Abstract
In this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.
Funding source: China University of Mining and Technology
Award Identifier / Grant number: YC150003
Funding statement: We express our sincere thanks to the Editor and Reviewers for their valuable comments. This work is supported by the Fundamental Research Funds for Talents Cultivation Project of China University of Mining and Technology (Project No. YC150003).
Acknowledgements
We express our sincere thanks to the Editor and Reviewers for their valuable comments. This work is supported by the Fundamental Research Funds for Talents Cultivation Project of China University of Mining and Technology (Project No. YC150003).
References
[1] R. Hirota, Direct Methods in Soliton Theory, Springer, Berlin 2004.10.1017/CBO9780511543043Search in Google Scholar
[2] X. B. Hu, C. X. Li, J. J. C. Nimmo, and G. F. Yu, J. Phys. A Math. Gen. 38, 195 (2005).10.1088/0305-4470/38/1/014Search in Google Scholar
[3] P. J. Olver, Application of Lie Groups to Differential Equations. Springer-Verlag, New York 1993.10.1007/978-1-4612-4350-2Search in Google Scholar
[4] L. V. Ovsiannikov, Group Analysis of Differential Equations. New York, Academic Press 1982.10.1016/B978-0-12-531680-4.50012-5Search in Google Scholar
[5] N. H. Ibragimov, CRC Haandbook of Lie Group Analysis of Differential Equation. Vol. 1−3, CRC Press, Bocaraton 1994.Search in Google Scholar
[6] G. M. Bluman, A. Cheviakov, and S. Anco, Applications of Symmetry Methods to Partial Differential Equations. Springer, New York 2010.10.1007/978-0-387-68028-6Search in Google Scholar
[7] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for Solving the Korteweg-de Vries Equation, Phys. Rev. Lett. 19, 1095 (1967).10.1103/PhysRevLett.19.1095Search in Google Scholar
[8] V. B. Matveev and M. A. Salle, Darboux Transformations and Sliton, Springer, Berlin 1991.10.1007/978-3-662-00922-2Search in Google Scholar
[9] C. Rogers and W. K. Schief, Bäcklund and Darboux Transformation Geometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge 2002.10.1017/CBO9780511606359Search in Google Scholar
[10] M. Singh, Nonlinear Dyn. 84, 875 (2016).10.1007/s11071-015-2533-zSearch in Google Scholar
[11] M. Singh and R. K. Gupta, Nonlinear Dyn. 86, 1171 (2016).10.1007/s11071-016-2955-2Search in Google Scholar
[12] M. Singh and R. K. Gupta, Commun. Nonlinear Sci. Numer. Simul. 37, 362 (2016).10.1016/j.cnsns.2016.01.023Search in Google Scholar
[13] R. K. Gupta, M. Singh, Nonlinear Dyn. Doi 10.1007/s11071-016-3132-3.Search in Google Scholar
[14] S. Y. Lou and X. B. Hu, J. Math. Phys. 38, 6401 (1997).10.1063/1.532219Search in Google Scholar
[15] S. Y. Lou, Phys. Rev. Lett. 71, 4099 (1993).10.1103/PhysRevLett.71.4099Search in Google Scholar PubMed
[16] X. Y. Tang, S. Y. Lou, and Y. Zhang, Phys. Rev. E 66, 046601 (2002).10.1103/PhysRevE.66.046601Search in Google Scholar PubMed
[17] C. Z. Qu and C. R. Zhang, J. Phys. A: Math. Theor. 42, (2009) 475201.10.1088/1751-8113/42/47/475201Search in Google Scholar
[18] Y. Chen, X. R. Hu, Z. Naturforsch. A 64, 8 (2009).10.1515/zna-2009-1-202Search in Google Scholar
[19] X. R. Hu, S. Y. Lou, and Y. Chen, Phys. Rev. E 85, 056607 (2012).10.1103/PhysRevE.85.056607Search in Google Scholar PubMed
[20] Z. Dong, F. Huang, Y. Chen, Z. Naturforsch. A. 66, 75 (2011).10.1515/zna-2011-1-212Search in Google Scholar
[21] H. C. Hu, S. Y. Lou, Z. Naturforsch. A. 59, 337 (2004).10.1515/zna-2004-0604Search in Google Scholar
[22] H. C. Ma, S. Y. Lou, Z. Naturforsch. A. 60, 229 (2005).10.1515/zna-2005-0403Search in Google Scholar
[23] M. C. Nucci, P. A. Clarkson, Phys. Lett. A 164, 49 (1992).10.1016/0375-9601(92)90904-ZSearch in Google Scholar
[24] D. Levi and C. Scimiterna, App. Anal. 89, 507 (2010).10.1080/00036810903329969Search in Google Scholar
[25] S. F. Tian, T. T. Zhang, P. L. Ma, X. Y. Zhang, J. Nonlinear Math. Phys. 22, 180 (2015).10.1080/14029251.2015.1023562Search in Google Scholar
[26] S. F. Tian, Y. F. Zhang, B. L. Feng, and H. Q. Zhang, Chin. Ann. Math. B 36, 543 (2015).10.1007/s11401-015-0908-6Search in Google Scholar
[27] S. C. Anco and G. W. Bluman, Phys. Rev. Lett. 78, 2869 (1997).10.1103/PhysRevLett.78.2869Search in Google Scholar
[28] E. Nother, Transp Theory Stat Phys. 1, 186 (1971).10.1080/00411457108231446Search in Google Scholar
[29] N. H. Ibragimov, J. Math. Anal. Appl. 333, 311 (2007).10.1016/j.jmaa.2006.10.078Search in Google Scholar
[30] B. A. Kupershmidt, Commun. Math. Phys. 99, 51 (1985).10.1007/BF01466593Search in Google Scholar
[31] S. I. Svinolupov, V. V. Soklov, R. I. Yamilov, Sov. Math. Dokl. 28, 165 (1983).Search in Google Scholar
[32] S. L. Zhang, B. Wu, S. Y. Lou, Phys. Lett. A 300, 40 (2002).10.1016/S0375-9601(02)00688-6Search in Google Scholar
[33] Y. Chen, Q. Wang. Appl. Math. Comput. 168, 1189 (2005).10.1016/j.amc.2004.10.012Search in Google Scholar
[34] X. D. Zheng, Y. Chen, and H. Q. Zhang, Phys. Lett. A 311, 145 (2003).10.1016/S0375-9601(03)00451-1Search in Google Scholar
[35] E. Yomba, Phys. Lett. A 336, 463 (2005).10.1016/j.physleta.2005.01.027Search in Google Scholar
[36] Z. Y. Yan and H. Q. Zhang, Phys. Lett. A 252, 291 (1999).10.1016/S0375-9601(98)00956-6Search in Google Scholar
[37] Y. Chen and Q. Wang, Phys. Lett. A 347, 215 (2005).10.1016/j.physleta.2005.08.015Search in Google Scholar
[38] Y. B. Zhou and L. Chao, Commun. Theor. Phys. 51, 664 (2009).10.1088/0253-6102/51/4/17Search in Google Scholar
[39] M. Song, J. Cao, X. L. Guan, Math. Comput. Model. 55, 688 (2012).10.1016/j.mcm.2011.08.043Search in Google Scholar
[40] S. F. Tian, F. B. Zhou, S. W. Zhou, and T. T. Zhang, Mod. Phys. Lett. B 30, 1650100 (2016).10.1142/S0217984916501001Search in Google Scholar
[41] S. F. Tian and P. L. Ma, Commun. Theor. Phys. 62, 245 (2014).10.1088/0253-6102/62/2/12Search in Google Scholar
[42] L. Wang, Y. J. Zhu, F. H. Qi, M. Li, and R. Guo, Chaos 25, 063111 (2015).10.1063/1.4922025Search in Google Scholar PubMed
[43] L. Wang, J. H. Zhang, C. Liu, M. Li, F. H. Qi, Phys. Rev. E 93, 062217 (2016).10.1103/PhysRevE.93.062217Search in Google Scholar PubMed
[44] S. F. Tian and H. Q. Zhang, J. Phys. A: Math. Theor. 45, 055203 (2012).10.1088/1751-8113/45/5/055203Search in Google Scholar
[45] S. F. Tian and H. Q. Zhang, Stud. Appl. Math. 132, 212 (2014).10.1111/sapm.12026Search in Google Scholar
[46] X. B. Wang, S. F. Tian, C. Y. Qin, and T. T. Zhang, EPL, 114, 20003 (2016).10.1209/0295-5075/114/20003Search in Google Scholar
[47] S. F. Tian, Proc. R. Soc. Lond. A. 472, 20160588 (2016).Search in Google Scholar
[48] S. F. Tian, J. Differ. Equa. 262, 506 (2017).10.1016/j.jde.2016.09.033Search in Google Scholar
[49] N. H. Asmar, Partial Differential Equations with Fourier Series and Boundary Value Problems, 2nd ed. China Machine Press, Beijing 2005.Search in Google Scholar
[50] H. Liu and J. Li, Lie Symmetry Analysis and Exact Solutions for the Short Pulse Equation. Nonlinear Anal. 71, 2126 (2009).10.1016/j.na.2009.01.075Search in Google Scholar
[51] R. Naz, F. M. Mahomed, and T. Hayat, Conservation Laws for Third-Order Variant Boussinesq System. Appl. Math. Lett. 23, 883 (2010).10.1016/j.aml.2010.04.003Search in Google Scholar
©2017 Walter de Gruyter GmbH, Berlin/Boston
