R-matrix for a geodesic flow associated with a new integrable peakon equation

We use the r-matrix formulation to show the integrability of geodesic flow on an N -dimensional space with coordinates qk, with k = 1, ..., N , equipped with the co-metric gij = e−|qi−qj |(2 − e−|qi−qj |). This flow is generated by a symmetry of the integrable partial differential equation (pde) mt + umx + 3mux = 0, m = u − αuxx (α is a constant). This equation – called the Degasperis-Procesi (DP) equation – was recently proven to be completely integrable and possess peakon solutions by Degasperis, Holm and Hone [5]. The isospectral eigenvalue problem associated with the integrable DP equation is used to find a new Lmatrix, called the Lax matrix, for the geodesic dynamical flow. By 1 employing this Lax matrix we obtain the r-matrix for the integrable geodesic flow.