Symmetry reductions, exact solutions and conservation laws of a class of forth-order partial differential equations
-
摘要: 利用李群分析研究了一类变系数四阶偏微分方程，求出方程的李点对称，把偏微分方程约化为常微分方程，然后结合（G'/G）展开法及椭圆函数展开法，对约化后的常微分方程求其精确解，从而得到原方程的精确解.进一步，给出这类变系数偏微分方程的守恒律.Abstract: The partial differential equation with constant coefficients can merely approximately reflect the law of motion of substances. Relatively the partial differential equation with variable coefficients can reflect the complex movement of substances more accurately. Therefore, it is more important to study the partial differential equations with variable coefficients. This paper investigates a class of variable coefficient partial differential equations. By using Lie symmetry analysis, the symmetries of the equations are obtained, Then the partial differential equations are reduced to ordinary differential equations. Moreover, we combine with (G'/G) expansion method and elliptic function expansion, so exact solutions to the original equation are obtained. Furthermore, the conservation laws of this kind of variable coefficient differential equations are given.
-
Key words:
- variable coefficient equation /
- Lie symmetry analysis /
- exact solution /
- conservation law
-
图 1 当$\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in[0,1]$时, $u_{4}$为双孤子解
Fig. 1 When $\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in [0,1]$, $u_{4}$ is double solitonsolution
图 2 当$\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in[0,1]$时, $u_{5}$为凹尖峰孤子解
Fig. 2 When $\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in [0,1]$, $u_{5}$ is concave peak soliton solution
图 3 当$\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in [0,1]$时, $u_{6}$为双孤子解
Fig. 3 When $\sqrt[3]{\frac{24\beta}{k}}=2$, $c=2$, $x\in [-10,20]$, $t\in [0,1]$, $u_{6}$ is double soliton solution
-
[1] 田畴.李群及其在微分方程中的应用[M].北京:科学出版社, 2001. [2] OLVER P. Applications of Lie Groups to Differential Equations[M]. New York:Springer, 1993. [3] BLUMAN G, ANCO S. Symmetry and Integration Methods for Differential Equations[M]. New York:SpringerVerlag, 2002. [4] HIROTA R, SATSUMA J. A variety of nonlinear network equations generated form the Bäcklund transformation for the Tota lattice[J]. Suppl Prog Theor Phys, 1976, 59:64-100. doi: 10.1143/PTPS.59.64 [5] LIU H Z, LI J B, CHEN F J. Exact periodic wave solutions for the mKdV equations[J]. Nonlinear Anal, 2009, 70:2376-2381. doi: 10.1016/j.na.2008.03.019 [6] WANG G W, XU T Z, LIU X Q. New explicit solutions of the fifth-order KdV equation with variable cóefficients[J]. Bull Malays Math Sci Soc 2014, 37(3):769-778. https://www.sciencedirect.com/science/article/pii/S2211379716305563 [7] 胡晓瑞. 非线性系统的对称性与可积性[D]. 上海: 华东师范大学, 2012, 43-77. http://cdmd.cnki.com.cn/Article/CDMD-10269-1012435961.htm [8] 刘大勇, 夏铁成.齐次平衡法寻找Caudrey-Dodd-Gibbon-Kaeada方程的多孤子解[J].应用数学和计算数学学报, 2011, 25(2):205-212. http://www.cqvip.com/QK/90585X/201102/40374856.html [9] 刘丽环, 常晶, 冯雪.求非线性发展方程行波解的(G'/G)展开法[J].吉林大学学报(理学版), 2013, 51(2):183-186. http://www.cqvip.com/QK/95191X/201302/45269786.html [10] 张辉群.齐次平衡方法的扩展及应用[J].数学物理学报, 2001, 21A(3):321-325. http://d.wanfangdata.com.cn/Periodical/sxwlxb200103005 [11] YAO Q L. Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem[J]. Nonlinear Analysis, 2005, 63:237-246. doi: 10.1016/j.na.2005.05.009 [12] WANG M L, LI X Z, ZHANG J L. The (G'/G) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys Lett A, 2008, 372:417-423. doi: 10.1016/j.physleta.2007.07.051 [13] 赵烨, 徐茜.一类耦合Benjamin-Bona-Mahony型方程组的新精确解[J].纯粹数学与应用数学, 2015, 31:12-17. doi: 10.3969/j.issn.1008-5513.2015.01.002 [14] LI K H, LIU H Z. Lie symmetry analysis and exact solutions for nonlinear LC circuit equation[J]. Chinese Journal of Quantum Electronics, 2016, 33:279-286. http://d.wanfangdata.com.cn/Periodical/lzdzxb201603003 [15] 杨春艳, 李小青.一类四阶偏微分方程的对称分析及级数解[J].纯粹数学与应用数学, 2016, 32:432-440. doi: 10.3969/j.issn.1008-5513.2016.04.011 [16] 徐兰兰, 陈怀堂.变系数(2+1)维Nizhnik-Novikov-Vesselov的三孤子新解[J].物理学报, 2013, 62(9):090204(1-6). [17] 魏帅帅, 李凯辉, 刘汉泽.展开法在Riccati方程中的应用[J].河南科技大学学报. 2015, 36:92-96. doi: 10.3969/j.issn.1672-6871.2015.02.022 [18] IBRAGIMOV N H. Integrating factors, adjoint equations and Lagrangians[J]. J Math Anal Appl, 2006, 38:742-757. http://d.wanfangdata.com.cn/NSTLQK/NSTL_QKJJ027993877/ [19] IBRAGIMOV N H. A new conservation theorem[J]. J Math Anal Appl, 2007, 333:311-328. doi: 10.1016/j.jmaa.2006.10.078 [20] IBRAGIMOV N H. Nonlinear self-adjointness and conservation laws[J]. J Phys A, 2011, 44:432002(899). http://d.wanfangdata.com.cn/NSTLQK/NSTL_QKJJ0232859170/ [21] ROSA R, GANDARIAS M L, BRUZON M S. Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping[J]. Nonlinear Dyn, 2016, 84:135-141. doi: 10.1007/s11071-015-2254-3 [22] YOMBA E. On exact solutions of the coupled Klein-Gordon-Schrödinger and the complex coupled KdV equations using mapping method[J]. Chaos, Solitons and Fractals, 2004, 21:209-229. doi: 10.1016/j.chaos.2003.10.028 -
下载:
点击查看大图
图(4)
计量
- 文章访问数: 228
- HTML全文浏览量: 50
- PDF下载量: 335
- 被引次数: 0