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Coexistence of multifarious explicit nonlinear wave solutions for modified forms of Camassa-Holm and Degaperis-Procesi equations. (English) Zbl 1176.34005
Summary: We consider two nonlinear equations ${u}_{t}-{u}_{xxt}+3{u}^{2}{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx}$ and ${u}_{t}-{u}_{xxt}+4{u}^{2}{u}_{x}=3{u}_{x}{u}_{xx}+u{u}_{xxx}$ which are called modified forms of Camassa-Holm and Degaperis-Procesi equations, respectively. For given constant wave speed, we investigate the coexistence of multifarious explicit nonlinear wave solutions, solitary wave solution, peakon wave solution, singular wave solution, smooth periodic wave solution and singular periodic wave solution. Not only is the coexistence shown, but the concrete expressions are presented via the bifurcation method of dynamical systems. From our work it can be seen that these two equations possess a lot of similar properties, and the types of their explicit nonlinear wave solutions are more than that of Camassa-Holm and Degaperis-Procesi equations. Many previous results are our special cases.
##### MSC:
 34A05 Methods of solution of ODE 35Q53 KdV-like (Korteweg-de Vries) equations 34C27 Almost and pseudo-almost periodic solutions of ODE 35Q51 Soliton-like equations 34C25 Periodic solutions of ODE