Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

doi:10.1088/1674-1056/23/8/080201

中国物理B ›› 2014, Vol. 23 ›› Issue (8): 80201-080201.doi: 10.1088/1674-1056/23/8/080201

• GENERAL • 下一篇

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

殷久利, 邢倩倩, 田立新

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China

收稿日期:2013-12-12 修回日期:2014-02-12 出版日期:2014-08-15 发布日期:2014-08-15
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11101191).

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

Yin Jiu-Li (殷久利), Xing Qian-Qian (邢倩倩), Tian Li-Xin (田立新)

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Received:2013-12-12 Revised:2014-02-12 Online:2014-08-15 Published:2014-08-15
Contact: Yin Jiu-Li E-mail:yjl@ujs.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11101191).

摘要/Abstract

摘要： In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently.

关键词: Melnikov method, compacted solitary waves, control threshold

Abstract: In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently.

Key words: Melnikov method, compacted solitary waves, control threshold

中图分类号: (Partial differential equations)

02.30.Jr

02.60.Lj (Ordinary and partial differential equations; boundary value problems)

殷久利, 邢倩倩, 田立新. Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation[J]. 中国物理B, 2014, 23(8): 80201-080201.

Yin Jiu-Li (殷久利), Xing Qian-Qian (邢倩倩), Tian Li-Xin (田立新). Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation[J]. Chin. Phys. B, 2014, 23(8): 80201-080201.

参考文献 33

[1]	Gaididei Y, Flytzanis N and Mertens F G 1995 Physica D: Nonlinear Phenomenon 82 229
[2]	Luo Q H and Melrose D 2007 Planetary & Space Science 55 2281
[3]	Ziegler V, Dinkel J, Setzer C and Lonngren K E 2001 Chaos, Solitons and Fractals 12 1719
[4]	Peng G H 2013 Commun. Nonlinear Appl. Math. Comput. 18 2801
[5]	Salas A H 2010 Appl. Math. Comput. 216 2792
[6]	Anco S C, Mohiuddin M and Wolf T 2012 Appl. Math. Comput. 219 679
[7]	Angulo J, Lopes O and Neves A 2008 Nonlinear Anal. 69 1870
[8]	Song M, Hou X J and Cao J 2011 Appl. Math. Comput. 217 5942
[9]	Anco S C, Ngatat N T and Willoughby M 2011 Physica D: Nonlinear Phonomenon 240 1378
[10]	Guo S M, Mei L Q, Fang Y and Qiu Z Y 2012 Phys. Lett. A 376 158
[11]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[12]	Inc M 2007 Appl. Math. Comput. 184 631
[13]	Seadawy A R 2011 Appl. Math. Comput. 62 3741
[14]	Yin J L and Tian L X 2010 Nonlinear Anal.: Theory, Method and Application 73 465
[15]	Dereli Y 2012 Engineering Analysis with Boundary Elements 36 1416
[16]	Biswas A and Song M 2013 Commun. Nonlinear Sci. Numer. Simul. 18 1676
[17]	Song M, Liu Z R, Zerrad E and Biswas A 2013 Frontiers of Math. China 8 191
[18]	Song M, Bouthina A, Zerrad E and Biswas A 2013 Chaos 23 033115
[19]	Biswas A, Song M and Essaid Z 2013 Int. J. Nonlinear. Sci. Numer. Simul. 14 317
[20]	Song M, Ahmed B and Biswas A 2013 J. Appl. Math. 2013 972416
[21]	Antonova M and Biswas A 2009 Commun. Nonlinear Sci. Numer. Simul. 14 734
[22]	Houria T, Zlatko J and Biswas A 2014 Appl. Math. & Informat. Sci. 8 113
[23]	Polina R, Bouthina A and Biswas A 2014 Appl. Math. & Informat. Sci. 8 485
[24]	Zhang Z Y, Liu Z H, Miao X J and Chen Y Z 2010 Appl. Math. & Comput. 216 3064
[25]	Abdul H K, Houria T and Biswas A 2013 Appl. Math. & Informat. Sci. 7 877
[26]	Zhang Z Y, Li Y X, Liu Z H and Miao X J 2011 Commun. Nonliear Sci. Numer. Simul. 16 3097
[27]	Zhang Z Y, Liu Z H, Miao X J and Chen Y Z 2011 Phys. Lett. A 375 1275
[28]	Miao X J and Zhang Z Y 2011 Commun. Nonlinear Sci. Numer. Simul. 16 4259
[29]	Zhang Z Y, Xia F L and Li X P 2013 Pramana J. Physics 80 41
[30]	Wang Y F, Lou S Y and Qian X M 2010 Chin. Phys. B 19 050202
[31]	Zhang Z Y, Gan X Y, Yu D M, Zhang Y H and Li X P 2012 Commun. Theor. Phys. 57 764
[32]	Wazwaz A M 2004 Appl. Math. & Comput. 147 439
[33]	Wang F Z, Chen Z Q, Wu W J and Yuan Z Z 2007 Chin. Phys. 16 3238

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

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