Homogeneous cosmological solutions of the Einstein equation

Homogeneous solutions in the framework of general relativity form the basis to understand the properties of gravitation on global scale. Presently favoured models describe the evolution of the universe by an expansion of space, governed by a scale function, which depends on a global time parameter. Dropping the restriction that a global time parameter exists, and instead assuming that the time scale depends on spatial distance, leads to static solutions, which exhibit no singularities, need no unobserved dark energy and which can explain the cosmological red shift without expansion. In contrast to the expanding world model energy is globally conserved. Observations of high energy emission and absorption from the intergalactic medium, which can scarcely be understood in the ‘concordance model’, find a natural explanation.     

Vacuum Cosmological Models in Einstein and Barber Theories

An attempt has been made to solve the field equations with perfect fluid in an inhomogeneous space-time governed by the metric

Cosmological vacuum solutions in Brans and Dicke's scalar-tensor theory

The exact cosmological vacuum solutions of Brans and Dicke's scalas-tensor theory are derived when a power law is valid between the gravitational constant κ and the radius of curvatureR of the universe. There exist even in the case of the closed 3-dimensional space of positive curvature solutions with increasingR and κ with respect to the age of the universe, whereby the freely available parameter ω of the scalar-tensor theory can take all values greater than −3/2. Such solutions are contrary to Dirac's hypothesis as well as to Einstein-Mach's principle.     

Exact two-soliton solutions to the Einstein gravitational field equations

Following our previous work on the existence of 1-soliton solution to the Einstein gravitational field equations in the presence of a spherically-symmetric static background field, we have found six sets of analytical 2-soliton solutions to the Einstein field equations under a certain ansatz in the absence of the stated background field. Numerical analysis shows that if the two solitons of the transverse nature are injected at space variable z±, the longitudinal field componentg ₃₃ will acquire non-zero values for a bounded spatial region at later time. The nature of the solitons becomes rather complex when they interact. The amplitudeg of each soliton may change its magnitude resulting from the interaction. We have found that we might interpret the evolution of one field component as the gravitational instanton in our solutions. We must remark also that the total energy of the interacting solitons remains constant, as expected, at all time. These solutions correspond to the situation where the Riemann tensor is in general non-zero and are truly non-trivial solutions.     

Cosmological solutions of the Friedmann type in the tensor-scalar theory of gravitation

Erevan State University. Translated from Astrofizika, Vol. 36, No. 4, pp. 583–592, October–December, 1993.     

Power-series solutions of the Lane-Emden equation

In this paper the problem of representing the numerical solutions of the Lane-Emden equation analytically by means of a convergent power series has been considered. Our results show that it is possible to represent the numerical solutions of the Lane-Emden equation by means of a power series which can be convergent in the whole interior of a polytropic model.     

Toroidal solutions of the gegenbauer equation

New solutions are found for the Laplacian and Grad-Shafranov operators in toroidal coordinates suitable for the study of gravitational and plasma problems.     

Self-consistent solutions of the semi-classical Einstein equations with cosmological constant and perfect fluid

Exact solutions of the semi-classical Einstein equations with cosmological constant for conformally invariant free quantum fields in a spatially flat Robertson-Walker metric are found when a classical perfect fluid is present. The stability of the asymptotically Friedmann-de Sitter solutions is studied. The formers are found to be stable if < 0, while the stability of the latter depends of the signs of , and A.Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas.Supported by scholarship of the Universidad de Buenos Aires.     

Certain particular solutions of the clairaut equation

A study has been conducted to ascertain conditions on the density distribution within a fluid, rotating planet in order that the deformation of its outer shell be expressible in terms of the Bessel functions of the first kind and the Gaussian hypergeometric series.It has been established that (1) the density must be a solution to an ordinary differential equation of the Riccati type, and (2) two already considered density distributions are the only closed-form solutions of physical relevance.     

Solitons and other solutions to the quantum Zakharov-Kuznetsov equation

In this paper, we present G′/G-expansion method, exp-function method, modified F-expansion method as well as the traveling wave hypothesis for finding the exact traveling wave solutions of the quantum Zakharov-Kuznetsov equation which arises in quantum magneto-plasmas. By these methods, rich families of exact solutions have been obtained, including soliton solutions. This work continues to reinforce the idea that the proposed methods, with the help of symbolic computation, provide a powerful mathematical tool for solving nonlinear partial differential equations.     

The general Kepler equation and its solutions

My father K. Stumpff (1947, 1949, 1951, 1959, 1962) developed a transcendental equation which replaces the original Kepler equation but is valid for all types of orbits. Other advantages over the classical methods are: a) the independent arguments of the equation follow from the vectors of position and velocity at any instant T_o, where T_o is not necessarily the perihelion time; b) an explicit knowledge of the classical orbital elements is not required; c) transformations of coordinate systems are avoided. The present paper discusses the properties of the general Kepler equation in a wide range of its independent arguments, and it is shown that analytic solutions, existing in special cases, can be used for the numerical solution of general cases. The theory is generalized insofar as it now can handle not only attracting forces but also repulsive ones. As a result of this investigation, FORTRAN subroutines were developed which can be used in connection with any two-body problem for the computation of position and velocity as function of time along any unperturbed orbit.     

Monoenergetic-source solutions of the steady-state cosmic-ray equation of transport

A non spherically-symmetric monoenergetic-point-source solution of the steady-state equation of transport for cosmic-rays in the interplanetary region, in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by a Laplace transform method. The solution is for a spherically-symmetric model of the propagating region incorporating anisotropic diffusion, with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speedV. The spherically-symmetric monoenergeticsource solution of Webb and Gleeson (1973) and of Toptygin (1973) is regained from the spherically-symmetric component of the point-source solution.     

Cosmological models with viscous fluid and time- dependent equation of state in Wesson’s theory

Assuming the time-dependent equation of state p=λ(t)ρ, five dimensional cosmological models with viscous fluid for an open universe (k=−1) and flat universe (k=0) are presented. Exact solutions in the context of the rest mass varying theory of gravity proposed by Wesson (Astron. Astrophys. 119, 145, 1983) are obtained. It is found that the phenomenon of isotropisation takes place in this theory, i.e. the mass scale factor A(t) which characterizes the rest mass of a typical particle is evolving with cosmic time just as the spatial scale factor R(t). It is further found that rest mass is approximately constant in the present universe.     

Exact solutions of the Lane–Emden-type equation

We classify the Lane–Emden-type equation xy^″+ny^′+x^νf(y)=0 with respect to the standard Lagrangian according to the Noether point symmetries it admits. First integrals of the various cases, which admit Noether point symmetries, and reduction to quadratures for these cases are obtained. Six cases result in new solutions.     

On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

Note on the single-shock solutions of the Korteweg-de Vries-Burgers equation

The well-known shock solutions of the Korteweg-de Vries-Burgers equation are revisited, together with their limitations in the context of plasma (astro)physical applications. Although available in the literature for a long time, it seems to have been forgotten in recent papers that such shocks are monotonic and unique, for a given plasma configuration, and cannot show oscillatory or bell-shaped features. This uniqueness is contrasted to solitary wave solutions of the two parent equations (Korteweg-de Vries and Burgers), which form a family of curves parameterized by the excess velocity over the linear phase speed.     

One of the solutions for the Laplace equation and its physical interpretation

In this paper it is confirmed once more that there exists the general solution of Laplace's equation in ellipsoidal coordinates which satisfies the Stäckel theorem and which was derived earlier by M. Jarov-Jarovoi and S. J. Madden. The author interprets physically the general solution in real space as potentials of layers of charge and double layers in which the distribution of densities is defined by Green's formula.