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 \(q_k\), with \(k=1,...,N\), equipped with the co-metric \(g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)\). This flow is generated by a symmetry of the integrable partial diffe...

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Bibliographic Details
Published inarXiv.org
Main Authors Holm, Darryl D, Qiao, Zhijun
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.03.2023
Subjects
Eigenvalues
Partial differential equations
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Summary: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 \(g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)\). This flow is generated by a symmetry of the integrable partial differential equation (pde) \(m_t+um_x+3mu_x=0, m=u-\alpha^2u_{xx}\) (\(\al \) 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.
ISSN:2331-8422