The multiple soliton and peakon solutions of the Dullin-Gottwald-Holm equation

Explicit multi-soliton and multi-peakon solutions of the Dullin-Gottwald-Holm equation are constructed via Darboux transformation and direct computation, respectively. To this end we first map the Dullin-Gottwald-Holm equation to a negative order KdV equation by a reciprocal transformation. Then we use the Darboux matrix approach to derive multi-soliton solutions of the Dullin-Gottwald-Holm equation from the solutions of the negative order KdV equation. Finally, we find multi-peakon solutions of the Dullin-Gottwald-Holm equation in weak sense. For $\alpha=\frac{1}{2}$ and $\alpha\neq \frac{1}{2}$, several types of two-peakon solutions are discussed in detail. Moreover, the dynamic behaviors of the obtained solutions are illustrated through some figures.