Integrable reductions of Einstein–Maxwell equations and nonlinear interaction of strong gravitational and electromagnetic waves

The structures of completely integrable reductions of Einstein–Maxwell equations and construction of the corresponding spectral problems for spacetimes which admit an abelian two-dimensional isometry group will be described. We describe also a construction of exact solutions of these equations using the basic ideas of inverse scattering and monodromy transform approaches as well as some soliton theory methods. As an application, we present a new class of exact solutions of Einstein - Maxwell equations which describe propagation and nonlinear interaction of gravitational and electromagnetic waves of arbitrary amplitudes on the homogeneous anysotropically expanding universe [1]. The explicit expressions show many features of this interaction such as an evolution of a soliton passing through another wave, nonlinear scattering of colliding soliton and non-soliton waves, mutual transformations of gravitational and electromagnetic waves during their collision, creation of gravitational waves in the process of collision of two pure electromagnetic waves and some other wave phenomena.
(Preliminary knowledge of any basic notions of General relativity theory as well as of the inverse scattering approach to solution of integrable nonlinear equations and soliton theory are not expected.)
[1] G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe”, Phys. Rev. D, 93:6 (2016), 061501; http://lanl.arxiv.org/pdf/1511.03335v2 (text with figures)