@article {2010,
title = {On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system},
journal = {Int. Math. Res. Not. (2010) 2010:279-296},
number = {arXiv.org;0902.0953v2},
year = {2010},
publisher = {Oxford University Press},
abstract = {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.},
doi = {10.1093/imrn/rnp130},
url = {http://hdl.handle.net/1963/3800},
author = {Claudio Bartocci and Gregorio Falqui and Igor Mencattini and Giovanni Ortenzi and Marco Pedroni}
}
@article {2006,
title = {On a Camassa-Holm type equation with two dependent variables},
number = {SISSA;35/2005/FM},
year = {2006},
abstract = {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.},
doi = {10.1088/0305-4470/39/2/004},
url = {http://hdl.handle.net/1963/1721},
author = {Gregorio Falqui}
}
@article {2006,
title = {Quantisation of bending flows},
journal = {Czechoslovak Journal of Physics 56 (2006), n. 10-11, 1143-1148},
number = {arXiv.org;nlin/0610003},
year = {2006},
abstract = {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.},
doi = {10.1007/s10582-006-0415-9},
url = {http://hdl.handle.net/1963/2537},
author = {Gregorio Falqui and Fabio Musso}
}
@article {2006,
title = {On Separation of Variables for Homogeneous SL(r) Gaudin Systems},
number = {SISSA;106/2003/FM},
year = {2006},
abstract = {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.},
doi = {10.1007/s11040-006-9012-1},
url = {http://hdl.handle.net/1963/2538},
author = {Gregorio Falqui and Fabio Musso}
}
@article {2005,
title = {Gel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited},
number = {SISSA;29/2005/FM},
year = {2005},
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.},
doi = {10.1070/RD2005v010n04ABEH000322},
url = {http://hdl.handle.net/1963/1689},
author = {Gregorio Falqui and Marco Pedroni}
}
@article {2004,
title = {A geometric approach to the separability of the Neumann-Rosochatius system},
journal = {Differential Geom. Appl. 21 (2004) 349-360},
number = {arXiv.org;nlin/0307021},
year = {2004},
abstract = {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.},
doi = {10.1016/j.difgeo.2004.07.001},
url = {http://hdl.handle.net/1963/2541},
author = {Claudio Bartocci and Gregorio Falqui and Marco Pedroni}
}
@article {2003,
title = {Gaudin models and bending flows: a geometrical point of view},
journal = {J. Phys. A: Math. Gen. 36 (2003) 11655-11676},
number = {SISSA;45/2003/FM},
year = {2003},
publisher = {IOP Publishing},
abstract = {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.},
doi = {10.1088/0305-4470/36/46/009},
url = {http://hdl.handle.net/1963/2884},
author = {Gregorio Falqui and Fabio Musso}
}
@article {2003,
title = {Poisson Pencils, Integrability, and Separation of Variables},
number = {SISSA;90/2003/FM},
year = {2003},
institution = {SISSA},
abstract = {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.},
url = {http://hdl.handle.net/1963/3026},
author = {Gregorio Falqui}
}
@article {2003,
title = {Separation of variables for Bi-Hamiltonian systems},
journal = {Math. Phys. Anal. Geom. 6 (2003) 139-179},
number = {SISSA;27/2002/FM},
year = {2003},
publisher = {SISSA Library},
abstract = {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.},
doi = {10.1023/A:1024080315471},
url = {http://hdl.handle.net/1963/1598},
author = {Gregorio Falqui and Marco Pedroni}
}
@article {2002,
title = {On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds},
journal = {Rep.Math.Phys.50 (2002), no.3, 395},
number = {SISSA;31/2002/FM},
year = {2002},
publisher = {SISSA Library},
doi = {10.1016/S0034-4877(02)80068-4},
url = {http://hdl.handle.net/1963/1602},
author = {Gregorio Falqui and Marco Pedroni}
}
@article {2001,
title = {Bihamiltonian geometry and separation of variables for Toda lattices},
journal = {J. Nonlinear Math. Phys. 8 (2001), suppl., 118-127},
number = {SISSA;139/99/FM},
year = {2001},
publisher = {SISSA Library},
url = {http://hdl.handle.net/1963/1354},
author = {Gregorio Falqui and Franco Magri and Marco Pedroni}
}
@article {2001,
title = {Lax representation and Poisson geometry of the Kowalevski top},
journal = {J. Phys. A 34 (2001) 2077-2085},
number = {SISSA;70/2000/FM},
year = {2001},
publisher = {IOP Publishing},
abstract = {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.},
doi = {10.1088/0305-4470/34/11/301},
url = {http://hdl.handle.net/1963/3244},
author = {Gregorio Falqui}
}
@article {2001,
title = {A note on the super Krichever map},
journal = {J. Geom. Phys. 37 (2001), no. 1-2, 169-181},
number = {SISSA;36/00/FM},
year = {2001},
publisher = {SISSA Library},
abstract = {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{\textquoteleft}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.},
doi = {10.1016/S0393-0440(00)00037-1},
url = {http://hdl.handle.net/1963/1494},
author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa}
}
@article {2000,
title = {A bi-Hamiltonian theory for stationary KDV flows and their separability},
journal = {Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52},
number = {SISSA;137/99/FM},
year = {2000},
publisher = {SISSA Library},
doi = {10.1070/rd2000v005n01ABEH000122},
url = {http://hdl.handle.net/1963/1352},
author = {Gregorio Falqui and Franco Magri and Marco Pedroni and Jorge P. Zubelli}
}
@article {2000,
title = {An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy},
journal = {Theor. Math. Phys. 122 (2000) 17-28},
number = {SISSA;117/98/FM},
year = {2000},
publisher = {Springer},
abstract = {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.},
doi = {10.1007/BF02551166},
url = {http://hdl.handle.net/1963/3223},
author = {Gregorio Falqui and Franco Magri and Marco Pedroni and Jorge P. Zubelli}
}
@article {2000,
title = {Reduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy},
journal = {Theor. Math. Phys. 122 (2000) 176-192},
number = {SISSA;123/98/FM},
year = {2000},
publisher = {Springer},
abstract = {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.},
doi = {10.1007/BF02551195},
url = {http://hdl.handle.net/1963/3219},
author = {Gregorio Falqui and Franco Magri and G. Tondo}
}
@article {2000,
title = {Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy},
journal = {J. Geom. Phys. 35 (2000), no. 2-3, 239-272},
number = {SISSA;152/99/FM},
year = {2000},
publisher = {SISSA Library},
doi = {10.1016/S0393-0440(00)00007-3},
url = {http://hdl.handle.net/1963/1367},
author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa}
}
@inbook {1999,
title = {A bihamiltonian approach to separation of variables in mechanics},
booktitle = {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},
year = {1999},
publisher = {World Scientific},
organization = {World Scientific},
abstract = {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.},
url = {http://hdl.handle.net/1963/3222},
author = {Gregorio Falqui and Franco Magri and Marco Pedroni}
}
@inbook {1999,
title = {The method of Poisson pairs in the theory of nonlinear PDEs},
booktitle = {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)},
number = {SISSA;135/99/FM},
year = {1999},
publisher = {Springer},
organization = {Springer},
abstract = {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.},
doi = {10.1007/b13714},
url = {http://hdl.handle.net/1963/1350},
author = {Gregorio Falqui and Franco Magri and Marco Pedroni}
}
@article {1999,
title = {A note on fractional KDV hierarchies. II. The bihamiltonian approach},
number = {SISSA;6/99/FM},
year = {1999},
publisher = {SISSA Library},
url = {http://hdl.handle.net/1963/1220},
author = {Paolo Casati and Gregorio Falqui and Marco Pedroni}
}
@article {1997,
title = {Krichever maps, Fa{\`a} di Bruno polynomials, and cohomology in KP theory},
journal = {Lett. Math. Phys. 42 (1997) 349-361},
number = {SISSA;37/97/FM},
year = {1997},
publisher = {Springer},
abstract = {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.},
doi = {10.1023/A:1007323118991},
url = {http://hdl.handle.net/1963/3539},
author = {Gregorio Falqui and Cesare Reina and Alessandro Zampa}
}
@mastersthesis {1990,
title = {Moduli Spaces and Geometrical Aspects of Two-Dimensional Conformal Field Theories},
year = {1990},
school = {SISSA},
keywords = {Algebraic curves},
url = {http://hdl.handle.net/1963/5552},
author = {Gregorio Falqui}
}
@article {1990,
title = {N=2 super Riemann surfaces and algebraic geometry},
journal = {J. Math. Phys. 31 (1990), no.4, 948-952},
number = {SISSA;47/89/FM},
year = {1990},
publisher = {American Institute of Physics},
abstract = {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.},
doi = {10.1063/1.528775},
url = {http://hdl.handle.net/1963/807},
author = {Cesare Reina and Gregorio Falqui}
}
@article {1990,
title = {A note on the global structure of supermoduli spaces},
journal = {Comm.Math.Phys. 31 (1990), no.4, 948},
number = {SISSA;46/89/FM},
year = {1990},
publisher = {SISSA Library},
url = {http://hdl.handle.net/1963/806},
author = {Cesare Reina and Gregorio Falqui}
}
@article {1988,
title = {Susy-curves and supermoduli},
number = {SISSA;169/88/FM},
year = {1988},
publisher = {SISSA Library},
url = {http://hdl.handle.net/1963/761},
author = {Gregorio Falqui and Cesare Reina}
}