Gravitational and electromagnetic solitons Monodromy transform approach Solution of the characteristic initial value problem; Colliding gravitational and electromagnetic waves Many “languages” of integrability Solutions for black holes in the external fields

mathematical context: - infinite hierarchies of exact solutions, - initial and boundary value problems, - asymptotical behaviour Integrable cases: - Vacuum gravitational fields - Einstein – Maxwell - Weyl fields - Ideal fluid with - some string gravity models physical context: - supeposition of stat. axisymm. fields, - nonlinear interacting waves, - inhomogeneous cosmological models

3 Associated linear systems and ``spectral’’ problems Infinite-dimensional algebra of internal symmetries Solution generating procedures (arbitrary seed): -- Solitons, -- Backlund transformations, -- Symmetry transformations Infinite hierarchies of exact solutions -- Meromorfic on the Riemann sphere -- Meromorfic on the Riemann surfaces (finite gap solutions) Prolongation structures Geroch conjecture Riemann – Hielbert and Homogeneous Hilbert problems, Various linear singular integral equation methods Initial and boundary value problems -- Characteristic initial value problems -- Boundary value problems for stationary axisymmetric fields Twistor theory of the Ernst equation

4

5 SU(2,1) – symmetric form of dynamical equations Einstein – Maxwell fields: the Ernst-like equations 1) W.Kinnersley, J. Math.Phys. (1973) 1)

6 Isometry group with 2-surface –orthogonal orbits: The Einstein’s field equations: -- the “constraint” equations -- the “dynamical” equations

7 Geometrically defined coordinates: Generalized Weyl coordinates:

8 Belinski – Zakharov vacuum solitons Einstein – Maxwell solitons Examples of soliton solutions Integrable reductions of Einstein equations

9 Belinski – Zakharov form of reduced vacuum equations Kinnersley self-dual form of the reduced vacuum equations 2x2-matrix form of self-dual reduced vacuum equations Ernst vacuum equation

10 Associated spectral problem V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1) Dynamical equations for vacuum “Dressing” method for constructing solutions

11 Riemann problem for dressing matrix Linear singular integral equations Constraints for dressing matrix: V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1) Formulation of the matrix Riemann problem 1)

12 V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1) ( - solitons )Vacuum solitons 1) Soliton ansatz for dressing matrix 2N-soliton solution:

13 GA, Sov.Phys.Dokl. (1981) ; 1) Stationary axisymmetric solitons on the Minkowski background: a set of 4 N arbitrary real or pairwise complex conjugated constants

14 Integrable reductions of Einstein-Maxwell equations Spacetime metric and electromagnetic potential :

15 Ernst potentials : Ernst equations:

16 3x3-matrix form of Einstein – Maxwell equaations

17 1) GA, JETP Lett.. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1) For vacuum:

18 ( w - solitons ) Soliton ansatz for dressing matrix GA, JETP Lett. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1) Dressing matrix : --- a set of 3 N arbitrary complex constants

19 -- Superextreme part of the Kerr-Newman solution -- Interaction of two superextreme Kerr-Newman sources -- mass -- NUT-parameter -- angular momentum -- electric charge -- magnetic charge GA, Proc. Steklov Inst. Math. (1988); Physica D. (1999) 1)

20 -- Interaction of two superextreme Kerr-Newman sources

21