In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.

中文翻译：

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b族方程的一个例子的爆破解

在本文中，我们研究了一个 b 族方程组的爆破解。利用微分方程的定性理论和动力系统的分岔方法，我们得到五种类型的膨胀解：双曲膨胀解、分数膨胀解、三角膨胀解、第一椭圆膨胀解-up 解决方案，以及第二个椭圆爆炸解决方案。不仅给出了这些爆破解决方案的表达式，而且还发现了它们之间的关系。特别是，发现两个有界孤立解是从椭圆膨胀解中分岔出来的。