### Differential invariants and exact solutions of variable coefficients Benjamin-Bona-Mahony-Burgers equation

LI Hui-hui, LIU Xi-qiang*, XIN Xiang-peng

1. School of Mathematical Science, Liaocheng University, Liaocheng 252059, Shandong, China
• Received:2017-11-24 Online:2018-10-20 Published:2018-10-09

Abstract: The Lie symmetry method is performed for the variable coefficients Benjamin-Bona-Mahony-Burgers(BBMB)equation and the continuous equivalence transformations are obtained. Starting with the equivalent algebra, the differential invariants of order one are constructed. It is found that there is no zero-order differential invariant for this equation, but there are eight first-order invariants that are independent of each other. Using the obtained first-order differential invariants, we make the group classification. Finally, the general variable coefficient BBMB equations are mapped to the constant coefficient BBM equation or Burgers equation or BBMB equation by the given equivalent transformation. And then a series of new exact solutions of those variable coefficient equations are obtained. The images of the exact solution of the special BBM equation with variable coefficients and Burgers equation of the exact solution are made.

CLC Number:

• O175.2
 [1] 张金良, 王跃明, 王明亮, 等. 变系数(2+1)维Broer-Kaup方程的精确解[J].原子与分子物理学报, 2003, 20(1):92-94. ZHANG Jinliang, WANG Yueming, WANG Mingliang, et al. The exact solution of variable coefficient(2+1)dimensional Broer-Kaup equation[J]. Journal of Atomic and Molecular Physics, 2003, 20(1):92-94. [2] 庞晶, 靳玲花, 应孝梅. 利用(G'/G)展开法求解广义变系数Burgers方程[J]. 量子电子学报, 2011, 28(6): 674-681. PANG Jing, JIN Linghua, YING Xiaomei. Solving generalized Burgers equation with variable coefficients by(G'/G)-expansion[J]. Chinese Journal of Quantum Electronics, 2011, 28(6): 674-681. [3] MAO Jiejian, YANG Jianrong. New solitary-wave-like solutions and exact solutions to variable coefficient generalized KdV equation[J]. Acta Phys Sin, 2007, 56(9):5049-5053. [4] SOPHOCLEOUS C. Transformation properties of a variable-coefficient Burgers equation[J]. Chaos, Solitons & Fractals, 2004, 20(5):1047-1057. [5] 田贵辰, 刘希强. 含外力项的广义变系数KdV方程的精确解[J]. 量子电子学报, 2005, 22:339-343. TIAN Guichen, LIU Xiqiang. Exact solutions of the general variable coefficient KdV equation with external force term[J]. Chinese Journal of Quantum Electronics, 2005, 22:339-343. [6] 卢殿臣, 洪宝剑, 田立新. 带强迫项变系数组合KdV方程的显式精确解[J]. 物理学报, 2006, 55(11):5617-5622. LU Dianchen, HONG Baojian, TIAN Lixin. Explicit and exact solutions to the variable coefficient combined KdV equation with forced term[J]. Acta Phys Sin, 2006, 55(11):5617-5622. [7] 套格图桑, 斯仁道尔吉. 变系数(3+1)维Zakharov-Kuznetsov方程的Jacobi椭圆函数精确解[J]. 量子电子学报, 2010, 27(1):6-14. Taogetusang, Sirendaoerji. Jacobi-like elliptuc function exact solutions of(3+1)-dimensional Zakharov-Kuznetsov equation with variable coefficients[J]. Chinese Journal of Quantum Electronics, 2010, 27(1):6-14. [8] MOLATI M, KHALIQUE C M. Symmetry classification and invariant solutions of the variable coefficient BBM equation[J]. Appl Math Comput, 2013, 219(15):7917-7922. [9] GAO Xinyi. Backlund transformation and shock-wave-type solutions for a generalized(3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics[J]. Ocean Engineering, 2015, 96:245-247. [10] WAZWAZ A M, TRIKI H. Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients[J]. Commun Nonlinear Sci, 2011, 16(3):1122-1126. [11] WANG Mingliang, LI Xiangzheng, ZHANG Jinliang. The(G'/G)expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys Lett A, 2008, 374(4):417-423. [12] KUMAR V, GUPTA R K, JIWARI R. Painleve analysis, lie symmetries and exact solutions for variable coefficients Benjamin-Bona-Mahony-Burgers(BBMB)equation[J]. Commun Thero Phys, 2013, 60(8):175-182. [13] 郭美玉, 刘希强, 高洁. 广义变系数KdV-Burgers 方程的微分不变量及群分类[J]. 量子电子学报, 2009, 26(2):139-147. GUO Meiyu, LIU Xiqiang, GAO Jie. Differential invariants and group classification of KdV-Burgers equation[J]. Chinese Journal of Quantum Electronics, 2009, 26(2):139-147. [14] Taogetusang, Sirendaoerji. A method for constructing exact solutions of nonlinear evolution equation with variable coefficients[J]. Acta Phys Sin, 2009, 58(4):2121-2126. [15] 孙玉真, 王振立, 王岗伟, 等. 广义变系数五阶KdV和BBM方程的孤立子解[J]. 量子电子学报, 2013, 30(4):398-404. SUN Yuzhen, WANG Zhengli, WANG Gangwei, et al. Soliton solutions for generalized fifth-order KdV and BBM equations with variable coefficients[J]. Chinese Journal of Quantum Electronics, 2013, 30(4):398-404. [16] 雍雪林, 张鸿庆. 推广的投影Riccati方程法及其应用[J]. 物理学报, 2005, 54(6):2514-2519. YONG Xuelin, ZHANG Hongqin. Extended projective Riccati equations method and its application[J]. Acta Phys Sin, 2005, 54(6):2514-2519. [17] WEISS J, TABOR M, CARNEVALE G. The painleve property for partial differential equation[J]. J Math Phys, 1983, 24(3):522-526. [18] OLVER P J. Application of Lie group to differential equations[M]. Berlin: Springer-verlag, 1986. [19] 洪宝剑, 卢殿臣, 赵康生. Burgers-BBM方程新的精确解[J]. 应用数学, 2007, 20(1):134-139. HONG Baojian, LU Dianchen, ZHAO Kangsheng. New exact solutions of Burgers-BBM equation[J]. Mathematica Applicata, 2007, 20(1):134-139. [20] 刘式适, 付遵涛, 刘式达, 等. 求某些非线性偏微分方程特解的一个简洁方法[J]. 应用数学和力学, 2001, 22(3):281-286. LIU Shishi, FU Zuntao, LIU Shida, et al. A concise method for finding particular solutions of some nonlinear partial differential equations[J]. Appl Math Mech, 2001, 22(3):281-286. [21] 祁新雷, 李金花.(1+1)维Burgers方程新的行波解[J]. 纯粹数学与应用数学, 2008, 24(4):709-712. QI Xinlei, LI Jinhua. The new travelling wave solutions of the(1+1)-dimensional Burgers equation[J]. Pure and Applied Mathematics, 2008, 24(4):709-712. [22] 套格图桑, 斯仁道尔吉. BBM方程和修正的BBM方程新的精确孤立波解[J]. 物理学报, 2004, 53(12):4052-4060. Taogetusang, Sirendaoerji. New exact solitary wave solutions to the BBM and mBBM equations[J]. Acta Phys Sin, 2004, 53(12):4052-4060.
 [1] LIU Yong, LIU Xi-qiang. Symmetry, reductions and exact solutions of the (2+1)-dimension Caudrey-Dodd-Gibbon equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 49-55.
Viewed
Full text

Abstract

Cited

Shared
Discussed