The Existence and Bifurcation of Peakon and Anti-Peakon to the n-Degree b-Equation

In this paper, we study the existence and bifurcation of peakon and anti-peakon to the n-degree b-equation with b > 1 and positive integer n. Using qualitative theory and bifurcation method of dynamical systems, for positive wave speed we confirm the following properties: (1)When n = 1, the equation has peakon, but no anti-peakon. (2)When n ≥ 2, the equation has not only peakon but also anti-peakon. There is a bifurcation wave speed for peakon and anti-peakon. (3)When n is even, there exists a maximum wave speed for peakon. (4)When n is odd, there is no maximum wave speed for peakon.