﻿ 一类空间分数阶非线性SchrO¨dinger方程的李群约化 Lie Group Reduction for a Kind of Space-Fractional Order Nonlinear SchrO¨dinger Equation

Vol.05 No.02(2016), Article ID:17707,10 pages
10.12677/AAM.2016.52039

Lie Group Reduction for a Kind of Space-Fractional Order Nonlinear Schrödinger Equation

Chunhong Zhou, Cuncai Hua

School of Mathematics, Yunnan Normal University, Kunming Yunnan

Received: May 10th, 2016; accepted: May 27th, 2016; published: May 30th, 2016

ABSTRACT

This paper will apply the Lie group reduction method to a kind of space-fractional order nonlinear Schrödinger equation. New single parameter solutions and reduced equations of Lie symmetry are obtained for the equation. Moreover, by solving the reduced equation of Lie symmetry, some group-invariant solutions and travelling wave solutions are given for the space-fractional order nonlinear Schrödinger equation.

Keywords:Space-Fractional Order Nonlinear Schrödinger Equation, Lie Group Reduction, Group-Invariant Solutions, Travelling Wave Solutions

1. 引言

2. 一类空间分数阶非线性Schrödinger方程的李对称分析

2.1. 一类空间分数阶非线性Schrödinger方程的引入

(1)

1997年，Saichev A和Zaslavsky G.M.在他们的著作“Fractional Kinetic Equations: Solutions and Applications” [23] 中将分数阶的拉普拉斯算子表示为：

(2)

(3)

(4)

, (5)

(6)

2.2. 方程(5)的李代数、李对称群及其单参数解

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

Table 1. The commutators among

(16)

(17)

(18)

(19)

(20)

(21)

2.3. 空间分数阶非线性Schrödinger方程的李对称约化

(22)

(23)

(24)

(25)

(26)

(27)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

，其中 (47)

(48)

(49)

3. 结论

Lie Group Reduction for a Kind of Space-Fractional Order Nonlinear SchrO¨dinger Equation[J]. 应用数学进展, 2016, 05(02): 310-319. http://dx.doi.org/10.12677/AAM.2016.52039

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