arXiv subjects

Andrea Savoldi

Publications and source records attributed to Andrea Savoldi.

3 recordsLinked to original sources

arXiv2015

Deformations of non semisimple Poisson pencils of hydrodynamic type

We study deformations of two-component non semisimple Poisson pencils of hydrodynamic type associated with Balinski\vı-Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. It turns out that one function is invariant with respect to the subgroup of Miura transformations preserving the dispersionless limit and another function is related to a one-parameter family of truncated structures. In two expectional cases the second order deformations are parametrized by four functions. Among them two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of deformations of n-component semisimple structures. This example suggests that deformations of non semisimple pencils corresponding to the lifted invariant parameters are unobstructed.

math-ph↗

arXiv2015

Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D

First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics $g$, $\tilde{g}$ and a pair of differential-geometric objects $b$, $\tilde{b}$. If the determinant of the pencil $g+λ\tilde{g}$ vanishes for all $λ$, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.

math-ph↗

arXiv2015

On deformations of one-dimensional Poisson structures of hydrodynamic type with degenerate metric

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this class of deformations is non-trivial, and depends on a certain number of arbitrary functions. This shows that the second Poisson-Lichnerowicz cohomology group does not vanish.

math-ph↗