Mokhov, O. I. The Liouville canonical form for compatible nonlocal Poisson brackets of hydrodynamic type and integrable hierarchies. (English. Russian original) Zbl 1043.37049 Funct. Anal. Appl. 37, No. 2, 103-113 (2003); translation from Funkts. Anal. Prilozh. 37, No. 2, 28-40 (2003). Summary: We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov-Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov-Ferapontov brackets. Cited in 4 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions Keywords:metric of constant curvature; integrable hierarchy; system of hydrodynamic type; bi-Hamiltonian system; compatible Poisson brackets; Liouville bracket PDFBibTeX XMLCite \textit{O. I. Mokhov}, Funct. Anal. Appl. 37, No. 2, 103--113 (2003; Zbl 1043.37049); translation from Funkts. Anal. Prilozh. 37, No. 2, 28--40 (2003) Full Text: DOI arXiv