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Novikov Algebras and a Classification of Multicomponent Camassa–Holm Equations



A class of multicomponent integrable systems associated with Novikov algebras, which interpolate between Korteweg–de Vries (KdV) and Camassa–Holm-type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multicomponent bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated with the Novikov algebras. The related bilinear forms generating cocycles of first, second, and third order are classified. Several examples, including known integrable equations, are presented.