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F -manifolds and integrable systems of hydrodynamic type

Paolo Lorenzoni, Marco Pedroni, Andrea Raimondo (2011)

Archivum Mathematicum

We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F -manifold with compatible connection generalizing a structure introduced by Manin.

Fano manifolds of degree ten and EPW sextics

Atanas Iliev, Laurent Manivel (2011)

Annales scientifiques de l'École Normale Supérieure

O’Grady showed that certain special sextics in 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

Fibration of the phase space for the Korteweg-de Vries equation

Thomas Kappeler (1991)

Annales de l'institut Fourier

In this article we prove that the fibration of L 2 ( S 1 ) by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to N -gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.

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