TY - JOUR
T1 - On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system
JF - Int. Math. Res. Not. (2010) 2010:279-296
Y1 - 2010
A1 - Claudio Bartocci
A1 - Gregorio Falqui
A1 - Igor Mencattini
A1 - Giovanni Ortenzi
A1 - Marco Pedroni
AB - We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed.
PB - Oxford University Press
UR - http://hdl.handle.net/1963/3800
U1 - 8
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - RPRT
T1 - Gel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited
Y1 - 2005
A1 - Gregorio Falqui
A1 - Marco Pedroni
AB - 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.
JF - Regul. Chaotic Dyn. 10 (2005) 399-412
UR - http://hdl.handle.net/1963/1689
U1 - 2444
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A geometric approach to the separability of the Neumann-Rosochatius system
JF - Differential Geom. Appl. 21 (2004) 349-360
Y1 - 2004
A1 - Claudio Bartocci
A1 - Gregorio Falqui
A1 - Marco Pedroni
AB - We study the separability of the Neumann-Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system.
UR - http://hdl.handle.net/1963/2541
U1 - 1578
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Separation of variables for Bi-Hamiltonian systems
JF - Math. Phys. Anal. Geom. 6 (2003) 139-179
Y1 - 2003
A1 - Gregorio Falqui
A1 - Marco Pedroni
AB - We address the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called omega-N manifolds, to give intrisic tests of separability (and Staeckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the omega-N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel\\\'fand-Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1598
U1 - 2520
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds
JF - Rep.Math.Phys.50 (2002), no.3, 395
Y1 - 2002
A1 - Gregorio Falqui
A1 - Marco Pedroni
PB - SISSA Library
UR - http://hdl.handle.net/1963/1602
U1 - 2516
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Bihamiltonian geometry and separation of variables for Toda lattices
JF - J. Nonlinear Math. Phys. 8 (2001), suppl., 118-127
Y1 - 2001
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - Marco Pedroni
PB - SISSA Library
UR - http://hdl.handle.net/1963/1354
U1 - 3101
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A bi-Hamiltonian theory for stationary KDV flows and their separability
JF - Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52
Y1 - 2000
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - Marco Pedroni
A1 - Jorge P. Zubelli
PB - SISSA Library
UR - http://hdl.handle.net/1963/1352
U1 - 3103
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy
JF - Theor. Math. Phys. 122 (2000) 17-28
Y1 - 2000
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - Marco Pedroni
A1 - Jorge P. Zubelli
AB - We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.
PB - Springer
UR - http://hdl.handle.net/1963/3223
U1 - 1078
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - CHAP
T1 - A bihamiltonian approach to separation of variables in mechanics
T2 - Proceedings of the workshop on nonlinearity, integrability and all that : twenty years after NEEDS \\\'79, Lecce, Italy, July 1 - 10, 1999 / ed. by M. Boiti. - Singapore : World Scientific, 2000. - p. 258-266
Y1 - 1999
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - Marco Pedroni
AB - This paper is a report on a recent approach to the theory of separability of the Hamilton-Jacobi equations from the viewpoint of bihamiltonian geometry.
JF - Proceedings of the workshop on nonlinearity, integrability and all that : twenty years after NEEDS \\\'79, Lecce, Italy, July 1 - 10, 1999 / ed. by M. Boiti. - Singapore : World Scientific, 2000. - p. 258-266
PB - World Scientific
UR - http://hdl.handle.net/1963/3222
U1 - 1079
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - CHAP
T1 - The method of Poisson pairs in the theory of nonlinear PDEs
T2 - Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632)
Y1 - 1999
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - Marco Pedroni
AB - The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg-de Vries (KdV) equation.\\nIn these lectures we touch the following subjects:\\ni) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;\\nii) the theoretical basis of the method of Poisson pairs;\\niii) the Gel\\\'fand-Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;\\niv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite-dimensional Grassmannian manifold.\\nv) the reduction technique(s) and its use to construct classes of solutions;\\nvi) the role of the technique of separation of variables in the study of the reduced systems;\\nvii) some relations intertwining the method of Poisson pairs with the method of Lax pairs.
JF - Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632)
PB - Springer
UR - http://hdl.handle.net/1963/1350
U1 - 3105
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A note on fractional KDV hierarchies. II. The bihamiltonian approach
Y1 - 1999
A1 - Paolo Casati
A1 - Gregorio Falqui
A1 - Marco Pedroni
PB - SISSA Library
UR - http://hdl.handle.net/1963/1220
U1 - 2723
U2 - Mathematics
U3 - Mathematical Physics
ER -