含色散项的Zabolotskaya-Khokhlov方程的对称、约化、精确解和守恒律

J4 ›› 2015, Vol. 32 ›› Issue (1): 46-52.

含色散项的Zabolotskaya-Khokhlov方程的对称、约化、精确解和守恒律

冯春明，刘庆松

1.聊城大学东昌学院，山东聊城 252059;2.聊城大学学报编辑部，山东聊城 252059

收稿日期:2014-04-28 修回日期:2014-11-07 出版日期:2015-01-28 发布日期:2015-02-04
通讯作者: 刘庆松（1980-）山东人，编辑，硕士，从事微分方程理论及应用，编辑等。 E-mail:17939697@qq.com
作者简介:冯春明（1974-），男，山东，讲师，硕士，从事非线性发展方程研究和教学。
基金资助:
国家自然科学基金和中国工程物理研究院联合基金资助(11076015)；聊城大学东昌学院课题（2013LG001）

Symmetries, reductions, exact solutions and conservation laws of the Zabolotskaya-KhokhlovEquation with a Dissipative Term

FENG Chun-ming，LIU Qing-song

1.School of Dongchang, Liaocheng University, Liaocheng 252059,China;2. Editoral Department, Journal of Liaocheng University, Liaocheng 252059, China

Received:2014-04-28 Revised:2014-11-07 Published:2015-01-28 Online:2015-02-04

摘要/Abstract

摘要： 利用修正的CK直接方法得到了含色散项的Zabolotskaya-Khokhlov(简写为DZK)方程的对称、约化和一些精确解,包括双曲函数解，有理函数解，三角函数解等。同时得到了该方程的守恒律。

关键词: 修正的CK直接方法, DZK方程, 对称、约化, 精确解, 守恒律

Abstract: Using the modified Clarkson–Kruskal direct method, the symmetries reductions of the Zabolotskaya-KhokhlovEquation with a Dissipative Term are obtained. By solving the reduction equations, a great many of solutions are derived, including the rational functions solution, the trigonometric functions, hyperbolic functions and so on. The conservation laws are given at last.

Key words: the modified CK’s direct method, the DZK equation, symmetry reduction, exact solutions, conservation laws

中图分类号:

O175.29

冯春明，刘庆松. 含色散项的Zabolotskaya-Khokhlov方程的对称、约化、精确解和守恒律[J]. J4, 2015, 32(1): 46-52.

FENG Chun-ming，LIU Qing-song. Symmetries, reductions, exact solutions and conservation laws of the Zabolotskaya-KhokhlovEquation with a Dissipative Term[J]. J4, 2015, 32(1): 46-52.

参考文献

[1] Chen Mei, Liu Xiqiang. Symmetries and exact solutions of the breaking soliton equation [J]. Communications in Theoretical Physics，2011，56(11)：851-855．
[2] Biuman G W, Kumei S. Symmetries and Differential Equations[M]. Berlin: Springer Velag ,1989.
[3] Matveev V B, Salle M A. Darboux Transformation and Solitons[M]. Berlin: Springer Velag , 1991.
[4] Rogers C, Schief W K. Ba¨cklund and Darboux Transformations, Geometry and Morden Applications in Soliton Theory, Cambridge University Press, Cambridge,2002.
[5] Hirota R. The Direct Method in Soliton Theory, Cambridge University Press, Cambridge ,2004.
[6] Wang Mingliang, Zhou Yubin, Li Zhibin. Application of a homogeneous banlance method to exact solutions of nonlinear equations in mathematical physics[J].Phys. Lett. A, 1996,216(1):67-75
[7] Liu Xiqiang, Jiang Song. The sech-Tanh method and its applications [J].Phys. Lett. A, 2002,8(4):253-258.
[8] 陈美，刘希强. Konopelchenko-Dubrovsky 方程组的对称，精确解和守恒律[J]. 纯粹数学与应用数学,2011,27(4):533-539
[9] Xin Xiangpeng, Liu Xiqiang, Zhang Linlin. Symmetry Reduction, Exact Solutions and Conservation Laws of the Modified Kadomtzev–Patvishvili-II Equation, Phys.Lett.,2011,28(2): 020201-1---020201-4.
[10]张颖元，刘希强，王岗伟．Bogoyavlenskii-Kadontsev-Petviashili方程的新的显式解[J]．昆明学院学报，2012，34(3)：55-59．
[11]Zhang Li hua,Liu Xiqiang. New exact solutions and conservation Laws to (3+1)-Dimensional　Potential- YTSF equation[J]．Communications in Theoretical Physics，2006，45(3)：487-492．
[12]Clarkson P A，Kruskal M D．New similarity solutions of the Boussinesq equation[J]．Journal of Mathematical Physics，1989，30：2201-2213．
[13] Li B Q, Li S, Ma Y L. New eExact novel time solitons for the dissipative Zabolotskaya -Khokhlov equation from nonlinear acoustics[J]. Zeit- schrift für Naturforschung A, 2012, 67: 601-607.
[14] Liu Xiajie, Mei Chunliang, Ma Songhua. Traveling wave solutions and kind wave excitations for the (2 + 1)-dimensional dissipative Zabolotskaya-Khokhlov equation[J]. Applied Mathematics, 2013, 4: 1595-1598.
[15] Masayoshi T, Similarity reductions of the Zabolotskaya-Khokhlov equation with a dissipative term[J]. Nonlinear Mathematical Physics 1995, 2: 392–397.
[16] Lu D C, Hong B J．New exact solutions for the (2+1)-dimensional Generalized Broer-Kaup system. Applied Mathematics and Computation 2008; 199:572-580．
[17] Ibragimov N H. A new conservation theorem[J]. Journal of Mathematical Analysis Application, 2007, 333: 311–328.