Exact self-consistent plane-symmetric solutions of the equations of two interacting fields (spinor field and scalar field)

A spinor field interacting with a zero-mass neutral scalar field is considered for the case of the simplest type of direct interaction, where the interaction Lagrangian has the formL _int =1/2 ϕ_αϕ_,α F(S) whereF(S) is an arbitrary function of the spinor field invariantS=ψψ. Exact solutions of the corresponding systems of equations that take into account the natural gravitational field in a plane-symmetric metric are obtained. It is proved that the initial system of equations has regular localized soliton-type solutions only if the energy density of the zero-mass scalar field is negative as it “disengages” from interaction with the spinor field. In two-dimensional space-time the system of field equations we are studying describes the configuration of fields with constant energy densityT ₀₀ , i.e., no soliton-like solutions exist in this case. Russian People’s Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 69–75, July, 1998.     

Exact, self-consistent, plane-symmetric solutions of the equation for a spinor field with a nonlinear term that depends on the invariantS 2+P 2

Exact solutions of the Einstein-Dirac equations are found and it is shown that when gravitation is excluded, no solution of the soliton type exists. Russian University of People's Friendship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 42, No. 2, pp. 55–61, February, 1999.     

非线性Schroedinger方程组的精确解组

运用Maple语言程序，在没有假设的条件下，得到了具有耦合特性的非线性Schrfidinger方程组的行波精确解组及其约束条件方程，它们的表达式涵盖了所有的耦合解组与非耦合解组，具有任意性。耦合解组的算例函数及其特性分析，解释了α螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应，揭示了增加、稳定和控制蛋白质活性和功能的方向。文章的研究方法，为求解耦合的非线性微分方程组的行波精确解组探索了蹊径。     

Exact solutions of some nonlinear equations

We obtain exact solutions of three nonlinear diffusive equations and of the KdV-Burger equation by making an ansatz for the solution in each case.     

两类非线性方程的精确解

利用行波约化方法，并借助于一维立方非线性Klein-Gordon方程的精确解，求出了（1+1）维Zakharov方程组、变系数Korteweg-de Vries方程的一些精确解- 关键词： 行波约化方法 一维立方非线性Klein-Gordon方程 （1+1）维Zakharov方程组 变系数Korteweg-de Vries方程     

Cosmological solutions foe a spinor field in the general theory of relativity

We consider models of the universe containing linear and nonlinear spinor matter. It is assumed that the linear spinor matter is described by the generally covariant Dirac equation, and the nonlinear by the generally covariant Ivanenko-Heisenberg equation.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 68–71, December, 1973.     

Self-similar solutions of spinor field equations in de sitter space

Two types of exact self-similar solutions of spinor field equations in De Sitter space are obtained. A condition is found under which the solutions vanish at the initial moment of time.IBRAÉ, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 26–30, January, 1993.     

Exact solutions to Einstein field equations

In a recent paper Reboucas and d'Olival obtain an ordinary differential equation for a Bianchi type II metric with a rotating timelike congruence of geodesics, and obtain a particular solution of the differential equation. This paper completely integrates the differential equation.     

Exact solutions of the Poincaré gauge theory from its linearized field equations

It is shown that a class of linearized solutions of the Poincaré gauge theory also solves the exact field equations of the same theory. Utilizing this property, some new solutions are given.Alexander von Humboldt Fellow.     

Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.     

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term dependent on the invariant P2

Exact self-consistent plane-symmetric solutions of the spinor-field equation with zero mass parameter and a nonlinear term that is an arbitrary function of the invariant , are obtained in gravitation theory. An equation with power-law nonlinearity in which the nonlinear term in the spinor-field Lagrangian has the form L_N=λP²ⁿ, where λ is the nonlinearity parameter and n=const, is investigated in detail. It is shown that λ=−Λ²<0, n>1, the original system of Einstein and nonlinear spinor-field equations has regular solutions with a localized spinor-field energy density. Here the soliton-like configuration of the fields possesses a negative energy. Exact solutions are also obtained for the above spinor-field equation in flat spacetime, and it is demonstrated that there are no soliton-like solutions in that case. Thus it is established that the proper gravitational field plays a decisive, controlling role in the formation of soliton-type solutions of the above nonlinear spinor-field equation. Russian International Friendship University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–53, July, 1997.     

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得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

Classical theory of the interaction between a spinor field and the gravitational field: First-order field equations

The classical theory of the interaction of a neutral spin-1/2 Dirac field and the gravitational field is studied. For the purely gravitational part of the Lagrangian, written in terms of a vierbein and the local connection coefficient _ab , (regarded as independent field variables), the usual first-order form is adopted. For the Dirac part, however, a different choice is made, in which the covariant derivative of is built with the aid of the vierbein instead of with _ab . This still yields a first-order formalism, but one in which _ab is related to the vierbein in the same way as it would be in the absence of. This ensures that the global connection remains symmetric in andv in the presence of. The way in which the vierbein field equation leads to a familiar Einstein equation with a symmetric and conserved stress tensor on its right side is also analyzed.     

非线性不可压缩霍尔磁流体方程组的精确解

得到了不可压缩理想霍尔磁流体方程组的平面波精确解,这些平面波是斜传播的左旋圆极化波或右旋圆极化波,并且涨落速度和涨落磁场通过波数联系起来。讨论了这些平面波解的叠加性质。当波的传播方向平行时,任意两支圆极化平面波都可以叠加;而当波的传播方向不平行时,只有极化方向相同,波数相同,且各自的旋度与自身都是同方向(或都是反方向)的两支圆极化平面波才可以叠加。     

Exact static solutions of nonlinear spinor-field equations in a Hedel universe

Exact static solutions of spinor-field equations with nonlinear terms that are arbitrary functions of the invariant S=ψψ are obtained in the external gravitational field of a Hedel universe. The specific type of nonlinear Lagrangian that produces regular and localized distributions of spinor-field energy density is discussed. Exact solutions of the original equations are also obtained in plane spacetime. Here it is shown that irrespective of the form of the nonlinear Lagrangian, the energy density of the spinor field is constant, i.e., there is no localization. This means that the external gravitational field of a Hedel universe has a definite role in forming soliton-like configurations of the nonlinear spinor field. Russian University of International Amity. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 111–116, July, 1996.     

非线性Schrödinger方程组的精确解组

运用Maple语言程序,在没有假设的条件下,得到了具有耦合特性的非线性Schr(o)dinger方程组的行波精确解组及其约束条件方程,它们的表达式涵盖了所有的耦合解组与非耦合解组,具有任意性.耦合解组的算例函数及其特性分析,解释了a螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应,揭示了增加、稳定和控制蛋白质活性和功能的方向.文章的研究方法,为求解耦合的非线性微分方程组的行波精确解组探索了蹊径.     

Republication of: Exact solutions of the field equations of the general theory of relativity

This is an English translation of a paper by Pascual Jordan, Jürgen Ehlers and Wolfgang Kundt, first published in 1960. The original paper was part 1 of a five-part series of articles containing the first summary of knowledge about exact solutions of Einstein’s equations found until then. (The other parts of the series will be printed as Golden Oldies in the future.) The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. It is accompanied by an editorial note written by G. F. R. Ellis, and by the biographies of the authors: P. Jordan (written by A. Krasiński) and W. Kundt (written by himself). The biography of J. Ehlers is contained elsewhere in the same issue of GRG, which is devoted to his memory. An editorial note to this paper and a biography can be found in this issue preceding this Golden Oldie and online via doi:. Original paper: Pascual Jordan, Jürgen Ehlers, Wolfgang Kundt, Strenge L?sungen der Feldgleichungen der Allgemeinen Relativit?tstheorie. Akademie der Wissenschaften und der Literatur, Abhandlungen der Mathematisch-naturwissenschaftliche Klasse Nr 2S (1960), pp. 21–105. Reprinted with the kind permission of the Academy of Sciences and Literature, Mainz, and of the authors: Jürgen Ehlers and Wolfgang Kundt. Translated by Anita Ehlers, Anita.Ehlers@t-online.de, and by Manfred Trümper, manfred@truemper.fr, with ample help from Wolfgang Kundt. P. Jordan (Deceased July 31, 1980) J. Ehlers (Deceased May 20, 2008)

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Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonlinear convective term

Attempts have been made to look for the exact solutions of certain types of nonlinear diffusion-reaction equations which involve not only the quadratic and quartic nonlinearities but also a time-dependent nonlinear convective flux term. In particular, the solitary wave solutions are found. Such equations arise in a variety of contexts in physical and biological problems.     

Exact solutions for nonlinear partial fractional differential equations

In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved(G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.