00807nas a2200157 4500008004300000245008900043210006900132260002800201520027900229100002200508700002100530700002100551700002200572700001900594856003600613 2010 en_Ud 00aOn the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system0 ageometric origin of the biHamiltonian structure of the CalogeroM bOxford University Press3 aWe 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.1 aBartocci, Claudio1 aFalqui, Gregorio1 aMencattini, Igor1 aOrtenzi, Giovanni1 aPedroni, Marco uhttp://hdl.handle.net/1963/380001054nas a2200097 4500008004300000245006500043210005900108520073200167100002100899856003600920 2006 en_Ud 00aOn a Camassa-Holm type equation with two dependent variables0 aCamassaHolm type equation with two dependent variables3 aWe 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.1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/172100738nas a2200109 4500008004300000245003400043210003400077520044300111100002100554700001700575856003600592 2006 en_Ud 00aQuantisation of bending flows0 aQuantisation of bending flows3 aWe 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.1 aFalqui, Gregorio1 aMusso, Fabio uhttp://hdl.handle.net/1963/253700519nas a2200109 4500008004300000245006800043210006300111520016100174100002100335700001700356856003600373 2006 en_Ud 00aOn Separation of Variables for Homogeneous SL(r) Gaudin Systems0 aSeparation of Variables for Homogeneous SLr Gaudin Systems3 aBy 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.1 aFalqui, Gregorio1 aMusso, Fabio uhttp://hdl.handle.net/1963/253800711nas a2200109 4500008004300000245009600043210006900139520031700208100002100525700001900546856003600565 2005 en_Ud 00aGel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited0 aGelfandZakharevich Systems and Algebraic Integrability the Volte3 aIn 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.1 aFalqui, Gregorio1 aPedroni, Marco uhttp://hdl.handle.net/1963/168900797nas a2200121 4500008004300000245007900043210006900122520038600191100002200577700002100599700001900620856003600639 2004 en_Ud 00aA geometric approach to the separability of the Neumann-Rosochatius system0 ageometric approach to the separability of the NeumannRosochatius3 aWe 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.1 aBartocci, Claudio1 aFalqui, Gregorio1 aPedroni, Marco uhttp://hdl.handle.net/1963/254101156nas a2200121 4500008004300000245006500043210006400108260001900172520076900191100002100960700001700981856003600998 2003 en_Ud 00aGaudin models and bending flows: a geometrical point of view0 aGaudin models and bending flows a geometrical point of view bIOP Publishing3 aIn 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.1 aFalqui, Gregorio1 aMusso, Fabio uhttp://hdl.handle.net/1963/288400961nas a2200109 4500008004300000245006400043210006200107260001000169520061500179100002100794856003600815 2003 en_Ud 00aPoisson Pencils, Integrability, and Separation of Variables0 aPoisson Pencils Integrability and Separation of Variables bSISSA3 aIn 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.1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/302600998nas a2200121 4500008004100000245005500041210005400096260001800150520063200168100002100800700001900821856003600840 2003 en d00aSeparation of variables for Bi-Hamiltonian systems0 aSeparation of variables for BiHamiltonian systems bSISSA Library3 aWe 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.1 aFalqui, Gregorio1 aPedroni, Marco uhttp://hdl.handle.net/1963/159800365nas a2200109 4500008004100000245006500041210005500106260001800161100002100179700001900200856003600219 2002 en d00aOn a Poisson reduction for Gel\\\'fand-Zakharevich manifolds0 aPoisson reduction for GelfandZakharevich manifolds bSISSA Library1 aFalqui, Gregorio1 aPedroni, Marco uhttp://hdl.handle.net/1963/160200417nas a2200121 4500008004100000245007300041210006900114260001800183100002100201700001800222700001900240856003600259 2001 en d00aBihamiltonian geometry and separation of variables for Toda lattices0 aBihamiltonian geometry and separation of variables for Toda latt bSISSA Library1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco uhttp://hdl.handle.net/1963/135400848nas a2200109 4500008004300000245006600043210006600109260001900175520048700194100002100681856003600702 2001 en_Ud 00aLax representation and Poisson geometry of the Kowalevski top0 aLax representation and Poisson geometry of the Kowalevski top bIOP Publishing3 aWe 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.1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/324400873nas a2200133 4500008004100000245003800041210003600079260001800115520050900133100002100642700001800663700002200681856003600703 2001 en d00aA note on the super Krichever map0 anote on the super Krichever map bSISSA Library3 aWe 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.1 aFalqui, Gregorio1 aReina, Cesare1 aZampa, Alessandro uhttp://hdl.handle.net/1963/149400454nas a2200133 4500008004100000245007600041210006900117260001800186100002100204700001800225700001900243700002200262856003600284 2000 en d00aA bi-Hamiltonian theory for stationary KDV flows and their separability0 abiHamiltonian theory for stationary KDV flows and their separabi bSISSA Library1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco1 aZubelli, Jorge P. uhttp://hdl.handle.net/1963/135200863nas a2200145 4500008004300000245008500043210006900128260001300197520039100210100002100601700001800622700001900640700002200659856003600681 2000 en_Ud 00aAn elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy0 aelementary approach to the polynomial taufunctions of the KP Hie bSpringer3 aWe 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.1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco1 aZubelli, Jorge P. uhttp://hdl.handle.net/1963/322300781nas a2200133 4500008004300000245011000043210006900153260001300222520032300235100002100558700001800579700001400597856003600611 2000 en_Ud 00aReduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy0 aReduction of biHamiltonian systems and separation of variables a bSpringer3 aWe 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.1 aFalqui, Gregorio1 aMagri, Franco1 aTondo, G. uhttp://hdl.handle.net/1963/321900454nas a2200121 4500008004100000245010700041210006900148260001800217100002100235700001800256700002200274856003600296 2000 en d00aSuper KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy0 aSuper KP equations and Darboux transformations another perspecti bSISSA Library1 aFalqui, Gregorio1 aReina, Cesare1 aZampa, Alessandro uhttp://hdl.handle.net/1963/136700585nas a2200133 4500008004300000245006900043210006700112260002100179520015700200100002100357700001800378700001900396856003600415 1999 en_Ud 00aA bihamiltonian approach to separation of variables in mechanics0 abihamiltonian approach to separation of variables in mechanics bWorld Scientific3 aThis paper is a report on a recent approach to the theory of separability of the Hamilton-Jacobi equations from the viewpoint of bihamiltonian geometry.1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco uhttp://hdl.handle.net/1963/322201398nas a2200133 4500008004100000245006400041210006000105260001300165520099200178100002101170700001801191700001901209856003601228 1999 en d00aThe method of Poisson pairs in the theory of nonlinear PDEs0 amethod of Poisson pairs in the theory of nonlinear PDEs bSpringer3 aThe 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.1 aFalqui, Gregorio1 aMagri, Franco1 aPedroni, Marco uhttp://hdl.handle.net/1963/135000417nas a2200121 4500008004100000245007300041210006900114260001800183100001800201700002100219700001900240856003600259 1999 en d00aA note on fractional KDV hierarchies. II. The bihamiltonian approach0 anote on fractional KDV hierarchies II The bihamiltonian approach bSISSA Library1 aCasati, Paolo1 aFalqui, Gregorio1 aPedroni, Marco uhttp://hdl.handle.net/1963/122001059nas a2200133 4500008004300000245007500043210007000118260001300188520062700201100002100828700001800849700002200867856003600889 1997 en_Ud 00aKrichever maps, Faà di Bruno polynomials, and cohomology in KP theory0 aKrichever maps Faà di Bruno polynomials and cohomology in KP the bSpringer3 aWe 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.1 aFalqui, Gregorio1 aReina, Cesare1 aZampa, Alessandro uhttp://hdl.handle.net/1963/353900394nas a2200109 4500008004100000245008600041210006900127260001000196653002100206100002100227856003600248 1990 en d00aModuli Spaces and Geometrical Aspects of Two-Dimensional Conformal Field Theories0 aModuli Spaces and Geometrical Aspects of TwoDimensional Conforma bSISSA10aAlgebraic curves1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/555200855nas a2200121 4500008004100000245005400041210005300095260003400148520047700182100001800659700002100677856003500698 1990 en d00aN=2 super Riemann surfaces and algebraic geometry0 aN2 super Riemann surfaces and algebraic geometry bAmerican Institute of Physics3 aThe 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.1 aReina, Cesare1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/80700355nas a2200109 4500008004100000245005700041210005500098260001800153100001800171700002100189856003500210 1990 en d00aA note on the global structure of supermoduli spaces0 anote on the global structure of supermoduli spaces bSISSA Library1 aReina, Cesare1 aFalqui, Gregorio uhttp://hdl.handle.net/1963/80600306nas a2200109 4500008004100000245003200041210003100073260001800104100002100122700001800143856003500161 1988 en d00aSusy-curves and supermoduli0 aSusycurves and supermoduli bSISSA Library1 aFalqui, Gregorio1 aReina, Cesare uhttp://hdl.handle.net/1963/761