1. Gravitational and electromagnetic solitons Stationary axisymmetric solitons; soliton waves Monodromy transform approach Solutions for black holes in the external fields + Addendum to Lecture 2: How to calculate … + Solving of the characteristic initial value problems Colliding gravitational and electromagnetic waves Many “languages” of integrability

2. How to calculate : monodromy data  metric end potentials How to calculate : metric and potentials  monodromy data

3. GA and V.Belinski Phys.Rev. D (2007) 1) In equilibrium 1)

4. (No constraints)(Constraint: field equations) The space of local solutions: Free space of the mono- dromy data functions: “Direct’’ problem: “ Inverse’’ problem :

5. 5 Monodromy data map of some classes of solutions Solutions with diagonal metrics: static fields, waves with linear polarization: Stationary axisymmetric fields with the regular axis of symmetry are described by analytically matched monodromy data:: For asymptotically flat stationary axisymmetric fields with the coefficients expressed in terms of the multipole moments. For stationary axisymmetric fields with a regular axis of symmetry the values of the Ernst potentials on the axis near the point of normalization are For arbitrary rational and analytically matched monodromy data the solution can be found explicitly.

6. 6 Map of some known solutions Minkowski space-time Rindler metric Bertotti – Robinson solution for electromagnetic universe, Bell – Szekeres solution for colliding plane electromagnetic waves Melvin magnetic universe Kerr – Newman black hole Kerr – Newman black hole in the external electromagnetic field Symmetric Kasner space-time Khan-Penrose and Nutku – Halil solutions for colliding plane gravitational waves

8. 8 General structure of the matrices U, V, W

9. The symmetric vacuum Kazner solution is For this solution the matrix takes sthe form The monodromy data functions

10. The simplest example of solutions arise for zero monodromy data -- stationary axisymmetric or with cylindrical symmetry -- Kazner form -- accelerated frame (Rindler metric) This corresponds to the Minkowski space-time with metrics The matrix for these metrics takes the following form (where ):

11. 11 Calculation of the metric components and potentials

12. 12 Infinite hierarchies of exact solutions Analytically matched rational monodromy data: Hierarchies of explicit solutions :

13. 13 Inversion formulae for the Cauchy type integrals:

14. 14 NxN-matrix spectral problems

15. 15 Solving of the characteristic initial value problems for Einstein’s field equations with symmetries Characteristic initial value problem for colliding plane gravitational, electromagnetic, etc. waves Integral “evolution” equations as a new integral equation form of integrable reductions of Einstein’s field equations

16. 16 1) Analytical data:

17. 17 Irregular behaviour of Weyl coordinates on the wavefronts Generalized integral ``evolution’’ equations (decoupled form):

18. 18 GA & J.B.Griffiths, PRL 2001; CQG 20041)

19. 19 Space-time geometry and field equations Matching conditions on the wavefronts: -- are continuous

20. 20 Initial data on the left characteristic from the left wave -- u is chosen as the affine parameter -- arbitrary functions, provided and -- v is chosen as the affine parameter -- arbitrary functions, provided and Initial data on the right characteristic from the right wave

21. 21 1) Boundary values for on the characteristics: Scattering matrices and their properties: GA, Theor.Math.Phys. 2001; GA & J.B.Griffiths, PRL 2001; CQG 2004

22. 22 Dynamical monodromy data and : Derivation of the integral ``evolution’’ equations

23. 23 Coupled system of the integral ``evolution’’ equations: Decoupled integral ``evolution’’ equations:

24. 24