广义拟线性双曲型方程的对称势及其守恒定律

基于Lie群方法,研究广义拟线性双曲型方程的对称势和不变解.为了得到显式的不变解,关注物理上有趣的有对称势的情况.然后,利用局部的Lagrange函数逼近,在3种物理上引起注意的情况下,得到该方程的守恒定律.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

(2+1)-维破裂孤子方程的群叶状方法和显式解

利用等变活动标架理论,研究(2+1)-维破裂孤子方程的群叶状方法和显式解.原方程的对称群的无穷维部分被用来产生整个解空间的叶状结构,于是分解系统就继承了对称群的有限维部分.求解的过程完全符号化和算法化.利用群叶状方法,破裂孤子方程的一些显式精确解被得到,这些解关于无穷维对称子群封闭.     

Landau-Lifshitz方程的群不变解和守恒律

利用经典李群方法得到了Landau-Lifshitz方程不变群的无穷小生成元,验证其对换位运算构成一个七维的李代数,得到了对应的群不变解,建立了Landau-Lifshit,z新解和旧解之间的关系.同时利用对称和共轭方程组求得了Landau-Lifshitz方程的守恒律.     

微分方程(组)对称向量的吴-微分特征列算法及其应用

给出（偏）微分方程（组）（PDEs）对称向量的吴-微分特征列集（消元）算法理论．把古典和非古典PDEs对称问量的计算问题统-在吴-微分特征列理论框架之下处理．给出了产生PDEs对称向量的无穷小方程和验证已知向量为PDES对称向量的机械化原理,理论上彻底克服了传统算法中的缺陷并为计算PDEs对称向量提供了一种新算法．用计算机代数系统mathematica编制了相应的软件包,具体实现了该算法．作为应用给出了Burgers方程的非古典对称向量的完整解答．     

矩阵方程AX—XB=C的显式解：——纪念导师郭仲衡教授

现有关于矩阵方程ＡＸ－ＸＢ＝Ｃ的显式解的几乎所有结论都是在Ａ与Ｂ无公共特征值的条件下获得的。本利用特征投影给出了方程在Ａ与Ｂ均对称或反对称的一般解的显式形式。我们所得到的结果不仅适用于任何特征值重数情形，而且可以用来讨论该方程的一般情形。     

矩阵方程AX－XB＝C的显式解──纪念导师郭仲衡教授

现有关于矩阵方程ＡＸ－ＸＢ＝Ｃ的显式解的几乎所有结论都是在Ａ与Ｂ无公共特征值的条件下获得的。本文利用特征投影给出了方程在Ａ与Ｂ均对称或反对称时一般解的显式形式。我们所得到的结果不仅适用于任何特征值重数情形，而且可以用来讨论该方程的一般情形。     

KdV-Burgers-Kuramoto方程的多种类型新精确解及其分析

首先采用Riccati方程的解的性质和试探函数法找到了 Riccati方程的八种类型的显式新精确解.其次运用李群分析法获得了 KdV-Burgers-Kuramoto方程的约化方程和群不变解.然后利用Riccati方程的八种类型的显式新精确解和广义Tanh函数法给出了约化方程的多种类型的显式新精确解.最后将Riccat...     

形变Boussinesq方程的对称群及其行波解

讨论具有方程组形式的形变Boussinesq方程的对称群及其行波解.通过研究方程组所允许的Lie对称群得到该方程组的解有行波解,并将方程组约化为非线性的常微分方程组,再利用广义-Tanh方法,得到形变Boussinesq方程的行波解.     

用Riccati方程的新解求Fitzhugh-Nagumo方程的新行波解

本文研究了Riccati方程和Fitzhugh-Nagumo方程的新精确解的构造.利用试探函数法找到了Riccati方程的八种类型的新显式精确解.用广义Tanh函数法结合Riccati方程的新精确解,获得了Fitzhugh-Nagumo方程、Huxley方程、广义KPP方程及Newell-Whitehead方程的许多新...     

Symmetry analysis of the option pricing model with dividend yield from financial markets

In this work, the option pricing Black–Scholes model with dividend yield is investigated via Lie symmetry analysis. As a result, the complete Lie symmetry group and infinitesimal generators of the one-dimensional Black–Scholes equation are derived. On the basis of these infinitesimal generators, the similarity variables and newly explicit solutions of the Black–Scholes equation are obtained by solving the corresponding characteristic equations. Finally, figures for an explicit solution with different dividend yields are presented to demonstrate the novel properties.     

EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION

In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.     

Symmetries for a family of Boussinesq equations with nonlinear dispersion

In this paper, we make a full analysis of a family of Boussinesq equations which include nonlinear dispersion by using the classical Lie method of infinitesimals. We consider travelling wave reductions and we present some explicit solutions: solitons and compactons.For this family, we derive nonclassical and potential symmetries. We prove that the nonclassical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We write the equations in a conserved form and we obtain a new class of nonlocal symmetries. We also obtain some Type-II hidden symmetries of a Boussinesq equation.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.     

Nonclassical symmetry of nonlinear diffusion equation and factorization method

We present the nonclassical symmetry of a nonlinear diffusion equation whose a nonlinear term is an arbitrary function. Generally, there is no guarantee that we can always determine the nonclassical symmetries admitted by the given equation because of the nonlinearity included in the determining equations. Accordingly, constructing invariant solutions is also generally difficult. In this paper, we apply the factorization method to nonclassical symmetry analysis for the nonlinear diffusion equation. Applying this method simplifies the determining equations and leads to their invariant solution automatically.     

Lie group classifications and exact solutions for two variable-coefficient equations

In this paper, the Lie symmetry analysis and group classifications are performed for two variable-coefficient equations, the hanging chain equation and the bond pricing equation. The symmetries for the two equations are obtained, the exact explicit solutions generated from the similarity reductions are presented. Moreover, the exact analytic solutions are considered by the power series method.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

Investigation of new solutions for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Nonclassical symmetry reductions of the Boussinesq equation

In this paper we discuss symmetry reductions and exact solutions of the Boussinesq equation using the classical Lie method of infinitesimals, the direct method due to Clarkson and Kruskal and the nonclassical method due to Bluman and Cole. In particular, we compare and contrast the application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and differential Gröbner bases as a technique for solving the overdetermined systems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exact solutions of partial differential equations are also mentioned. To conclude we describe some of the important open problems in the field of symmetry analysis of differential equations.