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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
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• 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
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SU(2,1) – symmetric form of dynamical equations
Einstein – Maxwell fields: the Ernst-like equations
1)
W.Kinnersley, J. Math.Phys. (1973) 1)
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1)
Isometry group with 2-surface –orthogonal orbits:
The Einstein’s field equations:
-- the “constraint” equations
-- the “dynamical” equations
-- the “dynamical” equations
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Geometrically defined coordinates:
Generalized Weyl coordinates:
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Belinski – Zakharov vacuum solitons
Einstein – Maxwell solitons
Examples of soliton solutions
Integrable reductions of Einstein equations
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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
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Associated spectral problem
V.Belinski & V.Zakharov,, JETP 1978; 1979 ; 1)
1)
Dynamical equations for vacuum
“Dressing” method for constructing solutions
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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)
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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
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Integrable reductions of Einstein-Maxwell equations
Spacetime metric and electromagnetic potential:
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Ernst potentials :
Ernst equations:
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3x3-matrix form of Einstein – Maxwell equaations
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1)
GA, JETP Lett.. (1980); Proc. Steklov Inst. Math. (1988); Physica D. (1999)1)
For vacuum:
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(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
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-- 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)
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-- Interaction of two superextreme Kerr-Newman sources
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