﻿ 一类(1 + 2)-维非线性薛定谔方程的Lie-对称分析 The Lie-Symmetry Analysis of (1 + 2)-Coupled Nonlinear Schrodinger Equations

Dynamical Systems and Control
Vol.05 No.01(2016), Article ID:16766,13 pages
10.12677/DSC.2016.51003

The Lie-Symmetry Analysis of (1 + 2)-Coupled Nonlinear Schrodinger Equations

Dongdong Xu1, Chaolu Temuer2*

1Inner Mongolia University, Hohhot Inner Mongolia

2Colledge of Arts and Sciences, Shanghai Maritime University, Shanghai

Received: Dec. 20th, 2015; accepted: Jan. 10th, 2016; published: Jan. 14th, 2016

ABSTRACT

For a class of (1 + 2)-dimensional nonlinear Schrödinger equations, 8-dimensional subalgebra of the infinite Lie algebra is found and its one optimal system is constructed. By further reduction with its symmetry we obtain the corresponding ordinary differential equations. Solving the ordinary differential equations, one finds some exact invariant solutions of the Schrödinger equations.

Keywords:Nonlinear Schrodinger Equation, Lie Algebra, Optimal System, Invariant Solutions

1内蒙古大学，内蒙古 呼和浩特

2上海海事大学文理学院，上海

1. 引言

(1)

2. 方程组(1)的LIE代数

(2)

(3)

(4)

3. Lie代数L8的1-维优化系统

， (5)

Table 1. Commutation of the Lie algebra [7]

，进一步可使前的系数为或0。这样我们得到，其中或0。

Table 2. Adjoint representation of basis element of Lie algebra

(6)

4. 方程组(1)的关于优化系统(6)的第一次约化

， (7)

, , ,.

, (8)

(9)

Table 3. The first reductions of (1) by optimal system (6)

5. 通过表3中约化方程第二次约化2D-CNLS方程组(1)

(11)

(12)

(13)

Table 4. Commutators of (13)

Table 5. Commutators of (13)

(14)

6. 方程组(1)的部分精确不变解

(15)

Table 6. The second reductions of (1) with

，其中是实函数，带入(15)中可得

(16)

(17)

，上式化简为：

，得到

(18)

Figure 1. Graph of

Figure 2. Graph of

Figure 3. Graph of

Figure 4. Graph of or

Figure 5. Graph of and

7. 结束语

The Lie-Symmetry Analysis of (1 + 2)-Coupled Nonlinear Schrodinger Equations[J]. 动力系统与控制, 2016, 05(01): 18-30. http://dx.doi.org/10.12677/DSC.2016.51003

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*通讯作者。