Variable Separation for (1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1+1)-dimensional nonlinear evolution equations with mixed partial derivatives. As an application, we classify equations u_xt=A(u,u_x)u_xxx+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.     

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation u_tt=Au,u_xu_xx+Bu,u_x,u_t which admits the derivative-dependent functional separable solutions DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches.     

完整力学系统的高阶运动微分方程

从质点系的牛顿动力学方程出发,引入系统的高阶速度能量,导出完整力学系统的高阶Lagrange方程、高阶Nielsen方程以及高阶Appell方程,并证明了完整系统三种形式的高阶运动微分方程是等价的.结果表明,完整系统高阶运动微分方程揭示了系统运动状态的改变与力的各阶变化率之间的联系,这是牛顿动力学方程以及传统分析力学方程不能直接反映的.因此,完整系统高阶运动微分方程是对牛顿动力学方程及传统Lagrange方程、Nielsen方程、Appell方程等二阶运动微分方程的进一步补充. 关键词： 高阶速度能量 高阶Lagrange方程 高阶 Nielsen方程 高阶Appell方程     

Covariant Field Equations in Supergravity

Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations.     

Plasmon-exciton self-induced transparency

The possibility of forming stable bound plasmon-polariton states in an extended metallic cylinder surrounded by a two-level medium has been investigated. The dynamics of plasmons is described in the hydrodynamic approximation. It has been shown that the equations of motion of charge-density bunches and the Bloch equations for the two-level medium are reduced in certain approximations to integrable equations for both transverse and longitudinal plasmons. In the former case, the initial system of equations after the application of the slow-envelope approximation is reduced to equations equivalent to the Maxwell-Bloch equations. In the latter case, the equations describe wave dynamics beyond the slow-envelope approximation. In the approximation of unidirectional wave propagation, the initial system of equations is reduced to equations related to the reduced Maxwell-Bloch equations. Soliton and breather-like solutions of the derived equations describe plasmon-exciton self-induced transparency.     

Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus

We realize affine Weyl group symmetries on the Schr?dinger equations for the quantum Painlevé equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schr?dinger equations for the quantum Painlevé equations. In other words, since the Schr?dinger equations for the quantum Painlevé equations are equivalent to the Knizhnik?CZamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik?CZamolodchikov equations.     

Fundamental Equations for Submanifolds

We notice that in the embedding of submanifolds, the fundamental equations are only the Gauss equations for the tangent vectors while the Weingarten equations for the normal equations can essentially be determined by them, and furthermore that the integrability condition of the Weingarten equations, the Ricci equations, are consitently satisfied under that of the Gauss equations. Therefore, the Weingarten and the Ricci equations do not describe essentially independent conditions for embedding. We demonstrate these facts by explicitly constructing the normal vectors from the tangent vectors.     

Conservation equations and equations of motion in the null formalism

The object of this paper is to relate three equations in the Newman-Penrose system of equations to the conservation laws and, hence, to the equations of motion. To do so, the corresponding result is first obtained using the Einstein equations in a null coordinate system. The Newman-Penrose equations are then analyzed. They are separated into hypersurface, propagation, supplementary, and conservation equations. When all field equations except the three conservation equations have been appropriately satisfied, the desired result follows.Research supported by the National Science Foundation under Grant GP-34641X.     

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

THE PROBLEM OF MOTION IN THE KALUZA THEORY

It is shown that the equations of motion for a charged massive particle are consequences of the field equations in Kaluza unification theory of gravitation and electromagnetism, i.e., the equations of motion for the particle can be deduced from Kaluza field equations, just as that in Einstein's theory of motion of general relativity the equations of motion for a massive particle are consequences of the Einstein equations. Furthermore, the Lorentz equations for a particle maving in the Maxwell electromagnetic field on the Minkowskian space-time can also be obtained from the Maxwell equations by means of the Kaluze mechanism of the Maxwell theory.     

ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.     

一类非完整奇异系统的Lie对称性与守恒量

利用微分方程在无限小变换群下的不变性,研究一类非完整奇异系统的Lie对称性.给出Lie对称性的确定方程、限制方程、附加限制方程和结构方程,并给出守恒量的形式 关键词： 奇异系统 非完整约束 Lie对称性 守恒量     

Kaluza理论的运动定律

本文指出,如同在广义相对论中粒子运动方程是场方程的推论一样,在引力场与电磁场的Kaluza统一理论中,粒子的运动方程也是场方程的一个推论,即带电粒子在引力场和电磁场中的运动方程可以从Kaluza统一理论中的场方程推导出来。本文进而在Minkowski时空的条件下,借助Maxwell理论的Kaluza形式,得到Maxwell方程也包含了带电粒子运动方程的结论。 关键词：     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

Einstein's Equations and Associated Linear Systems

The relationship between Einstein's field equations and classical higher spin field equations is investigated using two-component spinor valued differential forms. Linear systems of equations associated to both the vacuum and coupled gravitational matter field equations are constructed. The latter equations are shown to be the integrability conditions of the linear systems.     

Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

Octonionic Lorenz-like condition

In this study, the octonion algebra and its general properties are defined by the Cayley–Dickson’s multiplication rules for octonion units. The field equations, potential equations and Maxwell equations for electromagnetism are investigated with the octonionic equations and these equations can be compared with their vectorial representations. The potential and wave equations for fields with sources are also provided. By using Maxwell equations, a Lorenz-like condition is newly suggested for electromagnetism. The existing equations including the photon mass provide the most acknowledged Lorenz condition for the magnetic monopole and the source.     

Three-order pseudo-Hamilton canonical equations

Based on the three-order Lagrangian equations, Hamilton's function of acceleration $H^\ast$ and generalized acceleration momentum P_\alpha ^\ast are defined, and pseudo-Hamilton canonical equations corresponding to three-order Lagrangian equations are obtained. The equations are similar to Hamilton's canonical equations of analytical mechanics in form.     

EQUIVALENCE CLASSES OF THE SECOND ORDER ODEs WITH THE CONSTANT CARTAN INVARIANT

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.