Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II

Abstract

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-Kähler metrics.

Final version, to be published in Physics Letters A

References in corpus (2)

• Einstein-Weyl geometry, the dKP equation and twistor theory
• Anti-self-dual four-manifolds with a parallel real spinor

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