2N+1阶KdV型方程的孤波解

获得了２Ｎ＋１阶ＫｄＶ型方程的显式精确孤波解．作为特例，讨论了高阶广义ＫｄＶ型方程、高阶广义ＭＫｄＶ型方程和高阶广义Ｓｃｈａｍｅｌ的ＭＫｄＶ型方程．还研究了２Ｎ＋１阶ＫＰ型方程 关键词：     

Poincar-Chetaev变量下变质量非完整动力学系统的运动方程

研究Poincaré-Chetaev变量下,变质量非线性非完整力学系统的运动方程.首先,由变质量力学系统的D’Alembert-Lagrange原理导出Chaplygin型方程、Nielsen型方程及Appell型方程.其次,研究Chaplygin方程与Appell方程的等价性问题.最后,举例说明新结果的应用.     

Poincaré—Chetaev变量下变质量非完整动力学系统的运动方程

研究Poincaré-Chetaev变量下,变质量非线性非完整力学系统的运动方程.首先,由变质量力学系统的D'Alembert-Lagrange原理导出Chaplygin型方程、Nielsen型方程及Appell型方程．其次,研究Chaplygin方程与Appell方程的等价性问题.最后,举例说明新结果的应用.     

求sine-Gordon 型方程孤波解的一种统一方法

借助于一个辅助常微分方程的解，提出了一种求sine-Gordon 型方程孤波解的统一方法，并用该法简洁地求得了三个著名的sine-Gordon 型方程，即单sine-Gordon 方程、双sine-Gordon 方程和mKdV-sine-Gordon方程的精确孤波解。     

sine-Gordon型方程的无穷序列新精确解

为了获得sine-Gordon型方程的无穷序列精确解,给出三角函数型辅助方程和双曲函数型辅助方程及其Bäcklund变换和解的非线性叠加公式,借助符号计算系统Mathematica,构造了sine-Gordon方程、mKdV-sine-Gordon方程、(n+1)维双sine-Gordon方程和sinh-Gordon方程的无穷序列新精确解.其中包括无穷序列三角函数解、无穷序列双曲函数解、无穷序列Jacobi椭圆函数解和无穷序列复合型解. 关键词： sine-Gordon型方程 解的非线性叠加公式 辅助方程 无穷序列精确解     

双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解

在辅助方程法的基础上利用两种函数变换和一种双曲函数型辅助方程，通过符号计算系统Mathematica构造了在力学当中一个重要的模型，有5次强非线性项的波方程的新三角函数型和双曲函数型精确孤波解.这种方法寻找其他具5次强非线性项的非线性发展方程的新精确解方面具有普遍意义. 关键词： 双曲函数型辅助方程 函数变换 具5次强非线性项的波方程 精确孤波解     

统一的对流扩散型可压缩流体力学方程与解法

流体力学的动量方程、能量方程、湍动能方程和耗散方程都具有对流扩散方程的形式,但连续方程却不是对流扩散型的。对于可压缩问题,本文通过合理的数学推导,不作任何近似、假定与简化,得到一个全新的连续方程形式．该连续方程以压力为未知变量,并具有对流扩散型形式,使得所有的流体动力学方程组都具有完全统一的方程形式,给出了这种三维对流扩散方程组的有限精确差分计算格式。对流体力学的进一步发展具有一定意义．     

在非线性三波耦合的不稳定饱和过程中出现的Lorenz型混沌

我们证明了在等离子体中，三波耦合方程同构于Ｌｏｒｅｎｚ型方程，预言了在此过程中会出现Ｌｏｒｅｎｚ型混沌     

杨诺赛波动方程作为原子核或元粒子理论的数学形式

本文指出杨诺赛波动方程作为电子的个体理论的困难;指出这一类型波动方程可以作为原子核系综理论的数学形式,也可以作为元粒子系综理论的数学形式。本文并讨论了这一类型波动方程的第二级量子化问题。     

组合KdV方程的双扭结孤立波及其稳定性研究

利用扩展的双曲函数法得到了combined KdV-mKdV (cKdV)方程的几类精确解,其中一类为具有扭结—反扭结状结构的双扭结单孤子解.在不同的极限情况下,该解分别退化为cKdV方程的扭结状或钟状孤波解.理论分析表明,cKdV方程既有传播型孤立波解,也有非传播型孤立波解.文中对双扭结型孤立波解的稳定性进行了数值研究,结果表明,cKdV方程既存在稳定的双扭结型孤立波,也存在不稳定的双扭结型孤立波. 关键词： cKdV方程 双扭结单孤子 稳定性     

Approximate radiative solutions of the Yang-Mills field equations

In this paper we look for the asymptotic radiative solutions of the Yang-Mills field equations. Considering the potential of the Yang-Mills field as a connection in a principal fibre bundle gives us a fully covariant formalism similar to the formalism of the General Relativity. Then we apply directly the results obtained by Mme Choquet-Bruhat for the gravitational field by means of the W.K.B. method. After deriving the equations for the asymptotic waves and interpreting the zero-order conditions as the initial conditions, we consider some known trivial solutions of the Yang-Mills field equations as the background field and construct the asymptotic waves explicitly. All the solutions considered turn out to be of the electromagnetic type, with some extra restrictions of the algebraic type.     

ON THE ADM EQUATIONS FOR GENERAL RELATIVITY

The Arnowitt–Deser–Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in . An example of this construction is given in the case of a closed Friedmann–Lemaitre–Robertson–Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.     

Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik–Zamolodchikov (KZ) equations. In the case of a principal series module, we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions.For the spin representation of the affine Hecke algebra of type C, the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-\({\frac{1}{2}}\) XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter’s 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik–Zamolodchikov–Bernard (KZB) equations.     

Nonlinear von neumann equations for quantum dissipative systems

For pure states nonlinear Schrödinger equations, the so-called Schrödinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed.Supported by Deutsche ForschungsgemeinschaftSupported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project no. 3569     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

Abstract

In this paper we employ a “direct method” to construct rank-k solutions, expressible in Riemann invariants, to hyperbolic system of first order quasilinear di!erential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions and applying the conditional symmetry reduction technique to the initial equations. We discuss in detail the necessary and su"cient conditions for existence of these type of solutions. We demonstrate our approach through several examples of hydrodynamic type systems; new classes of solutions are obtained in a closed form.     

The Padé interpolation method applied to <Emphasis Type="Italic">q</Emphasis>-Painlevé equations II (differential grid version)

Recently, we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type \(E_7^{(1)}\), \(E_6^{(1)}\), \(D_5^{(1)}\), \(A_4^{(1)}\) and \((A_2+A_1)^{(1)}\). By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding q-Painlevé equations. It is natural that the q-Painlevé equations were derived by the interpolation method of q-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the q-Painlevé equation of type \(D_5^{(1)}\) by Ikawa. In this paper, we continue the above study and apply the Padé approximation method to the q-Painlevé equations of type \(E_6^{(1)}\), \(D_5^{(1)}\), \(A_4^{(1)}\) and \((A_2+A_1)^{(1)}\). Moreover, determinant formulae of the special solutions for q-Painlevé equation of type \(E_6^{(1)}\) are given in terms of the terminating q-Appell Lauricella function.     

On superembedding approach to multiple D-brane system. D0 story

We develop superembedding approach to multiple D-particle (D0-brane) system. In flat target D=10$D=10$ type IIA superspace this produces the supersymmetric and Lorentz covariant version of the Matrix model equations. The equations following from our superembedding approach to multiple D0 in curved type IIA superspace shows the Myers ‘dielectric brane effect’, i.e. interaction with higher form gauge fields which do not interact with a single D0-brane.     

Integration in the GHP Formalism II: An Operator Approach for Spacetimes with Killing Vectors, with Applications to Twisting Type N Spaces

Held has proposed a coordinate- and gauge-free integration procedure within the ghp formalism built around four functionally independent zero-weighted scalars constructed from the spin coefficients and the Riemann tensor components. Unfortunately, a spacetime with Killing vectors (and hence cyclic coordinates in the metric, and in all quantities constructed from the metric) may be unable to supply the full quota of four scalars of this type. However, for such a spacetime additional scalars may be supplied by the components of the Killing vectors. As an illustration we investigate the vacuum type N spaces admitting a Killing vector and a homothetic Killing vector. In a direct manner, we reduce the problem to a pair of ordinary differential operator master equations, making use of a new zero-weighted ghp operator. In two different ways, we show how these master equations can be reduced to one real third-order operator differential equation for a complex function of a real variable—but still with the freedom to choose explicitly our fourth coordinate. It is then easy to see there is a whole class of coordinate choices where the problem reduces essentially to one real third-order differential equation for a real function of a real variable. It is also outlined how the various other differential equations, which have been derived previously in work on this problem, can be deduced from our master equations.