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38
Euler equations on homogeneous spaces and Virasoro Orbits
 TO APPEAR IN ADV. MATH.
, 2002
"... We show that the following three systems related to various hydrodynamical approximations: the Korteweg–de Vries equation, the Camassa–Holm equation, and the Hunter–Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the ..."
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Cited by 46 (2 self)
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We show that the following three systems related to various hydrodynamical approximations: the Korteweg–de Vries equation, the Camassa–Holm equation, and the Hunter–Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different rightinvariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold’s approach to the Euler equations as geodesic flows of onesided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 44 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 38 (0 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Geometric phases, reduction and LiePoisson structure for the resonant threewave interaction
 Physica D
, 1998
"... Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These resul ..."
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Cited by 16 (5 self)
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Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the threewave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.
The symmetric representation of the rigid body equations and their discretization
 141–71 FEDEROV Y N 2005 INTEGRABLE FLOWS AND BACKLUND TRANSFORMATIONS ON EXTENDED STIEFEL VARIETIES WITH APPLICATION TO THE EULER TOP ON THE LIE GROUP SO(3) PREPRINT NLIN.SI/0505045 AQ2 GELFAND I M AND FOMIN S V 2000 CALCULUS OF VARIATIONS TRANSLATED BY R
, 1998
"... This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian pr ..."
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Cited by 11 (6 self)
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This paper analyzes continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the MoserVeselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in the present paper may be found in Bloch, Crouch, Marsden and Ratiu [1998].
Noncommutative and commutative integrability of generic Toda flows in simple Lie algebra
 Comm. Pure and Appl. Math
, 1999
"... 1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in th ..."
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Cited by 9 (2 self)
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1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s
Tri–hamiltonian vector fields, spectral curves, and separation coordinates
 Rev. Math. Phys. 14 (2002
"... We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro–geometr ..."
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Cited by 8 (2 self)
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We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro–geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1 − λP0) and (P2 − µP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The frameworks is applied to Lax equations with spectral parameter, for which not only it unifies the separation techniques of Sklyanin and of Magri, but also provides a more efficient “inverse ” procedure not involving the extraction of roots. 1
Lagrangian and Symplectic Techniques in Discrete Mechanics
, 1996
"... OF THE DISSERTATION Lagrangian and symplectic techniques in discrete mechanics by James William Gilliam Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, August, 1996 Professor John C. Baez, Chairperson By a "discrete mechanical system," we mean a system in w ..."
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Cited by 6 (0 self)
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OF THE DISSERTATION Lagrangian and symplectic techniques in discrete mechanics by James William Gilliam Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, August, 1996 Professor John C. Baez, Chairperson By a "discrete mechanical system," we mean a system in which time evolution proceeds in integer steps and the configuration space is a finite set. The most widely studied examples are cellular automata on finite lattices. We develop algebraic analogs of many of the basic concepts from differential geometry to overcome obstacles created by working over algebraic structures. We extend the methods of Lagrangian mechanics to treat discrete mechanical systems. In particular, we derive an algebraic analog of the EulerLagrange equation for discrete mechanics, starting from a variational principle. As an example of how this analog works, we prove a version of Noether's theorem applicable to this context. We relate our framework to Hamiltonian mechanics,...
COMPLETE FAMILIES OF COMMUTING FUNCTIONS FOR COISOTROPIC HAMILTONIAN ACTIONS
, 2005
"... In this paper we study symplectic and Poisson algebraic varieties. Let us start this the main definitions. Definition 1. LetA be a commutative associative algebra equipped with an additional anticommuting operation { , }:A×A →A called a Poisson bracket such that ..."
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Cited by 6 (2 self)
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In this paper we study symplectic and Poisson algebraic varieties. Let us start this the main definitions. Definition 1. LetA be a commutative associative algebra equipped with an additional anticommuting operation { , }:A×A →A called a Poisson bracket such that