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

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 Painlevé 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.

References in corpus (15)

• Einstein-Weyl geometry, the dKP equation and twistor theory
• The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation
• On the Einstein-Weyl and conformal self-duality equations
• Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II
• Screen bundles of Lorentzian manifolds and some generalisations of pp-waves
• On the central quadric ansatz: integrable models and Painleve reductions
• Einstein--Weyl geometry, dispersionless Hirota equation and Veronese webs
• Integrable Background Geometries
• Harmonic functions, central quadrics, and twistor theory
• On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
• On critical behaviour in generalized Kadomtsev--Petviashvili equations
• Shock formation in the dispersionless Kadomtsev-Petviashvili equation
• On supersymmetric Einstein-Weyl spaces
• On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks
• Parallel spinors on Lorentzian Weyl spaces

Cited by in corpus (1)

• Recurrent Lorentzian Weyl spaces