An inverse scattering approach is presented for computing n-peakon solutions of the Degasperis–Procesi equation (a modification of the Camassa–Holm shallow water equation). The associated non-self-adjoint spectral problem is a generalization of the Krein string to a third order case for which the n-peakon dynamics generates an isospectral deformation. As a result one obtains a nontrivial insight into the spectral properties of this non-self-adjoint problem. Moreover, the inverse problem can solved by a method generalizing the continued fraction solution of the inverse problem of the Krein string to a new situation involving simultaneous rational approximations, similar to Pad´e approximants, of two Weyl functions. This is joint work with H. Lundmark (Link¨oping, Sweden).