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: