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

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

2002-04-17

arxiv.org/abs/nlin/0204043v2

Phys.Lett. A303 (2002) 253-264

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Final version, to be published in Physics Letters A

Scientific paper

10.1016/S0375-9601(02)01258-6

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlev\'e 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-K\"ahler metrics.

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