Mikhail B. Sheftel

Feza Gursey Institute,
PO Box 6, Cengelkoy, 81220 Istanbul, Turkey
and
Department of Higher Mathematics, North Western State Technical University,
Millionnaya Str. 5, 191186, St. Petersburg, Russia

E-mail: sheftel@gursey.gov.tr

Method of group foliation and non-invariant solutions of invariant equations

Abstract:
Using the heavenly equation as an example, we propose a new method of  obtaining non-invariant solutions of partial
differential equations with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given
equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively
on the submanifold of the common solutions. By studying the integrability conditions of this automorphic system, i.e. the
resolving equations, one can provide an explicit foliation of the entire solution manifold of the considered equations.The new
important features of the method are:

  1. The presentation of the resolving system as a commutator algebra of the operators of invariant differentiation.
  2. The concept of invariant integration applied to the automorphic system.
The main physical result of this work are new exact analytical solutions of  the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation, of importance in the general theory of relativity. They determine the metrics which are new exact solutions of the Einstein field equations with only one Killing vector. Solutions are obtained with both Euclidean and ultra-hyperbolic signatures, the first  one being of interest for the theory of gravitational instantons.