A Novel Dynamics Equation of Elastic Rotation Shaft Possessing Twin Side Based on the Theory of Relativity

A novel dynamics equation of elastic rotation shaft possessing twin side based on the theory of relativity is built in this paper. The eql~tion is established in different coordinate systems, which can provide the foundation theoretically and methods for the similarity engineering and the similarity calibration of instruments used for measuring,observing, and controlling.     

在伽利略坐标变换下的二端面弹性转轴相似动力学方程

在非相对论性原理下建立了二端面弹性转轴的相似动力学方程.使弹性转轴的动力学方程与地面坐标下的动力学方程相似,为相似工程学和计量、测控仪器的相似标定提供理论依据及方法 关键词： 二端面弹性转轴 相似模拟 坐标变换 相似动力学方程 标定     

CONDITIONAL SIMILARITY REDUCTION APPROACH: JIMBO－MIWA EQUATION

The direct method developed by Clarkson and Kruskal (1989 J. Math. Phys. 30 2201) for finding the symmetry reductions of a nonlinear system is extended to find the conditional similarity solutions. Using the method of the Jimbo－Miwa (JM) equation, we find that three well-known (2+1)-dimensional models－the asymmetric Nizhnik--Novikov－Veselov equation, the breaking soliton equation and the Kadomtsev-Petviashvili equation－can all be obtained as the conditional similarity reductions of the JM equation.     

New Similarity Reduction Solutions for the(2+1)-Dimensional Nizhnik-Novikov-Veselov Equation

In this paper,some new formal similarity reduction solutions for the(2+1)-dimensional Nizhnik-Novikov-Veselov equation are derived.Firstly,we derive the similarity reduction of the NNV equation with the optimal system of the admitted one-dimensional subalgebras.Secondly,by analyzing the reduced equation,three types of similarity solutions are derived,such as multi-soliton like solutions,variable separations solutions,and KdV type solutions.     

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.     

Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.     

Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Bosonization approach is applied in solving the most general N=1 supersymmetric Korteweg de-Vries equation with an arbitrary parameter a (sKdV-a) equation. By introducing some fermionic parameters in the expansion of the superfield, the sKdV-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the sKdV-a equation are discussed which was not considered integrable previously.     

Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations

In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.     

Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations

In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.     

Similarity Reductions of Nonisospectral KP Equation by a Direct Method

On bases of the direct method developed by Clarkson and Kruskal [J. Math. Phys. 27 （1989） 2201], the （2＋1）-dimensional nonisospectral Kadomtsev-Petviashvili （KP） equation has been reduced to three types of （1 ＋1）- dimensional partial differential equations. We focus on solving the third type of reduction and dividing them into three subcases, from which we obtain rich solutions including some arbitrary functions.     

Symmetries and Exact Solutions of the Breaking Soliton Equation

With the aid of the classical Lie group method and nonclassical Lie group method, we derive the classical Lie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS) equation. Using the symmetries, we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation. Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.     

Conditional Similarity Reductions of Jimbo-Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

An extension of the direct method and similarity reductions of a generalized Burgers equation with an arbitrary derivative function

In this paper,we extend the well-known direct method proposed by Clarkson and Kruskal for finding similarity reductions of partial differential equations.It follows that some new similarity reductions of the generalized Burgers equation,such as travelling wave reduction,logarithmic reduction,power reduction,rational fractional reduction,etc,are derived,in which some of these cannot be obtained solely by using the direct method.The similarity reductions obtained are interpreted by the nonclassical symmetry Lie group.     

Similarity Reduction and Integrability for the Nonlinear Wave Equations from EPM Model

Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plastic-microstructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou.As a result,the nonlinear wave equation is not integrable.     

Analysis of the wave functions for accelerating Kerr-Newman metric

As is well-known, it is very difficult to solve wave equations in curved space-time. In this paper,we find that wave equations describing massless fields of the spins s≤2 in accelerating KerrNewman black holes can be written as a compact master equation. The master equation can be separated to radial and angular equations, and both can be transformed to Heun's equation,which shows that there are analytic solutions for all the wave equations of massless spin fields.The results not only demonstrate that it is possible to study the similarity between waves of gravitational and other massless spin fields, but also it can deal with other astrophysical applications, such as quasinormal modes, scattering, stability, etc. In addition, we also derive approximate solutions of the radial equation.     

Covariant Relativistic Statistical Mechanics of Many Particles

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of event linking (concatenations) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view.     

Approximate Homotopy Symmetry Reduction Method: Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.     

广义Zakharov-Kuznetsov方程的对称及精确解

利用改进的直接方法给出了一类广义Zakharov-Kuznetsov方程ut auux bu2ux cuxxx duxyy=0新显式解与旧显式解之间的关系,并且得到了该方程的对称.这些对称推广了已有文献中应用Steinberg s相似方法获得的结果.利用广义Zakharov-Kuznetsov方程新旧显式解之间的关系,本文在已有显式解的基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.