Group foliation and non-invariant solutions of the heavenly equation

Details Group foliation and non-invariant solutions of the heavenly equation Group foliation and non-invariant solutions of the heavenly equation

2001-08-15

arxiv.org/abs/math-ph/0108004v1

Physics

Mathematical Physics

32 pages, Latex, corrected typos, submitted to J. Phys. A: Math. Gen. on April 13, 2001

Scientific paper

10.1088/0305-4470/34/43/310

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining noninvariant solutions of partial differential equations with infinite dimensional symmetry groups. The method involves the 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 of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.

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