Group foliation and non-invariant solutions of the heavenly equation

TL;DR

This paper introduces a new method for deriving non-invariant solutions of the heavenly equation, providing exact solutions relevant to general relativity and advancing the mathematical techniques for PDEs with infinite symmetry groups.

Contribution

It presents a novel approach using automorphic differential constraints to find non-invariant solutions of PDEs with infinite-dimensional symmetry groups.

Findings

Derived explicit non-invariant solutions of the heavenly equation

Developed a method involving automorphic constraints and foliation of solution manifolds

Provided solutions of importance in the context of general relativity

Abstract

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.