Paolo Lorenzoni

On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations

We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of sl(2,ℝ). The reduction process is a bi-Hamiltonian reduction that can be canonically performed on every bi-Hamiltonian manifold.

On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper, we are interested in nontrivial bi-Hamiltonian deformations of the Poisson pencil . Deformations are generated by a sequence of vector fields {X2, X3, X4, ...}, where each Xk is homogeneous of degree k with respect to a grading induced by rescaling. Constructing recursively the vector fields Xk, one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality, the set of linear relations expresses the tangency of the vector fields Xk to the symplectic leaves of ω1 and this tangency condition is equivalent to the exactness of the pencil ωλ. Moreover, extending the results of Lorenzoni P (2002 J. Geom. Phys. 44 331–75), we construct the nontrivial deformations of the Poisson pencil ωλ, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of u up to that order.

On the local and nonlocal Camassa–Holm hierarchies

We construct the local and nonlocal conserved densities for the Camassa–Holm equation by solving a suitable Riccati equation. We also define a Kadomtsev–Petviashvili extension for the local Camassa–Holm hierarchy.

F-Manifolds with Eventual Identities, Bidifferential Calculus and Twisted Lenard-Magri Chains

Given an

On the gauge structure of classical mechanics

F

On integrable conservation laws.

-manifold with eventual identities we examine what this structure entails from the point of view of integrable PDEs of hydrodynamic type. In particular, we show that in the semisimple case the characterization of eventual identities recently given by David and Strachan is equivalent to the requirement that

Natural Connections for Semi-Hamiltonian Systems: The Case of the {\epsilon}-System

E\circ

Metastability and dispersive shock waves in the Fermi-Pasta-Ulam system

has vanishing Nijenhuis torsion. Moreover, after having defined new equivalence relations for connections compatible with respect to the

Flat bidifferential ideals and semi-Hamiltonian PDEs

F

Complex reflection groups, logarithmic connections and bi-flat F-manifolds

-product