﻿ 一类复合方程的古典和非古典对称分类 Classical and Nonclassical Symmetry Classification of a Composite Type Equation

Pure Mathematics
Vol.07 No.04(2017), Article ID:21401,9 pages
10.12677/PM.2017.74040

Classical and Nonclassical Symmetry Classification of a Composite Type Equation

Yuexing Bai, Bilige Sudao*

College of Sciences, Inner Mongolia University of Technology, Hohhot Inner Mongolia

Received: Jun. 25th, 2017; accepted: Jul. 9th, 2017; published: Jul. 19th, 2017

ABSTRACT

In this paper, the classifications of classical and nonclassical symmetries to a composite type equation are determined. Firstly, the classification of classical symmetries to the composite equation is determined based on the differential characteristic set algorithm. Secondly, the classification of nonclassical symmetries for the composite equation is determined. First step, adding invariant surface condition and the original equation composed a new system of partial differential equations (PDEs), and the determining equations (DTEs) of symmetry to PDEs are determined by using the symbolic computation software Mathematica. Second step, the nonclassical symmetries are classified by calculating DTEs, so we can obtain the specific form of F(u) which is the parameter of the composite equation. Third step, the invariant solutions and exact solutions of the corresponding nonclassical symmetry are determined. The invariant solutions and exact solutions cannot be obtained by classical symmetry, so enrich the exact solutions of the composite equation.

Keywords:Classical Symmetry, Nonclassical Symmetry, Symmetry Classification, Differential Characteristic Set Algorithm, The Composite Equation

1. 引言

Lie对称是一个较为普适性的方法 [1] [2] ，且偏微分方程组(PDEs)对称已有了广泛的应用 [1] [2] [3] 。为了更好的运用对称方法，人们扩充古典对称概念，提出了各种广义对称概念，如非古典对称 [4] 、势对称 [5] 、近似对称 [6] 、条件对称 [7] 等。这些广义对称得到了广泛的应用，并且理论正在蓬勃发展。其中非古典对称的计算与古典对称不同之处是添加一个不变曲面条件，再计算确定方程组；并且非古典对称的确定方程组(DTEs)是非线性PDEs，所以非古典对称的确定仍然是目前具有挑战性的问题。然而通过获得非线性PDEs的非古典对称，扩充方程的古典对称依然是目前研究的热门课题。目前国内外研究者对非古典对称进行了一些研究，推动了其发展 [8] - [13] 。

(1)

2. 复合方程的古典和非古典对称分类

(2)

(3)

2.1. 古典对称分类

(4)

(5)

2.1.1. 主对称

(6)

(7)

(8)

2.1.2. 扩充对称

2.2. 非古典对称分类

(9)

(10)

Table 1. Classical symmetry classification of composite Equation (1)

. (11)

2.2.1.的情况

(12)

(13)

(II.1) 当时，化简(13)得

(14)

(15)

(1) 取，原复合方程变为Burgers方程 [14] ，由表1知该情况具有非古典对称。

(2) 取，原复合方程变为BBM-Burgers方程 [15] ，由表1知该情况具有非古典对称。

(3) 取，原复合方程变为RLW-Burgers方程 [16] ，由表1知该情况具有非古典对称。

(II.2) 当时，化简(14)得

(16)

(II.2.1) 若，由(16)中的第一式得，因的函数，知。令，由(16)中的第二式得。其中是任意常数，且，最后我们得到

(17)

(II.2.2) 若，化简(14)得到

(18)

(II.3)，根据(10)的化简知必须满足条件，化简(10)得到

(19)

(20)

(II.3.1) 若，设，化简(20)得到

(21)

(22)

(II.3.2) 若，计算得，化简(20)有

(23)

(24)

2.2.2.的情况

3. 不变解

3.1. 非古典对称对应的不变解

(1) 当时，复合方程变为

Table 2. Nonclassical symmetry classification of composite Equation (1)

Table 3. Nonclassical symmetry classification, of composite Equation (1)

, (25)

. (26)

(2) 当时，复合方程变为

, (27)

(28)

，即可得到方程(27)的不变解

(3) 当时，复合方程变为

(29)

(30)

3.2. 精确解

1、下面首先确定当时，古典对称对应的Lie变换群，

(31)

(32)

.

2、下面首先确定当时，古典对称对应的Lie变换群为：

(33)

.

4. 本文结论

Classical and Nonclassical Symmetry Classification of a Composite Type Equation[J]. 理论数学, 2017, 07(04): 301-309. http://dx.doi.org/10.12677/PM.2017.74040

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