0
2013
Impact Factor

Gregorio Falqui

Via Roberto Cozzi 53, I-20125, Milano
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca

Publications:

Falqui G. G., Pedroni M.
Poisson Pencils, Algebraic Integrability, and Separation of Variables
2011, vol. 16, no. 3-4, pp.  223-244
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 set-up, 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.
Keywords: Hamilton–Jacobi equations, bihamiltonian manifolds, separation of variables, generalized Toda lattices
Citation: Falqui G. G., Pedroni M.,  Poisson Pencils, Algebraic Integrability, and Separation of Variables, Regular and Chaotic Dynamics, 2011, vol. 16, no. 3-4, pp. 223-244
DOI:10.1134/S156035471103004X
Falqui G. G., Perdoni M.
Gel'fand–Zakharevich systems and algebraic integrability: the Volterra lattice revisited
2005, vol. 10, no. 4, pp.  399-412
Abstract
In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton–Jacobi (H–J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov–Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton–Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel'fand–Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H–J equations. We will then show how this setting works by framing results by Veselov and Penskoï about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables.
Keywords: Hamilton–Jacobi equations, bi-Hamiltonian manifolds, separation of variables, generalized Toda lattices
Citation: Falqui G. G., Perdoni M.,  Gel'fand–Zakharevich systems and algebraic integrability: the Volterra lattice revisited , Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 399-412
DOI:10.1070/RD2005v010n04ABEH000322
Falqui G. G., Magri F., Pedroni M., Zubelli J. P.
A Bi-Hamiltonian Theory for Stationary KDV Flows and Their Separability
2000, vol. 5, no. 1, pp.  33-52
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 bi-Hamiltonian formulation. Then we prove the separability of these flows making use of their bi-Hamiltonian structure, and we show that the variables of separation are supplied by the Poisson pair.
Citation: Falqui G. G., Magri F., Pedroni M., Zubelli J. P.,  A Bi-Hamiltonian Theory for Stationary KDV Flows and Their Separability, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 33-52
DOI:10.1070/RD2000v005n01ABEH000122

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