New Families of Nontravelling Wave Solutions to Two (3+1)-Dimensional Equations

In this paper, two (3+1)-dimensional equations are investigated. Auto-Bäcklund transformation is obtained, which is used with some ansatze to seek new types of exact solutions including some arbitrary functions. When these arbitrary functions are taken as some special functions, these solutions possess abundant structures. These solutions contain soliton-like solutions and rational solutions.     

关于分离变量法的一个注记

从B-cklund变换出发，利用Cole-Hopf变换，一些有重要物理意义的(2+1)维非线性演化方程(组)被简化为具有两个分别关于(x,t),(y,t)的任意函数的单个线性偏微分方程.通过对该方程解的研究，获得了原方程一些新的精确解.其中，一些近年来被广泛研究的由分离变量法所获得的解也被重新得到. 关键词： B-cklund变换 Cole-Hopf变换 分离变量法 精确解     

The Qiao--Liu Equation with Self-Consistent Sources and Its Solutions

The Qiao--Liu equation with self-consistent sources (QLESCS) and its Lax representation are derived. A reciprocal transformation for the QLESCS is given. By making use of the reciprocal transformation and the solutions of the mKdV equation with self-consistent sources (mKdVSCS), the solutions of the QLESCS are presented.     

Double duality and hidden gauge freedom in the poincaré gauge theory of gravitation

We investigate the double duality ansatz of the Poincaré gauge theory of gravitation. It is shown that many known exact solutions belong, as special cases, to larger families of solutions. These families of solutions include several arbitrary functions and can be generated by a transformation which is a Lorentzian rotation of the connection with fixed tetrad. Several new spherically symmetric and wavelike exact solution are presented.     

Bäcklund Transformations,Solitary Waves,Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

Some simple special Bäcklund transformation theorems are proposed and utilized to obtain exact solutions for the (2+1)-dimensional Euler equation. It is found that the (2+1)-dimensional Euler equation possesses abundant soliton or solitary wave structures, conoid periodic wave structures and the quasi-periodic Bessel wave structures on account of the arbitrary functions in its solutions. Moreover, all solutions of the arbitrary two dimensional nonlinear Poisson equation can be used to construct exact solutions of the (2+1)-dimensional Euler equation.     

Integrability and Solutions of the (2+1)-Dimensional Broer-Kaup Equation with Variable Coefficients

The integrability of the (2+1)-dimensional Broer-Kaup equation with variable coefficients (VCBK) is verified by finding a transformation mapping it to the usual (2+1)-dimensional Broer-Kaup equation (BK). Thus the solutions of the (2+1)-dimensional VCBK are obtained by making full use of the known solutions of the usual (2+1)-dimensional BK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.     

耦合Schrödinger系统的周期振荡折叠孤子

引入对称延拓和非线性变换, 将(G'/G)展开法扩展到研究(1+1)维非线性耦合Schrödinger系统, 构造出该系统的一些分离变量形式的精确解. 通过对解中的任意函数进行适当的设置, 获得了两类周期振荡折叠孤子. 关键词： 耦合Schrö dinger系统 G'/G)展开法')" href="#">(G'/G)展开法 精确解 周期振荡折叠孤子     

Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation（s）

In this paper, we extend the mapping transformation method through introducing variable coefficients. By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.     

Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation(s)

In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.     

Integrability and Solutions of the (2+1)-Dimensional Broer-Kaup Equation with Variable Coefficients

The integrability of the(2+1)-dimensional Broer-Kaup equation with variable coefficients(VCBK) is verified by finding a transformation mapping it to the usual(2+1)-dimensional Broer-Kaup equation(BK).Thus the solutions of the(2+1)-dimensional VCBK are obtained by making full use of the known solutions of the usual(2+1)dimensional BK.Two new integrable models are given by this transformation,their dromion-like solutions and rogue wave solutions are also obtained.Further,the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.     

(2+1)维非线性Burgers方程变量分离解和新型孤波结构

利用变量分离方法，获得了(2+1)维非线性Burgers方程的变量分离解.由于在Bcklund变换和变量分离步骤中引入了作为种子解的任意函数, 因而精确解中含有三个任意函数(其中一个为条件函数),适当地选择任意函数，可以获得多种形状的扭状孤波解、周期性孤子解和格子型孤波解. 关键词： 变量分离解 非线性波方程 （2+1）维     

Generalization of Brittingham's localized solutions to the wave equation

A family of localized solutions of Brittingham's type is constructed for different cylindric coordinates. We use method of incomplete separation of variables with zero separation constant and, then, the Bateman transformation, which enables us to obtain solutions in the form of relatively undistorted progressing waves containing two arbitrary functions, each of which depends on a specific phase function. Received 23 March 2001     

A transformation theory for camouflaging arbitrary heat sources

Camouflage devices have attracted intensive research interest for their significant applications. However, most camouflage devices are specifically designed according to target heat sources. Here, by applying the transformation thermotics approach, we develop a coordinate transformation, and design an unspecific camouflage device which can camouflage arbitrary heat sources into a circular one with an anisotropic shell. We verify the ability of our unspecific camouflage device with both steady and transient simulations. We also find the “apparent negative thermal conductivity” under certain conditions without violating the second law of thermodynamics. To ensure the feasibility, we further put forward the effective medium approximation for sample fabrication, and only two natural materials are required. Our results have relevance to the different applications of infrared misleading, uniform heating, and so on; they may also provide guidance to the research on other diffusive fields, such as magnetostatic and electrostatic fields.     

Lie symmetry analysis,new group invariant for the (3 + 1)-dimensional and variable coefficients for liquids with gas bubbles models

The explored solutions described some different solutions as, Lump soliton, a solitary wave and exponential solutions. These solutions are investigated through some new Lie infinitesimals for the (3 + 1) dimensional variable coefficients Kudryashov-Sinelshchikov (VCKS). We used the fourth prolongation to investigate fifteen cases of Lie vectors. In each case, there is an infinite number of possibilities of vectors due to the unknown arbitrary functions and the variable coefficients for the considered model. We selected one case and examined the commutative product between multi unknown Lie infinitesimals for the (3 + 1) dimensional (VCKS) equation and this complicated process resulted from some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through three stages of Lie symmetry reduction using the equivalence transformation, (VCKS) equation is reduced to solvable nonlinear ODEs using various combinations of optimal Lie vectors. By solving these ODEs, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for (VCKS). Some selected solutions are illustrated through three-dimensional plots.     

Bäcklund Transformation for Supersymmetric Self-Dual Theories for Semisimple Gauge Groups and a Hierarchy of Ai Solutions

We present a Bdcklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ^4|4N for an arbitrary semisimple gauge group. For the case of anA ₁ gauge algebra we integrate the transformation starting with a given solution and iterating the process we construct a hierarchy of explicit solutions.     

Generalized Jordan-Wigner transformations

We introduce a new spin-fermion mapping, for arbitrary spin S generating the SU(2) group algebra, that constitutes a natural generalization of the Jordan-Wigner transformation for S = 1/2. The mapping, valid for regular lattices in any spatial dimension d, serves to unravel hidden symmetries. We illustrate the power of the transformation by finding exact solutions to lattice models previously unsolved by standard techniques. We also show the existence of the Haldane gap in S = 1 bilinear nearest-neighbor Heisenberg spin chains and discuss the relevance of the mapping to models of strongly correlated electrons. Moreover, we present a general spin-anyon mapping for the case d < or = 2.     

KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.     

Solving Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Solving Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Backlund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x,t} and {y,t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Soliton theoretic framework for generating multimonopoles

We present a systematic method using Bäcklund transformation for generating SU(2) Yang-Mills-Higgs monopoles of arbitrary charge. The purely algebraic iteration formula for our Bäcklund transformation is derived. Our method is based on the equivalence of the axially and mirror-symmetric Bogomolny equations and the Ernst equation. The properties of the Ernst equation that are relevant for monopoles are also discussed. the application of the method is illustrated for the example of the one- and two-monopole solutions.