On some exact solutions of heavenly equations in four dimensions

Abstract

Some new classes of exact solutions (so-called functionally-invariant solutions) of the elliptic and hyperbolic complex Monge-Amp$\grave{e}$re equations and of the second heavenly equation, mixed heavenly equation, asymmetric heavenly equation, evolution form of second heavenly equation, general heavenly equation, real general heavenly equation and one of the real sections of general heavenly equation, are found. Besides non-invariance of these found classes of solutions has been investigated. These classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to found classes, has been also formulated.

References in corpus (4)

• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• On the integrability of symplectic Monge-Ampére equations
• General heavenly equation governs anti-self-dual gravity
• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics