Non-homogeneous 1+0 Hamiltonian operators in Darboux form are characterized by a Lie algebra, a non-degenerate quadratic Casimir, and a 2-cocycle, with an explicit catalogue of all cases through dimension 6.
Classification of degenerate non-homogeneous Hamiltonian operators
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In this paper, the authors investigate non-homogeneous Hamiltonian operators composed of a first-order Dubrovin-Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of equations associated with a class of Hamiltonian scalar equations. Often, the involved operators are degenerate in the first-order term. For this reason, a complete classification of the operators with degenerate leading coefficient in systems of 2 and 3 components is presented.
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Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators
Non-homogeneous 1+0 Hamiltonian operators in Darboux form are characterized by a Lie algebra, a non-degenerate quadratic Casimir, and a 2-cocycle, with an explicit catalogue of all cases through dimension 6.