An effective method for finding special solutions of nonlinear differential equations with variable coefficients

There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method.     

Multi-order exact solutions to the Drinfel'd-Sokolov-Wilson equations

In this Letter, based on the Lamé function and Jacobi elliptic function, the perturbation method is applied to the classical Drinfel'd-Sokolov-Wilson (hereafter DSW for short) equations, and many multi-order solutions are derived. It is shown that different Lamé functions can exist in the first order solutions of DSW system.     

New Lamé function and its application to nonlinear equations

In this Letter, a new kind of Lamé functions are given. Based on the new Lamé functions and Jacobi elliptic function, the perturbation method is applied to the nonlinear equations, and many multi-order solutions of novel forms are derived. In addition, it is shown that different Lamé functions can exist in the first order solutions of nonlinear system.     

Stability of Motion of a Nonholonomic System

Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev＇s type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.     

Weyl metrics and the generating conjecture

By means of the Ernst complex potential formalism it is shown that previously studied static axially symmetric Einstein-Maxwell fields obtained though the application of the Horsky-Mitskievitch generating conjecture represent a combination of Kinnersley’s transformations [W. Kinnersley: J. Math. Phys.14 (1973) 651]. New theoretical background for the conjecture is suggested and commented.     

Nearly everywhere flat spaces

We discuss some spacetimes, which are flat everywhere except for a thin shell of matter or a string of matter, in the framework of the Israel formalism. First we study spherically symmetric universes with a single sheet of matter. Then we show that the construction of a cosmic string as a limit of various thin shell distributions of matter leads to identical results.     

Metric-affine f(R) theories of gravity

General Relativity assumes that spacetime is fully described by the metric alone. An alternative is the so called Palatini formalism where the metric and the connections are taken as independent quantities. The metric-affine theory of gravity has attracted considerable attention recently, since it was shown that within this framework some cosmological models, based on some generalized gravitational actions, can account for the current accelerated expansion of the universe. However we think that metric-affine gravity deserves much more attention than that related to cosmological applications and so we consider here metric-affine gravity theories in which the gravitational action is a general function of the scalar curvature while the matter action is allowed to depend also on the connection which is not a priori symmetric. This general treatment will allow us to address several open issues such as: the relation between metric-affine f(R) gravity and General Relativity (in vacuum as well as in the presence of matter), the implications of the dependence (or independence) of the matter action on the connections, the origin and role of torsion and the viability of the minimal-coupling principle.     

A new class of plane symmetric solution

A new class of static plane symmetric solution of Einstein field equation generated by a perfect fluid source is put forward. A special family of this new solution is investigated in detail. The constraints on the parameters by different energy conditions are studied. The classical stability of this solution is discussed. The junction conditions matching to Minkowski metric and Taub metric are analyzed respectively.     

Cosmological Models with Time Dependent G and Λ Coupling Scalars

A cosmological model in which the universe has its critical density and gravitational constants generalized as coupling scalars in Einstein's theory is considered. A general method of solving the field equations is given. An exact solution for matter distribution in cosmological models satisfying G=G₀(R/R₀)ⁿ is presented. Corresponding physical interpretations of the cosmological solutions are also discussed.     

Charged perfect fluid and scalar field coupled to gravity

Charged perfect fluid with vanishing Lorentz force and massless scalar field is studied for the case of stationary cylindrically symmetric spacetime. The scalar field can depend both on radial and longitudinal coordinates. Solutions are found and classified according to scalar field gradient and magnetic field relationship. Their physical and geometrical properties are examined and discussion of particular cases, directly generalizing Gödel-type spacetimes, is presented.     

Gravitating discs around a Schwarzschild black hole II

A sequence of exact spacetimes is obtained describing the fields of a Schwarzschild black hole surrounded by stable static axisymmetric thin discs having their inner rim at the least possible radius. In the previous paper we only required stability with respect to perturbations in the disc plane, while it turns out that for discs with relative mass >0.23 the perturbations in perpendicular direction are more dangerous. The discs of the resulting sequence have their inner rims just on, or very close to, circular geodesics marginally stable with respect to either of the perturbations. Redshift from static and Keplerian observers in the disc is computed. The inverted first Morgan-Morgan counter-rotating disc, used in superpositions, has a number of satisfactory physical properties, but it has turned out to have a curvature singularity at the inner rim. However, this is only a consequence of a too steep radial start of density, not present in (inverted) “higher” Morgan-Morgan solutions. Dedicated to Professor Jiří Bičák on the occasion of his 60th birthday.     

Bianchi Type-III Cosmological Models with Gravitational Constant G and the Cosmological Constant ∧

Einstein field equations with variable gravitational and cosmological constants are considered in the presence of perfect fluid for the Bianchi type-Ⅲ universe by assuming conservation law for the energy-momentum tensor. Exact solutions of the field equations are obtained by using the scalar of expansion proportional to the shear scalar θ∝σ, which leads to a relation between metric potential B = Cn, where n is a constant. The corresponding physical interpretation of the cosmological solutions are also discussed.     

Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method

In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.     

New Symmetries for a Model of Fast Diffusion

The new symmetries for a mathematical model of fast diffusion are determined. A new system method is given to search for new symmetries of differential equations written in a conserved form, several new symmetry generators and exact solutions are presented.     

Quasi-Local Energy Distribution of the Modified Reissner--Nordström Black Hole

We discuss the energy distribution of the modified Reissner-Nordstrom black hole. It is suggested that the quasilocal energy distribution from Einstein energy-momentum complex should be appropriate. For the Reissner- NordstrOm spacetime, it can be seen that the energies of the outer and inner regions are exactly equivalent.     

Exact self-gravitating N-body motion in the CGHS model

In the asymptotically flat two-dimensional dilaton gravity, we present an N-body particle action which has a dilaton coupled mass term for the exact solubility. This gives nonperturbative exact solutions for the N-body self-gravitating system, so the infalling particles form a black hole and their trajectories are exactly described. In our two-dimensional case, the critical mass for the formation of black holes does not exist, so even a single particle forms a black hole. The infalling particles give additional time-like singularities in addition to the space-like black hole singularity. However, the latter singularities can be properly cloaked by the future horizons within some conditions.     

The improved sub-ODE method for a generalized KdV-mKdV equation with nonlinear terms of any order

In this Letter, Li and Wang's sub-ODE method [X.Z. Li, M.L. Wang, Phys. Lett. A 361 (2007) 115] is improved and applied to the generalized KdV-mKdV equation with nonlinear terms of any order. As a result, more travelling wave solutions are obtained including not only all the known solutions found by Li and Wang but also other formal solutions. This improved sub-ODE method can be used for solving other nonlinear partial differential equations with nonlinear terms of any order in mathematical physics.     

Exact solutions for shells collapsing towards a pre-existing black hole

The gravitational collapse of a star is an important issue both for general relativity and astrophysics, which is related to the well-known “frozen star” paradox. This paradox has been discussed intensively and seems to have been solved in the comoving-like coordinates. However, to a real astrophysical observer within a finite time, this problem should be discussed in the point of view of the distant rest-observer, which is the main purpose of this Letter. Following the seminal work of Oppenheimer and Snyder (1939), we present the exact solution for one or two dust shells collapsing towards a pre-existing black hole. We find that the metric of the inner region of the shell is time-dependent and the clock inside the shell becomes slower as the shell collapses towards the pre-existing black hole. This means the inner region of the shell is influenced by the property of the shell, which is contrary to the result in Newtonian theory. It does not contradict the Birkhoff's theorem, since in our case we cannot arbitrarily select the clock inside the shell in order to ensure the continuity of the metric. This result in principle may be tested experimentally if a beam of light travels across the shell, which will take a longer time than without the shell. It can be considered as the generalized Shapiro effect, because this effect is due to the mass outside, but not inside as the case of the standard Shapiro effect. We also found that in real astrophysical settings matter can indeed cross a black hole's horizon according to the clock of an external observer and will not accumulate around the event horizon of a black hole, i.e., no “frozen star” is formed for an external observer as matter falls towards a black hole. Therefore, we predict that only gravitational wave radiation can be produced in the final stage of the merging process of two coalescing black holes. Our results also indicate that for the clock of an external observer, matter, after crossing the event horizon, will never arrive at the “singularity” (i.e. the exact center of the black hole), i.e., for all black holes with finite lifetimes their masses are distributed within their event horizons, rather than concentrated at their centers. We also present a worked-out example of the Hawking's area theorem.     

A new auxiliary equation method and its application to the Sharma-Tasso-Olver model

A new auxiliary equation method, constructed by a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term, is first proposed for exploring more exact solutions to nonlinear evolution equations. Being concise and straightforward, the method, with the aid of symbolic computation, is applied to the Sharma-Tasso-Olver model, and some new exact solitary wave solutions are obtained. The approach is also applicable to searches for exact solutions of other nonlinear evolution equations.