Abstract
The exact solitary wave solutions of simplified modified Camassa–Holm equation with any power are investigated by using the method of undetermined coefficient and qualitative theory of planar dynamical system. The existence and numbers of bell solitary wave solutions, kink solitary wave solutions and periodic wave solutions are analyzed with the help of Maple software and phase portraits. The four new exact expressions of bell solitary wave solutions and kink solitary wave solutions are obtained. By applying the theory of orbital stability proposed by Grillakis, Shatah and Strauss and the explicit expressions of discrimination d′′(c), the wave speed interval of orbital stable and unstable for bell solitary wave solutions with any power are given. Furthermore, we discuss the orbital stability of kink solitary wave solutions with first power and fractional power and deduce the wave speed interval of orbital unstable. Moreover, we simulate numerically the conclusion about orbital stability of the four solitary wave solutions obtained in this paper and show the orbital stable results visually.
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