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

Abstract

We use the r-matrix formulation to show the integrability of geodesic flow on an $N$-dimensional space with coordinates $q_k$, with $k=1,...,N$, equipped with the co-metric $g^{ij}=e^{-|q_i-q_j|}(2-e^{-|q_i-q_j|})$. This flow is generated by a symmetry of the integrable partial differential equation (pde) $m_t+u{m}_x+3m{u}_x=0,m=u-{\alpha }^2{u}_{xx}$ ($\al $ 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 (DHH[2002]). The isospectral eigenvalue problem associated with the integrable DP equation is used to find a new $L$-matrix, called the Lax matrix, for the geodesic dynamical flow. By employing this Lax matrix we obtain the $r$-matrix for the integrable geodesic flow.

Some errors in the previous versions have been corrected

References in corpus (1)

• An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion