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Symmetry reductions, exact equations and the conservation laws of the generalized (3+1) dimensional Zakharov-Kuznetsov equation

Zhang Li-Xiang, Liu Han-Ze, Xin Xiang-Peng
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• 摘要

运用李群分析，得到了广义（3+1）维Zakharov-Kuznetsov（ZK）方程的对称及约化方程，结合齐次平衡原理，试探函数法和指数函数法得到了该方程的群不变解和新精确解，包括冲击波解、孤立波解等. 进一步给出了广义（3+1）维ZK方程的伴随方程和守恒律.

Abstract

Because the nonlinear evolution equations can describe the complex phenomena of physical, chemical and biological field, many methods have been proposed for investigating such types of equations, and the Lie symmetry analysis method is one of the powerful tools for studying the nonlinear evolution equations. By using the Lie symmetry analysis method, we can obtain the symmetries, reduced equations, group invariant solutions, conservation laws, etc. In the reduction process, we can reduce the order and dimension of the equations, and a complex partial differential equations (PDE) can be reduced to ordinary differential equations directly, which simplifies the solving process. Meanwhile, the symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. For example, we can obtain a lot of new exact solutions by the known symmetries of the original equation; through the analysis of the special form of solution we can better explain some physical phenomena. In addition, the studying of conservation laws and symmetry groups is also the central topic of physical sciencein both classical mechanics and quantum mechanics. Lie symmetry analysis method is suitable for not only constant coefficient equations, but also variable coefficient equations and PDE systems. By using Lie symmetry analysis method, the symmetries and corresponding symmetry reductions of the (3+1) dimensional generalized Zakharov-Kuzetsov (ZK) equation are obtained. Combining the homogeneous balance principle, the trial function method and exponential function method, the group invariant solutions and some new exact explicit solutions are obtained, including the shock wave solutions, solitary wave solutions, etc. Then, we give the conservation laws of the generalized (3+1) dimensional ZK equation in terms of the Lagrangian and adjoint equation method.

作者及机构信息

通信作者: 刘汉泽, hnz_liu@aliyun.com
• 基金项目: 国家自然科学基金（批准号：11171041，11505090）资助的课题.

Authors and contacts

Corresponding author: Liu Han-Ze, hnz_liu@aliyun.com
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11171041, 11505090).

参考文献

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施引文献

•  [1] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer) pp186-206 [2] Tian C 2001 Applications of Lie Groups to Differential Equations (Beijing: Science Press) pp243-248 (in Chinese) [田畴 2001 李群及其在微分方程中的应用 (北京: 科学出版社) 第243-248页] [3] Cao L M, Si X H, Zheng L C 2016 J. Appl. Math. Mech. 37 433 [4] Li D S, Zhang H Q 2005 Acta Phys. Sin. 54 1569 (in Chinese) [李德生, 张鸿庆 2005 物理学报 54 1569] [5] Hirota R, Satsuma J 1976 Suppl. Prog. Theor. Phys. 59 64 [6] Weiss J, Tabor M, Carnevale G 1983 J. Math. Phys. 24 522 [7] Clarkson P 1989 J. Math. Phys. 30 2201 [8] Lou S Y, Ma H C 2005 J. Phys. A: Math. Gen. 38 L129 [9] Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417 [10] Pan J T, Gong L X 2007 Acta Phys. Sin. 56 5585 (in Chinese) [潘军廷, 龚伦训 2007 物理学报 56 5585] [11] Pang J, Bian C Q, Chao L 2010 Appl. Math. Mech. 30 884 (in Chinese) [庞晶, 边春泉, 朝鲁 2010 应用数学和力学 30 884] [12] Naher H, Abdullah F A 2014 Res. J. Appl. Sci. Eng. Technol. 7 4864 [13] Han Z, Zhang Y F, Zhao Z L 2013 Commun. Theor. Phys. 60 699 [14] Xu F, Yan W, Chen Y L 2009 Comput. Math. Appl. 58 2307 [15] Wazwaz A M 2005 Commun. Nonlinear Sci. Numer. Simul. 10 97 [16] Liu S S, Fu Z T, Liu S D, Zhao Q 2001 Appl. Math. Mech. 22 281 (in Chinese) [刘式适, 付遵涛, 刘式达, 赵强 2001 应用数学和力学 22 281] [17] He J H, Wu X H 2006 Chaos, Solitions and Fractals 30 700 [18] Zhang H Q 2001 J. Math. Phys. 21A 321 (in Chinese) [张辉群 2001 数学物理学报 21A 321] [19] Wang M L, Zhou Y Z, Li Z B 1996 Phys. Lett. A 216 67 [20] Wang M L, Li Z B, Zhou Y B 1999 J. Lanzhou Univ. 35 8 (in Chinese) [王明亮, 李志斌, 周宇斌 1999 兰州大学学报 35 8] [21] Ibragimov Z H 2006 J. Math. Anal. Appl. 318 742 [22] Ibragimov Z H 2007 J. Math. Anal. Appl. 333 311 [23] Xi X P, Chen Y 2013 Commun. Theor. Phys. 59 573 [24] Li K H, Liu H Z, Xin X P 2016 Acta Phys. Sin. 65 140201 (in Chinese) [李凯辉, 刘汉泽, 辛祥鹏 2016 物理学报 65 140201]
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出版历程
• 收稿日期:  2016-01-01
• 修回日期:  2017-01-02
• 刊出日期:  2017-04-05

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