With bifurcation method of dynamical system, we investigate the travelling wave solutions of the generalized Dullin–Gottwald–Holm equation (G-DGH) with parabolic law nonlinearity. Based on phase portraits, all possible exact expressions of traveling wave solutions are obtained including compactons, peakons, periodic peakons, periodic wave solutions, solitary wave solutions and kink (or anti-kink) wave solutions. The core of bifurcation analysis is the changes of parameters cause the change of topology of the traveling wave system, and so give different exact solutions. Finally, we summarize our results into a theorem.

中文翻译：

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具有抛物线非线性的广义 Dullin-Gottwald-Holm 方程的行波解

用动力系统的分岔法研究了具有抛物线非线性的广义Dullin-Gottwald-Holm方程(G-DGH)的行波解。基于相图，得到了所有可能的行波解的精确表达式，包括压缩波、峰波、周期波峰、周期波解、孤立波解和扭结（或反扭结）波解。分岔分析的核心是参数的变化引起行波系统拓扑结构的变化，从而给出不同的精确解。最后，我们将我们的结果总结为一个定理。