-Weyl spaces belonging to this class, and depending on one arbitrary function of one variable.