In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line.Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties:(i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation.Secondly, we perform numerical simulations to visualize the above properties.Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

中文翻译：

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一个简单方程的孤立波的分岔

在本文中，我们考虑一个简单的方程，它包含一个参数[公式：见正文]，它的行波系统有一条奇异线。首先，利用微分方程的定性理论和动力系统的分岔方法，我们证明了峰孤波和谷孤波的存在与分岔。特别地，我们发现了以下新的性质：（i）在行波系统中，存在无限多的周期轨道，相交于一点或两点并通过奇异线，并且在同宿轨道内没有奇异点。(ii) 当[公式：见正文]时，方程中存在三种谷孤波分岔，包括周期波、爆破波和双孤波。(iii) 当[公式：见正文]时，