82 research outputs found
Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlev\'e I and II
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.Comment: Final version, to be published in Physics Letters
- …