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- Title
Hamiltonian Operators of Dubrovin-Novikov Type in 2D.
- Authors
Ferapontov, Evgeny; Lorenzoni, Paolo; Savoldi, Andrea
- 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).
- Subjects
HAMILTONIAN operator; NOVIKOV conjecture; DIFFERENTIAL algebra; SKEWNESS (Probability theory); JACOBI identity; MATHEMATICAL symmetry
- Publication
Letters in Mathematical Physics, 2015, Vol 105, Issue 3, p341
- ISSN
0377-9017
- Publication type
Article
- DOI
10.1007/s11005-014-0738-6