Related papers: Solutions of Schlesinger system and Ernst equation…
We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of Schlesinger system, Ernst equation…
It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous…
The Schwarzschild solution is a complete solution of Einstein's field equations for a static spherically symmetric field. The Einstein's field equations solutions appear in the literature, but in different ways corresponding to different…
From a particularly simple solution of the Ernst equation, we build a solution of the vacuum stationary axisymmetric Einstein equations depending on three parameters. The parameters are associated to the total mass of the source and its…
We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for…
We generalize some classical results for the Schlesinger system of partial differential equations and give the explicit form of its solution, associated with rational matrix functions in general position.
We apply the energy-momentum tensor which is coordinate independent to calculate the energy content of the axisymmetric solutions. Our results are compared with what have been obtained before within the framework of Einstein general…
In this paper we construct explicit solutions and calculate the corresponding $\tau$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose…
The Einstein field equations are derived for a static cylindrically symmetric spacetime with elastic matter. The equations can be reduced to a system of two nonlinear ordinary differential equations and we present analytical and numerical…
A general formula for the metric as an explicit function of the generic energy-momentum tensor is given which satisfies static plane symmetric Einstein's equations with cosmological constant.In order to illustrate it, the solutions for the…
Contents: 1) Introduction and a few excursions [A word on the role of explicit solutions in other parts of physics and astrophysics. Einstein's field equations. "Just so" notes on the simplest solutions: The Minkowski, de Sitter and anti-de…
We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid.…
Einstein's equations for stationary axisymmetric fields are reformulated as the equations for affine geodesics in a two--dimensional space. The affine collineations of this space are investigated and used to relate explicit solutions of…
The Ernst equation is formulated on an arbitrary Riemann surface. Analytically, the problem reduces to finding solutions of the ordinary Ernst equation which are periodic along the symmetry axis. The family of (punctured) Riemann surfaces…
We consider the Schr\"odinger--Poisson system on the complete, simply-connected Riemannian manifolds of constant sectional curvature. We obtain closed-form stationary spherically-symmetric solutions for the homogeneous equations for certain…
The problem of comparison of the stationary axisymmetric vacuum solutions obtained within the framework of exact and approximate approaches for the description of the same general relativistic systems is considered. We suggest two ways of…
A procedure to find static axially symmetric solutions to the Einstein field equations is presented. We obtained two general solutions and five particular solutions, which depend on the existence conditions for circular and $z$ direction…
In this work we present an approximate solution of the Einstein equations describing a global model for the gravitational field generated by a bounded, self-gravitating stationary and axisymmetric body rotating rigidly with constant angular…
We discuss solution generating techniques treating stationary and axially symmetric metrics in the presence of a cosmological constant. Using the recently found extended form of Ernst's complex equation, which takes into account the…
Axisymmetric and stationary solutions are constructed to the Einstein--Vlasov and Vlasov--Poisson systems. These solutions are constructed numerically, using finite element methods and a fixed-point iteration in which the total mass is…