  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.
