一类四阶偏微分方程的对称分析及级数解

研究了一类四阶偏微分方程的李对称,构造了方程所容许的李对称的优化系统,进行了对称约化,得到了精确解.进一步,基于幂级数理论,得到了这类四阶偏微分方程的幂级数解.     

KdV-Burgers方程的对称与孤子解

考虑KdV-Burgers方程的一些简单对称及其构成的李代数,并利用对称约化方法将KdV-Burgers方程化为常微分方程,从而得到该方程的群不变解.此外,利用多项式展开式的方法去获得KdV-Burgers方程的新的孤子波解.     

Smoluchowski方程的对称与显式解析解

研究了一类带齐次核函数的偏微分一积分Smoluchowski方程.利用发展了的李群分析方法给出了带齐次核函数的Smoluchowski方程的决定方程的通解、对称、最优子李代数系统、约化的常微分—积分方程、群不变解和显式解析解.     

非线性演化方程的切对称群分析

将优化系统的概念推广应用至切代数,并以一个二阶非线性演化方程为例,给出了方程所容许的切对称,建立了切对称的一维优化系统.并利用优化系统对所研究的方程进行了对称约化,得到了与不等价对称相对应的约化方程和不变解.     

一类组合sinh-cosh-Gordon方程的对称约化、动力学性质与精确解（英文）

文章综合运用李对称分析、幂级数解法和动力系统法来求解组合sinh-cosh-Cordon方程的精确解.利用李对称分析得到了组合sinh-cosh-Cordon方程的向量场和相似变换,把难以求解的偏微分问题约化为常微分方程,利用幂级数解法求得了方程的精确解析解.然后用MATLAB画出了约化后方程的相图,最后利用动力系统法分析研究了解的动力学行为,并得到了方程的行波解.     

一类时空分数阶非线性偏微分方程的对称分析、对称约化、精确解和守恒律

借助对称分析方法研究了一类时空分数阶非线性偏微分方程及其特殊情形,建立了方程所允许的李代数,构造了相应的一维优化系统.进一步地,利用优化系统对所研究的方程进行了对称约化,得到了方程的群不变解.另外,利用新的守恒定律和推广的Noether算子,建立了时空分数阶微分方程的非局部守恒律.     

一类自伴随Lubrication方程的对称,守恒律,拉格朗日函数和精确解

本文借助李对称分析研究了一类自伴随的Lubrication方程,此类方程可用来描述液体薄膜动力学行为.基于非奇异的局域守恒律乘子和李对称方法,我们系统地推导出了此类方程的局域守恒律,非局域相关系统,李对称和一些有趣的精确解.此模型的非局域相关系统在本文中被首次研究,可用于寻找原方程更丰富的解空间.此外,基于局域守恒律和变分原则,我们推导出原方程的四类拉格朗日函数.     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

广义变系数Kdv方程的对称及其群不变解

利用经典李对称的方法对广义变系数Kdv方程进行研究,利用这种方法得到了该方程的一个新的精确解,这种方法的基本思路是通过对称约化将原来较难求解的偏微分方程转化为较易求解的常微分方程进行求解.实例证明这种方法具有一般性,适合于求一大类变系数的非线性演化方程.     

Group analysis of KdV equation with time dependent coefficients

We study the generalized KdV equation having time dependent variable coefficients of the damping and dispersion from the Lie group-theoretic point of view. Lie group classification with respect to the time dependent coefficients is performed. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are obtained. These subalgebras are then used to construct a number of similarity reductions and exact group-invariant solutions, including soliton solutions, for some special forms of the equations.     

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

In this paper, the Lie symmetry analysis are performed on the three nonlinear elastic rod (NER) equations. The complete group classifications of the generalized nonlinear elastic rod equations are obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system method is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.     

Lie symmetry analysis and exact explicit solutions for general Burgers’ equation

In this paper, the Lie symmetry analysis is performed for the general Burgers’ equation. The exact solutions and similarity reductions generated from the symmetry transformations are provided. Furthermore, the all exact explicit solutions and similarity reductions based on the Lie group method are obtained, some new method and techniques are employed simultaneously. Such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science.     

Conservation Laws and Exact Solutions to the Modified Hyperbolic Geometric Flow

In this paper, we investigate Lie symmetry group, optimal system, exact solutions and conservation laws of modified hyperbolic geometric flow via Lie symmetry method. Then, conservation laws of modified hyperbolic geometric flow are obtained by applying Ibragimov method.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

Lie Symmetry Analysis and Exact Solutions for?the?Extended mKdV Equation

By using Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation, the all exact solutions based on the Lie group method are given. Especially, the bifurcations and traveling wave solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlinear science.