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 - On a Camassa-Holm type equation with two dependent variables
Y1 - 2006
A1 - Gregorio Falqui
AB - We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced in [16]. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures\\non (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and\\nprovide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.
JF - J. Phys. A 39 (2006) 327-342
UR - http://hdl.handle.net/1963/1721
U1 - 2430
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Quantisation of bending flows
JF - Czechoslovak Journal of Physics 56 (2006), n. 10-11, 1143-1148
Y1 - 2006
A1 - Gregorio Falqui
A1 - Fabio Musso
AB - We briefly review the Kapovich-Millson notion of Bending flows as an integrable system on the space of polygons in ${\\\\bf R}^3$, its connection with a specific Gaudin XXX system, as well as the generalisation to $su(r), r>2$. Then we consider the quantisation problem of the set of Hamiltonians pertaining to the problem, quite naturally called Bending Hamiltonians, and prove that their commutativity is preserved at the quantum level.
UR - http://hdl.handle.net/1963/2537
U1 - 1582
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - RPRT
T1 - On Separation of Variables for Homogeneous SL(r) Gaudin Systems
Y1 - 2006
A1 - Gregorio Falqui
A1 - Fabio Musso
AB - By means of a recently introduced bihamiltonian structure for the homogeneous Gaudin models, we find a new set of Separation Coordinates for the sl(r) case.
JF - Math. Phys. Anal. Geom. 9 (2006), n. 3, 233-262 (2007)
UR - http://hdl.handle.net/1963/2538
U1 - 1581
U2 - Mathematics
U3 - Mathematical Physics
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 - Gaudin models and bending flows: a geometrical point of view
JF - J. Phys. A: Math. Gen. 36 (2003) 11655-11676
Y1 - 2003
A1 - Gregorio Falqui
A1 - Fabio Musso
AB - In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the \\\'standard\\\' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the \\\'bending flows\\\' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.
PB - IOP Publishing
UR - http://hdl.handle.net/1963/2884
U1 - 1816
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - RPRT
T1 - Poisson Pencils, Integrability, and Separation of Variables
Y1 - 2003
A1 - Gregorio Falqui
AB - In this paper we will 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 will discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the \\\\bih structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.
PB - SISSA
UR - http://hdl.handle.net/1963/3026
U1 - 1307
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 - Lax representation and Poisson geometry of the Kowalevski top
JF - J. Phys. A 34 (2001) 2077-2085
Y1 - 2001
A1 - Gregorio Falqui
AB - We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel\\\'fand-Zakharevich bi-Hamiltonian setting for integrable systems.
PB - IOP Publishing
UR - http://hdl.handle.net/1963/3244
U1 - 1457
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A note on the super Krichever map
JF - J. Geom. Phys. 37 (2001), no. 1-2, 169-181
Y1 - 2001
A1 - Gregorio Falqui
A1 - Cesare Reina
A1 - Alessandro Zampa
AB - We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based\\non the Fa`a di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1494
U1 - 2669
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 - JOUR
T1 - Reduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
JF - Theor. Math. Phys. 122 (2000) 176-192
Y1 - 2000
A1 - Gregorio Falqui
A1 - Franco Magri
A1 - G. Tondo
AB - We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel\\\'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.
PB - Springer
UR - http://hdl.handle.net/1963/3219
U1 - 1082
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy
JF - J. Geom. Phys. 35 (2000), no. 2-3, 239-272
Y1 - 2000
A1 - Gregorio Falqui
A1 - Cesare Reina
A1 - Alessandro Zampa
PB - SISSA Library
UR - http://hdl.handle.net/1963/1367
U1 - 3088
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 -
TY - JOUR
T1 - Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory
JF - Lett. Math. Phys. 42 (1997) 349-361
Y1 - 1997
A1 - Gregorio Falqui
A1 - Cesare Reina
A1 - Alessandro Zampa
AB - We study the geometrical meaning of the Faa\\\' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa\\\' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.
PB - Springer
UR - http://hdl.handle.net/1963/3539
U1 - 1162
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - THES
T1 - Moduli Spaces and Geometrical Aspects of Two-Dimensional Conformal Field Theories
Y1 - 1990
A1 - Gregorio Falqui
KW - Algebraic curves
PB - SISSA
UR - http://hdl.handle.net/1963/5552
U1 - 5395
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - JOUR
T1 - N=2 super Riemann surfaces and algebraic geometry
JF - J. Math. Phys. 31 (1990), no.4, 948-952
Y1 - 1990
A1 - Cesare Reina
A1 - Gregorio Falqui
AB - The geometric framework for N=2 superconformal field theories are described by studying susy2 curves-a nickname for N=2 super Riemann surfaces. It is proved that \\\"single\\\'\\\' susy2 curves are actually split supermanifolds, and their local model is a Serre self-dual locally free sheaf of rank two over a smooth algebraic curve. Superconformal structures on these sheaves are then examined by setting up deformation theory as a first step in studying moduli problems.
PB - American Institute of Physics
UR - http://hdl.handle.net/1963/807
U1 - 2984
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A note on the global structure of supermoduli spaces
JF - Comm.Math.Phys. 31 (1990), no.4, 948
Y1 - 1990
A1 - Cesare Reina
A1 - Gregorio Falqui
PB - SISSA Library
UR - http://hdl.handle.net/1963/806
U1 - 2985
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Susy-curves and supermoduli
Y1 - 1988
A1 - Gregorio Falqui
A1 - Cesare Reina
PB - SISSA Library
UR - http://hdl.handle.net/1963/761
U1 - 3030
U2 - Mathematics
U3 - Mathematical Physics
ER -