Gravitational and electromagnetic solitons Monodromy transform approach Solution of the...

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

1)

Isometry group with 2-surface –orthogonal orbits:

The Einstein’s field equations:

-- the “constraint” equations

-- the “dynamical” 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)

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 problem1)

12

V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1)

( - solitons)Vacuum solitons1)

Soliton ansatz for dressing matrix

2N-soliton solution:

13GA, Sov.Phys.Dokl. (1981) ; 1)

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)

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)

1)

20

-- Interaction of two superextreme Kerr-Newman sources

21