Marco Pedroni
Viale Marconi 5, I24044 Dalmine (BG)
Dipartimento di Ingegneria dellâ€™Informazione e Metodi Matematici, Universita di Bergamo
Publications:
Falqui G. G., Pedroni M.
Poisson Pencils, Algebraic Integrability, and Separation of Variables
2011, vol. 16, nos. 34, pp. 223244
Abstract
In this paper we review a recently introduced method for solving the Hamilton–Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical setup, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.

Falqui G. G., Magri F., Pedroni M., Zubelli J. P.
A BiHamiltonian Theory for Stationary KDV Flows and Their Separability
2000, vol. 5, no. 1, pp. 3352
Abstract
We present a fairly new and comprehensive approach to the study of stationary flows of the Korteweg–de Vries hierarchy. They are obtained by means of a double restriction process from a dynamical system in an infinite number of variables. This process naturally provides us with a Lax representation of the flows, which is used to find their biHamiltonian formulation. Then we prove the separability of these flows making use of their biHamiltonian structure, and we show that the variables of separation are supplied by the Poisson pair.

Pedroni M., Vanhaecke P.
A Lie algebraic generalization of the Mumford system, its symmetries and its multihamiltonian structure
1998, vol. 3, no. 3, pp. 132160
Abstract
In this paper we generalize the Mumford system which describes for any fixed $g$ all linear flows on all hyperelliptic Jacobians of dimension $g$. The phase space of the Mumford system consists of triples of polynomials, subject to certain degree constraints, and is naturally seen as an affine subspace of the loop algebra of $\mathfrak{sl}(2)$. In our generalizations to an arbitrary simple Lie algebra $\mathfrak{g}$ the phase space consists of $\mathrm{dim}\,\mathfrak{g}$ polynomials, again subject to certain degree constraints. This phase space and its multiHamiltonian structure is obtained by a Poisson reduction along a subvariety $N$ of the loop algebra $\mathfrak{g}((\lambda1))$ of $\mathfrak{g}$. Since $N$ is not a Poisson subvariety for the whole multiHamiltonian structure we prove an algebraic. Poisson reduction theorem for reduction along arbitrary subvarieties of an affine Poisson variety; this theorem is similar in spirit to the Marsden–Ratiu reduction theorem. We also give a different perspective on the multiHamiltonian structure of the Mumford system (and its generalizations) by introducing a master symmetry; this master symmetry can be described on the loop algebra $\mathfrak{g}((\lambda1))$ as the derivative in the direction of $\lambda$ and is shown to survive the Poisson reduction. When acting (as a Lie derivative) on one of the Poisson structures of the system it produces a next one, similarly when acting on one of the Hamiltonians (in involution) or their (commuting) vector fields it produces a next one. In this way we arrive at several multiHamiltonian hierarchies, built up by a master symmetry.
