Related papers: A hyperelliptic solution class for the hyperbolic …
We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert techniques.…
A program is outlined concerning the set of all solutions of the hyperbolic Ernst equation on a two-dimensional manifold whose underlying topological space is the same as the domain of all Ernst potentials for colliding plane gravitational…
The continuous point symmetry algebra of the hyperbolic Ernst equation is presented. In a second step the corresponding group transformations are considered. Accordingly, the solutions of the hyperbolic Ernst equation that are invariant…
The collision of two plane gravitational waves in Einstein's theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst…
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…
In a recent paper, we applied Riemann--Hilbert techniques to analyze the Goursat problem for the hyperbolic Ernst equation, which describes the interaction of two colliding gravitational plane waves. Here we generalize this approach to…
A multidimensional gravitational model with several scalar fields and form fields is considered. A wide class of generalized pp-wave solutions defined on a product of n+1 Ricci-flat spaces is obtained. Certain examples of solutions (e.g. in…
The inhomogeneous wave equations for the scalar, vector, and Hertz potentials are derived starting from retarded charge, current, and polarization densities and then solved in the reciprocal (or k-) space to obtain general solutions, which…
We enunciate and prove here a generalization of Geroch's famous conjecture concerning analytic solutions of the elliptic Ernst equation. Our generalization is stated for solutions of the hyperbolic Ernst equation that are not necessarily…
We show that, given an arbitrary shift, the lapse $N$ can be chosen so that the extrinsic curvature $K$ of the space slices with metric $\overline g$ in arbitrary coordinates of a solution of Einstein's equations satisfies a quasi-linear…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
We obtain a new family of exact vacuum solutions to quadratic gravity that describe pp-waves with two-dimensional wave surfaces that can have any prescribed constant curvature. When the wave surfaces are flat we recover the Peres waves…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
A first-order elliptic-hyperbolic system in extended projective space is shown to possess strong solutions to a natural class of Guderley-Morawetz-Keldysh problems on a typical domain.
Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing…
A new family of nonlinear partial differential equations is presented. They represent a generalization of the hyperbolic Ernst equations for an Einstein-Mawxell-Weyl field in general relativity. A B\"acklund transformation for the system of…
This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is…
Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the…
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the…
We present new periodic, kink-like and soliton-like travelling wave solutions to the hyperbolic generalization of Burgers equation. To obtain them, we employ the classical and generalized symmetry methods and the ansatz-based approach