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

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

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.

Keywords: Melnikov method compacted solitary waves control threshold

Received: 12 December 2013
Revised: 12 February 2014
Accepted manuscript online:

PACS:	02.30.Jr	(Partial differential equations)
	02.60.Lj	(Ordinary and partial differential equations; boundary value problems)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11101191).

Corresponding Authors: Yin Jiu-Li
E-mail: yjl@ujs.edu.cn

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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 2014 Chin. Phys. B 23 080201

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

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