On the tau-functions of the Degasperis-Procesi equation

The DP equation is investigated from the point of view of determinant-pfaffian identities. The reciprocal link between the Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system is used to construct the N-soliton solution of the DP equation. The N-soliton solution of the DP equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations, the identities between determinants and pfaffians, and the $\tau$-functions of the DP equation are obtained from the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system.