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–Ampère equations and of the second heavenly equation are found. Besides, non-invariance of the found classes of solutions of the hyperbolic complex Monge–Ampère equation and the second heavenly equation is investigated—these classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to the found classes is also formulated.

On symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation

“…has reduced to the investigation of the Monge-Ampère equations in the spaces of different dimensions and different types. Some details on this theme can be found in [33,1,34,37,38,24,21,26,39,22,2,15,40,25,36,23,35] (see, also, the references therein).…”

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

“…is reduced to the investigation of the Monge-Ampère equations in the spaces of different dimensions and different types. Some details on this theme can be found in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] (see, also, the references therein).…”