Hamiltonian Operators of Dubrovin-Novikov Type in 2D

Abstract

First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics ${g}$ and ${\tilde g}$ which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement that ${\tilde g}$ is a linear Killing tensor of g with zero Nijenhuis torsion. This allowed us to obtain a complete classification of n-component operators with n ≤ 4 (for n = 1, 2 this was done before). For 2D operators the Darboux theorem does not hold: the operator may not be reducible to constant coefficient form. All interesting (non-constant) examples correspond to the case when the flat pencil ${g, \tilde g}$ is not semisimple, that is, the affinor ${\tilde g g^{-1}}$ has non-trivial Jordan block structure. In the case of a direct sum of Jordan blocks with distinct eigenvalues, we obtain a complete classification of Hamiltonian operators for any number of components n, revealing a remarkable correspondence with the class of trivial Frobenius manifolds modelled on H ^*(CP ^n-1).