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 $n\le 4$ (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 $H^{\ast }(C{P}^{n-1})$.

40 pages