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The Traveling Wave Solutions and Their Bifurcations for the BBM-Like 𝐵(𝑚, 𝑛) Equations

Shaoyong Li
Organization: School of Mathematics and Information Sciences, Shaoguan University
Zhengrong Liu
Journal / Anthology

Journal of Applied Mathematics
Year: 2013

We investigate the travelingwave solutions and their bifurcations for the BBM-like 𝐵(𝑚, 𝑛) equations 𝑢 𝑡 +𝛼𝑢 𝑥 +𝛽(𝑢𝑚) 𝑥 −𝛾(𝑢𝑛) 𝑥𝑥𝑡 = 0 by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like B(3, 2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like B(4, 2) equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

*Applied Mathematics