Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

Abstract and Applied Analysis

Research Article

Zheng Tian¹  Xin Li¹  Jing Li¹  Yanping Ran² 

[1] College of Applied Science, Beijing University of Technology, Beijing 100124, China, bjut.edu.cn;College of Applied Science, Beijing University of Technology, Beijing 100124, China, bjut.edu.cn;School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China, tsnc.edu.cn

Others  :  1346872
DOI  :  10.1155/2015/820916

 received in 2014-06-27, accepted in 2014-07-15,  发布年份 2015

【 摘 要 】

Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation method of planar dynamical systems. The bifurcation regions in different subsets of the parameters space are obtained. According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions, and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.

【 授权许可】

CC BY   
Copyright © 2015 Yanping Ran et al. 2015

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【 参考文献 】

• [1]X. Geng, C. Cao, H. H. Dai. (2001). Quasi-periodic solutions for some -dimensional integrable models generated by the Jaulent-Miodek hierarchy. Journal of Physics A: Mathematical and General.34(5):989-1004. DOI: 10.1088/0305-4470/34/5/305.
• [2]X. Geng, Y. Ma. (2007). -soliton solution and its Wronskian form of a (3+1)-dimensional nonlinear evolution equation. Physics Letters A.369(4):285-289. DOI: 10.1088/0305-4470/34/5/305.
• [3]A. Wazwaz. (2009). Multiple kink solutions and multiple singular kink solutions for -dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Physics Letters A.373(21):1844-1846. DOI: 10.1088/0305-4470/34/5/305.
• [4]Y. Y. Zhang, X. Q. Liu, G. W. Wang. (2012). Symmetry reductions and exact solutions of the -dimensional Jaulent-Miodek equation. Applied Mathematics and Computation.219(3):911-916. DOI: 10.1088/0305-4470/34/5/305.
• [5]R. Hirota. (2004). The Direct Method in Soliton Theory.155. DOI: 10.1088/0305-4470/34/5/305.
• [6]W. Hereman, A. Nuseir. (1997). Symbolic methods to construct exact solutions of nonlinear partial differential equations. Mathematics and Computers in Simulation.43(1):13-27. DOI: 10.1088/0305-4470/34/5/305.
• [7]A. M. Wazwaz. (2009). Partial Differential Equations and Solitary Waves Theorem. DOI: 10.1088/0305-4470/34/5/305.
• [8]A. Wazwaz. (2012). Multiple soliton solutions for some (3 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Applied Mathematics Letters.25(11):1936-1940. DOI: 10.1088/0305-4470/34/5/305.
• [9]J. B. Li, G. R. Chen. (2012). Exact traveling wave solutions and their bifurcations for the Kudryashov-Sinelshchikov equation. International Journal of Bifurcation and Chaos.22(5):1-19. DOI: 10.1088/0305-4470/34/5/305.
• [10]J. Li, Y. Zhang, X. Zhao. (2009). On a class of singular nonlinear traveling wave equations (II): an example of Gckdv equations. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering.19(6):1995-2007. DOI: 10.1088/0305-4470/34/5/305.
• [11]H. Liu, F. Yan. (2011). The bifurcation and exact travelling wave solutions for -dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. International Journal of Nonlinear Science.11(2):200-205. DOI: 10.1088/0305-4470/34/5/305.
• [12]Y. Zhou, Q. Liu. (2007). Bifurcation of travelling wave solutions for a -dimensional nonlinear dispersive long wave equation. Applied Mathematics and Computation.189(1):970-979. DOI: 10.1088/0305-4470/34/5/305.