New integral equation form of integrable reductions of Einstein equations

TL;DR

This paper introduces a new integral equation formulation for integrable reductions of Einstein equations, incorporating evolving monodromy data, which simplifies the analysis of hyperbolic and elliptic cases.

Contribution

It develops a novel set of linear integral equations using dynamical monodromy data, extending the monodromy transform method for Einstein equations.

Findings

New integral equations include evolving monodromy data.

Field components can be expressed in quadratures using these equations.

Applicable to both hyperbolic and elliptic integrable reductions.

Abstract

A new development of the ``monodromy transform'' method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (``scattering'') matrices give rise to a new set of linear (quasi-Fredholm) integral equations equivalent to the symmetry reduced Einstein equations. Unlike previously derived singular integral equations constructed with the use of conserved (nonevolving) monodromy data on the spectral plane for the fundamental solutions of associated linear systems, the scalar kernels of the new equations include another kind of functional parameters -- the evolving (``dynamical'') monodromy data for the scattering matrices. For hyperbolic reductions, in the context of characteristic initial value problem these data are determined completely by the characteristic initial data for the fields. In terms of solutions of the new integral equations the field components are expressed in quadratures.