Axially symmetric solutions of einstein equations

Two exact axially symmetric solutions of the gravitational field equations, which depend on a number of arbitrary real constants, are derived.     

Exact traveling wave solutions to nonlinear diffusion and wave equations

By the introduction of some ansatz equations, we have obtained several new classes of traveling (solitary) wave solutions to the nonlinear diffusion equation $$f_1 (u)u_t + f_2 (u)u_x + f_3 (u)u_{xx} + f_4 (u)u_x^2 = f_5 (u)$$ and the nonlinear wave equation $$f_1 (u)u_u + f_2 (u)u_t + f_3 (u)u_{xx} + f_4 (u)u_x + f_5 (u)u_x^2 + \cdots = f_6 (u)$$ Some applications of these solutions are discussed.     

一类非线性波动方程的精确行波解

利用改进的G’/G展开方法,借助于计算机代数系统Mathematica成功获得了一大类非线性波动方程一系列新的含有多个参数的精确行波解.这些解包括孤立波解、双曲函数解、三角函数解.     

Exact travelling kink-like wave solutions to some nonlinear equations

New explicit and exact travelling wave solutions to some nonlinear equations are obtained. For some chosen parameters in the nonlinear equations, constants in the explicit solutions are given numerically.     

Exact travelling wave solutions to some fifth order dispersive nonlinear wave equations

Exact travelling wave solutions to some nonlinear equations of fifth order derivatives are derived by using some accurate ansatz methods.     

Soliton solutions to the einstein equations in five dimensions

We present a new class of solutions in odd dimensions to Einstein's equations containing either a positive or a negative cosmological constant. These solutions resemble the even-dimensional Eguchi-Hanson-(anti)-de Sitter [(A)dS] metrics, with the added feature of having Lorentzian signatures. They provide an affirmative answer to the open question as to whether or not there exist solutions with a negative cosmological constant that asymptotically approach AdS5/gamma but have less energy than AdS5/gamma. We present evidence that these solutions are the lowest-energy states within their asymptotic class.     

Exact and explicit solitary wave solutions to some nonlinear equations

Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative Φ⁴-model equation, the generalized Fisher equation, and the elastic-medium wave equation.     

Exact solutions of the Altarelli-Parisi equations

Evolution equations for single species parton distributions are solved analytically. For infrared-convergent theories the solutions can be expressed in a closed form as convolutions over Bessel functions. For infrared-divergent theories, such as QCD, the exact solution involves an infinite number of convolutions. The accuracy of various approximations to this solution is estimated.     

Exact solitary wave solutions to a class of generalized odd-order KdV equations

Using a proper ansatz, we have obtained a series of exact solitary wave solutions to a class of generalized odd-order KdV equations.     

Exact periodic steady solutions for nonlinear wave equations: A new approach

In many practical physical problems, the nonlinear wave equations of interest are nonintegrable. In some cases they may be close to an integrable equation, and in this case perturbation techniques are available. However, even in these cases and certainly in general, it is useful to construct exact explicit solutions. Here we introduce a new method for finding exact and explicit periodic travelling wave solutions, from which solitary wave solutions can be extracted if they exist.     

Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method

Based on the Exp-function method, exact solutions for some nonlinear evolution equations are obtained. The KdV equation, Burgers' equation and the combined KdV–mKdV equation are chosen to illustrate the effectiveness of the method.     

Polynomial solutions for coupledU(1)-gauge einstein equations

We point out numerical solutions in polynomial form to the field equations derived by Garfinkle and Laguna forU(1)-gauge cosmic strings. With these solutions we evaluate the second-order term for the deficit angle produced by the string.     

On some classes of exact solutions of the einstein equations. II

Certain types of spaces are investigated in a wave coordinate system in this paper. The superposition of a monochromaticity condition, as well as certain others, permitted integrating the Einstein equations completely. The solutions found can be interpreted as wave solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 56–59, May, 1978.     

Exact vacuum solutions of the Einstein equations

The vacuum Einstein equations for the Kerr-Schild metric are investigated. It is shown that they admit representation in the form of the double four-dimensional curl of the perturbation of the Euclidean metric, whereupon it is possible to note certain general directions in which to seek exact solutions. For spaces with a normal isotropic geodesic congruence the GR equations are rewritten with the application of a dyadic splitting of the metric; cases of two-dimensional subspaces of constant curvature are discussed. The investigation is illustrated by the exact nonstationary algebraic type N and anti-Schwarzschild solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 23–26, September, 1982.     

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 Einstein-conformal scalar equations

The massless scalar field which satisfies a conformally invariant equation is in some respects more interesting than the ordinary one. Unfortunately, few, if any, exact solutions of Einstein's equations for a conformal scalar stress-energy have appeared previously. Here we present a theorem by means of which one can generate two Einstein-conformal scalar solutions from a single Einstein-ordinary scalar solution (of which many are known). As an example we show how to obtain Weyl-like solutions with a conformal scalar field. We obtain and analyze in some detail two families of spherically symmetric static Einstein-conformal scalar solutions. We also exhibit a family of static spherically symmetric Einstein-Maxwell-conformal scalar solutions (parametrized by both electric and scalar charge), which have black-hole geometries but are not genuine black holes. Finally, we present all the Robertson-Walker cosmological models which contain both incoherent radiation and a homogeneous conformal scalar field. One class of these represents open universes which bounce and never pass through a singular state; they circumvent the “singularity theorems” by violating the energy condition.     

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.