Gravity on a Parallelizable Manifold. Exact Solutions

The wave type field equation^a =^a, where ^ais a coframe field on a space-time, was recentlyproposed to describe the gravity field. This equationhas a unique static, spherically symmetric, asymptotically flat solution, whichleads to the viable Yilmaz–Rosen metric. We showthat the wave type field equation is satisfied by thepseudo-conformal frame if the conformal factor isdetermined by a scalar 3D-harmonic function. This functioncan be related to the Newtonian potential of classicalgravity. So we obtain a direct relation between thenon-relativistic gravity and the relativistic model: every classical exact solution leads to asolution of the field equation. With this result weobtain a wide class of exact, static metrics. We showthat the theory of Yilmaz relates to thepseudo-conformal sector of our construction. We derive also aunique cosmological (time dependent) solution of thedescribed type.     

Tensor Expressions for Solving Einstein's Equations by the Method of Sequential Approximation

The main aim of this paper is to develop a mathematical tool for General Relativity (GR). For this purpose useful tensor expressions have been worked out, which considerably ease various calculations using the sequential approximation in Einstein's GR. Based upon these expressions, compact and explicit formulae have been worked out for the covariant and contravariant components of the metric tensor and its determinant.     

Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem

In a nongeometrical interpretation of gravity,the metric g(x) = + (x)is interpreted as an effective metric, whereas(x) is interpreted as afundamental gravitational field, propagated in spacetime which isactually flat. Some advantages and disadvantages of suchan interpretation are discussed. The main advantage isa natural resolution of the flatness problem.     

Symmetric Variations of the Metric and Extrema of the Action for Pure Gravity

Symmetries of generalized gravitational actions, yielding field equations which typically involve at most second-order derivatives of the metric, are considered. The field equations for several different higher-derivative theories in the first-order formalism are derived, and variations of a generic set of higher-order curvature terms appearing in string effective actions are studied. It is shown that there often exists a particular set of solutions to the field equations of pure gravity theories, consisting of different combinations of curvature tensors, which satisfies the vacuum equations with cosmological constant. Implications of generalized symmetries of the field equations derived from the superstring effective action for the cosmological constant problem are discussed.     

Qualitative Properties of Scalar-Tensor Theories of Gravity

The qualitative properties of spatiallyhomogeneous stiff perfect fluid and minimally coupledmassless scalar field models within general relativityare discussed. Consequently, by exploiting the formal equivalence under conformal transformations and field redefinitions of certain classes of theories ofgravity, the asymptotic properties of spatiallyhomogeneous models in a class of scalar-tensor theories of gravity that includes the Brans-Dicke theory can be determined. For example, exact solutions are presented, which are analogues of the general relativistic Jacobs stiff perfect fluid solutions andvacuum plane wave solutions, which act as past andfuture attractors in the class of spatially homogeneousmodels in Brans-Dicke theory.     

Hamiltonian Formulation of FRW Equations of Cosmology

We give a general Hamiltonian formulation of the Friedmann-Robertson-Walker (FRW) fundamental equations of a homogeneous and isotropic universe in the Standard Cosmological model. It gives some interesting insight on the 3-curvature of the universe, which is found to be obtained in terms of the scale factor (also termed radius of the universe) and its first derivative. It also leads to an easy determination of the Schrödinger equation for the (R) wave function of the universe, in the radiative era.     

Solutions and Conditional Lie-Backlund Symmetries of Quasi-linear Diffusion-Reaction Equations

New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.     

Letter: A Note on Kerr-Schild Metrics with Geodesic Null Vector

We calculate the Einstein and Landau-Lifshitzpseudo-tensors for Kerr-Schild metrics with geodesicnull vector and show that the first of these vanishesidentically but that the second does not. These results correct a statement by Gurses and Gursey whoclaimed that both were zero. We also show that thesub-class of metrics for which both pseudotensors vanishincludes the Kinnersley photon rocket metric for anarbitrarily accelerating point mass.     

A Class of Two-Component Adler-Bobenko-Suris Lattice Equations

We study a class of two-component forms of the famous list of the Adler-Bobenko-Suris lattice equations. The obtained two-component lattice equations are still consistent around the cube and they admit solutions with ＇jumping properties＇ between two levels.     

Deduction of the Quasi-linear Transport Equations of Hydro-thermodynamics from the Gyarmati Principle

From the universal form of Gyarmati's variational principle of thermodynamics the differential equations governing the internal energy and impulse transport of one component hydro-thermodynamic systems are derived. In our particular case Gyarmati's “supplementary theorem” is confirmed, by which the validity of the universal form of Gyarmati's variational principle is guaranted also in non-linear cases. Finally some problems of the Gyarmati principle and of non-linear thermodynamics are discussed.     

热传导方程的一类无网格方法

构造求解热传导方程的一类无网格方法,只要选择好每个节点的适当的邻点集合,便可利用节点信息顺利进行计算.作为特殊情形,也可在各种结构或非结构网格上进行计算.在矩形或均匀平行四边形网格上进行计算时具有二阶精度,当在任意的不规则四边形或三角形网格上计算时仍然是守恒的和相容的,且至少具有一阶精度.作为数值试验,将该方法用于在不规则四边形网格上及四边形与三角形混合网格上求解二维非线性抛物型方程,并在不规则四边形网格上求解二维三温辐射热传导方程,均获得了较为理想的数值结果.     

Bianchi A Cosmological Models as the Simplest Dynamical Systems in R4

The behaviour of the vacuum non-tilted Bianchimodels of class A is studied in terms of dynamicalsystems theory. We introduce phase variables in whichthe Hamiltonian constraint is solved algebraically. It is shown that in these variables BianchiVIII and Bianchi IX models assume the form of afour-dimensional autonomous system with a polynomialvector field defined on the phase space, whereas Bianchi I and Bianchi II world models can be presentedas a one- and two-dimensional system, respectively. TheBianchi VI₀ and Bianchi VII₀ worldmodels are represented as a three-dimensional dynamicalsystem.     

A Class of Solutions of Non-Linear Evolution Equations with Diffusion

The technique of generating functions developed by WILHELMSSON [1] for the diffusion equation with quadratic recombination term is improved to yield overall exact (special) solutions and discussed with respect to higher-order non-linearity. Some applications of our results are mentioned.     

A New Look at Relativity Transformations

A free system, considered to be a comparisonsystem, allows for the notion of objective existence andinertial frame. Transformations connecting inertialframes are shown to be either Lorentz or generalizedGalilei.     

Quasi-linear pulses in birefringent fibers

The evolution of quasi-linear optical pulses in birefringent fibers, with strong dispersion-management, is studied analytically. The nonlinear terms of the Gabitov–Turitsyn equations (GTE) are analyzed asymptotically. The total spectral intensity, for a lossless system, is found to be an invariant of propagation, while for a lossy system it is dependent on the relative position of the amplifier in the dispersion map.OCIS Codes. 060.2330; 060.4370; 060.5530; 190.4370; 260.2030     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

A Variational Iteration Solving Method for a Class of Generalized Boussinesq Equations

We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation. The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iteration method.     

Conservation Laws of a Class of Combined Equations

In this paper, we investigate conservation laws of a class of partial differential equations, which combines the nonlinear telegraph equations and the nonlinear diffusion-convection equations. Moreover, some special conservation laws of the combined equations are obtained by means of symmetry classifications of wave equations uxx = H （x）utt.     

FLRW Cosmology with Horava-Lifshitz Gravity: Impacts of Equations of State

Inspired by Lifshitz theory for quantum critical phenomena in condensed matter, Horava proposed a theory for quantum gravity with an anisotropic scaling in ultraviolet. In Horava-Lifshitz gravity (HLG), we have studied the impacts of six types of equations of state on the evolution of various cosmological parameters such as Hubble parameters and scale factor. From the comparison of the general relativity gravity with the HLG with detailed and without with non-detailed balance conditions, remarkable differences are found. Also, a noticeable dependence of singular and non-singular Big Bang on the equations of state is observed. We conclude that HLG explains various epochs in the early universe and might be able to reproduce the entire cosmic history with and without singular Big Bang.     

Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity

Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrodinger equation obtained from this non-relativistic limit, we can see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrodinger equation, which can explain the gravitational phase effects found in COW experiments.And because of this Newtonian gravitational potential, a quantum particle in the earth's gravitational field may form a gravitationally bound quantized state, which has already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are studied in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, and radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star.