一类非线性波动方程的势对称分类

先给出了含有一个任意函数的线性波动方程的古典和势对称的完全分类.然后,在此基础上给出了含有两个任意函数的一类非线性波动方程的两种情形势对称分类,得到了该方程的新势对称.在处理对称群分类问题的难点-求解确定方程组时我们提出了微分形式吴方法算法,克服了以往难于处理的困难.在整个计算过程中反复使用了吴方法,吴方法起到了关键的作用.     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

一般非齐次非线性扩散方程的等价变换和高维不变子空间

结合压力变换和不变子空间方法中的等价变换,给出了一般非齐次非线性扩散方程的等价方程,并给出了等价方程的高维不变子空间.由此构造了一般非齐次非线性扩散方程的广义分离变量解,并给出了几个例子解释这个过程.     

非齐次非线性扩散方程的高维不变子空间和等价变换

利用与不变子空间方法相关的等价变换和变换v=enu给出了非齐次非线性扩散方程的等价方程,并得到了等价方程的高维不变子空间.最后给出一些例子构造了非齐次非线性扩散方程的广义泛函分离变量解.     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

一类变系数非线性Schrodinger方程的对称群分类和容许变换

从对称群和容许变换的角度讨论一类变系数非线性Schrodinger方程,给出所考察方程的非平凡点对称群。     

一类变系数非线性Schrǒdinger方程的对称群分类和容许变换(英文)

从对称群和容许变换的角度讨论一类变系数非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程,给出所考察方程的非平凡点对称群     

(3+1)-维耦合高阶非线性Schrödinger方程的光学多畸形波

本文研究带有高阶项、时间色散项和非线性系数项的复杂（3+1）-维高阶耦合非线性Schrödinger（3DHCNLSE）方程的精确解. 首先,利用相似变换将非自治的方程转化为自治的耦合Hirota 方程; 其次,采用Darboux 变换方法得到耦合Hirota 方程带有任意常数的有理解; 最后,给出变系数3DHCNLSE方程带有任意常数的1 阶和2 阶多畸形波解. 本文获得的（3+1）-维（3D）多畸形波解可以用来描述深海动力学波和非线性光学纤维中出现的一些物理现象.     

(3+1)-维耦合高阶非线性Schrdinger方程的光学多畸形波

本文研究带有高阶项、时间色散项和非线性系数项的复杂(3+1)-维高阶耦合非线性Schrdinger(3DHCNLSE)方程的精确解.首先,利用相似变换将非自治的方程转化为自治的耦合Hirota方程;其次,采用Darboux变换方法得到耦合Hirota方程带有任意常数的有理解;最后,给出变系数3DHCNLSE方程带有任意常数的1阶和2阶多畸形波解.本文获得的(3+1)-维(3D)多畸形波解可以用来描述深海动力学波和非线性光学纤维中出现的一些物理现象.     

带阻尼项的二阶非线性中立型Emden-Fowler微分方程的振动准则

该文研究一类带有更广泛而不失物理意义阻尼项的二阶非线性中立型Emden-Fowler时滞微分方程的振动性.利用指数函数变换、Riccati变换和不等式技巧,获得了该类方程几个新的振动定理,推广、改进和丰富了已有文献中的研究结果,并逐一给出例子说明了相应定理的实用效果.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ans?tze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Given a class \(\mathcal{F(\theta)}\) of differential equations with arbitrary element θ, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member \(f\in\mathcal{F(\theta)}\) the structure of its Lie symmetry group G _f , conditional symmetry Q _f and conservation law \(\mathop {\rm CL}\nolimits _{f}\) under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u _tt =(g(x)H(u)u _x ) _x +h(x)K(u)u _x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.     

Equivalence Groups for First-Order Balance Equations and Applications to Electromagnetism

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.     

Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach

Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears to be a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way for the construction of exact solutions of mKdV-like equations using equivalence transformations is described.     

Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations

A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted infinitesimal equivalence transformations into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.     

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. é. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 119–142, 2005.     

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schrödinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schrödinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1+2)-dimensional cubic Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.     

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.