Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations

Abstract

In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called "monodromy transform approach", which provides some general base for the study of various integrable space - time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented.

8 pages, 1 figure, LaTeX2e; based on a talk given at the International Conference 'Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives', (Landau Institute for Theoretical Physics, Chernogolovka, Moscow region, Russia, August 3 -- 10, 1999); as submitted to Physica D

References in corpus (1)

• Soliton Solutions with Real Poles in the Alekseev formulation of the Inverse-Scattering method

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• Integrable and non-integrable structures in Einstein - Maxwell equations with Abelian isometry group ${\mathcal{G}}_2$
• Vacuum Solutions of Einstein's Equations in Parabolic Coordinates
• Solution generating methods as "coordinate" transformations in the solution spaces
• Comment on F.J.Ernst, V.S.Manko and E.Ruiz "On interrelations between Sibgatullin's and Alekseev's approaches to the construction of exact solutions of the Einstein-Maxwell equations" (J.Phys.:Conf.Ser.229(2010)012050; arXiv:1006.5118)