A hodograph transformation which applies to the heavenly equation

Abstract

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the heavenly equation (dispersionless Toda equation) as well as a family of explicit ultra-hyperbolic selfdual vacuum spaces admiting only one Killing vector which is not selfdual, we also give the corresponding explicit Einstein--Weyl structures.

arXiv commentLaTeX2e with amsmath package, 11 pages

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• Boyer-Finley equation and systems of hydrodynamic type
• Inverse Scattering Problem for Vector Fields and the Heavenly Equation
• Method of group foliation, hodograph transformation and non-invariant solutions of the Boyer-Finley equation
• A family of heavenly metrics