Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity. (July 2015)

• Record Type: Journal Article
• Authors: Yin, Jiuli; Xing, Qianqian; Tian, Lixin
• Is Part Of: Chaos, solitons and fractals. Volume 76(2015)
• Page Start: 40
• Page End: 46
• Publication Date: 2015-07
• Subjects: Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2015.03.002
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painlevé method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.