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Extension on bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. (English) Zbl 1259.35059
Summary: X. Fan et al. [Commun. Nonlinear Sci. Numer. Simul. 16, No. 10, 3956–3963 (2011; Zbl 1232.35125)] studied the bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. They gave a part of possible phase portraits and obtained some uncertain parametric conditions for solitons and kink (antikink) solutions. However, the exact explicit parametric conditions have not been given for the existence of solitons and kink (antikink) solutions. In this paper, we study the bifurcations for the two-component Fornberg-Whitham equation in detalis, present all possible phase portraits, and give the exact explicit parametric conditions for various solutions. In addition, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained. Our results extend the previous study.
MSC:
35C07Traveling wave solutions of PDE
35B32Bifurcation (PDE)