Gernot Neugebauer

General N-soliton solution of the AKNS class on arbitrary background

Abstract An explicit determinant formula for the N-fols Backlund transform of any pair of AKNS fields q(x, t), r(x, t) is derived. In particular this formula includes all N-soliton solutions of nonlinear evolution equations which belong to the AKNS class.

Einstein-Maxwell solitons

The application of the inverse scattering method to the Einstein-Maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the Ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution.

The superposition of two Kerr solutions

Abstract We present a new exact solution of Einsteins field equations: the nonlinear superposition of two Kerr-NUT solutions. The Tomimatsu-Sato δ = 2 solution is contained as a limiting case.

Zu axialsymmetrischen stationären Lösungen der Einsteinschen Feldgleichungen für das Vakuum

For axialsymmetric stationary vacuum fields Einstein equations reduce to a system, derivable from a simple Lagrangian. An investigation of its forminvariance leads to a method to construct from known solutions generalized solutions with one additional parameter.The method is applied to Weyls class and to Kerr metric.ZusammenfassungFür axialsymmetrische stationäre Vakuumfelder reduzieren sich die Einsteinschen Gleichungen auf ein System, das aus einer einfachen Lagrangefunktion herleitbar ist. Eine Untersuchung ihrer Forminvarianz führt zu einer Methode, die es erlaubt, aus bekannten Lössungen verallgemeinerte Lösungen mit einem zusätzlichen Parameter zu konstruieren.Das Verfahren wird auf die Weylsche Klasse und auf die Kerr-Metrik angewandt.

Progress in relativistic gravitational theory using the inverse scattering method

The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This article is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust (“galaxies”). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

The Einsteinian gravitational field of the rigidly rotating disk of dust

This paper presents the gravitational field of a uniformly rotating stationary and axisymmetric disk consisting of dust particles as a rigorous global solution to the Einstein equations. The problem is formulated as a boundary value problem of the Ernst equation and solved by means of inverse methods. The solution is given in terms of linear integral equations and depends on two parameters: the angular velocity Ω and the relative redshift z from the center of the disk. The Newtonian limit z<<1 represents the MacLaurin solution of a rotating fluid in the disk limit. For z→∞ the exterior solution is given by the extreme Kerr solution. This proves a conjecture of Bardeen & Wagoner (1969, 1971)

Non-existence of stationary two-black-hole configurations

We resume former discussions of the question, whether the spin–spin repulsion and the gravitational attraction of two aligned black holes can balance each other. To answer the question we formulate a boundary value problem for two separate (Killing-) horizons and apply the inverse (scattering) method to solve it. Making use of results of Manko, Ruiz and Sanabria-Gómez and a novel black hole criterion, we prove the non-existence of the equilibrium situation in question.

A general integral of the axially symmetric stationary Einstein equations

The Ernst function of an axially symmetric stationary asymptotically flat space-time involving an arbitrary harmonic function and an arbitrary number of constants is presented and discussed.

A NEW NONLINEAR SCHRODINGER EQUATION, ITS HIERARCHY AND N SOLITON SOLUTIONS

Abstract We consider a new spectral problem for which we show some of its associated nonlinear evolution equations, together with its Backlund transformations. As a reduction we obtain a new nonlinear Schrodinger equation. At the end, taking into account the transformation properties of this spectral problem we are able to recover the N-soliton solutions.

Vandermonde-like determinants and N-fold Darboux/Bäcklund transformations

We define Vandermonde-like determinants and analyze their structure. The resulting scheme is well-suited to achieve a remarkable compactness and transparency in N-soliton formulas or, more generally, in formulas for N-fold Darboux or Backlund transformations.