New integrable hierarchies from vertex operator representations of polynomial Lie algebras

Casati, Paolo; Ortenzi, Giovanni

doi:10.1016/j.geomphys.2005.02.010

Cited by 34 publications

(33 citation statements)

References 21 publications

(48 reference statements)

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“…[29] (for the particular value of k ¼ 0) (see also [30]) and is also included in the interesting study that relates integrable hierarchies with polynomial Lie algebras [31].…”

Section: A Parametric Coupled Kdv Systemmentioning

confidence: 99%

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Restuccia¹,

Sotomayor²

2017

Lagrangian Mechanics

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We present a general procedure to obtain the Lagrangian and associated Hamiltonian structure for integrable systems of the Helmholtz type. We present the analysis for coupled Korteweg-de Vries systems that are extensions of the Korteweg-de Vries equation. Starting with the system of partial differential equations it is possible to follow the Helmholtz approach to construct one or more Lagrangians whose stationary points coincide with the original system. All the Lagrangians are singular. Following the Dirac approach, we obtain all the constraints of the formulation and construct the Poisson bracket on the physical phase space via the Dirac bracket. We show compatibility of some of these Poisson structures. We obtain the Gardner ε-deformation of these systems and construct a master Lagrangian which describe the coupled systems in the weak ε-limit and its modified version in the strong ε-limit.

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“…[29] (for the particular value of k ¼ 0) (see also [30]) and is also included in the interesting study that relates integrable hierarchies with polynomial Lie algebras [31].…”

Section: A Parametric Coupled Kdv Systemmentioning

confidence: 99%

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Restuccia¹,

Sotomayor²

2017

Lagrangian Mechanics

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“…The system (1),(2) (introduced in [26]) for λ = corresponds to the ninth Hirota-Satsuma [1] coupled KdV system given in [6] (for the particular value of k = ) (see also [5]) and is also included in the interesting study which relates integrable hierarchies with polynomial Lie algebras [7].…”

Section: The Parametric Coupled Kdv Systemmentioning

confidence: 99%

“…Coupled KdV systems were also analyzed in [5][6][7]. An important area of interest of high energy physics related to coupled systems is provided by the supersymmetric extensions of KdV equations [8][9][10][11][12][13][14] and more generally by operator and Cli ord valued extensions of KdV equation [15,16].…”

Section: Introductionmentioning

confidence: 99%

Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

Restuccia

Sotomayor

2016

Open Physics

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Abstract:We obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four di erent real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose eld equations are the ϵ-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limit ϵ → the lagrangians reduce to the ones of the coupled KdV system while, after a suitable rede nition of the elds, in the strong limit ϵ → ∞ we obtain the lagrangians of the coupled modi ed KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ϵ → and ϵ → ∞.

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“…We follow the description of Casati and Ortenzi [1] and introduce 'polynomial Lie algebras'. For any positive integer n, let C (n) (λ) be the commutative algebra C[λ]/(λ) n+1 .…”

mentioning

confidence: 99%

“…Following [1] we define a representation of this algebra on n + 1 copies of the an infinite wedge space:…”

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confidence: 99%

Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra gl∞(n)

Leur

2007

Journal of Geometry and Physics

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New integrable hierarchies from vertex operator representations of polynomial Lie algebras

Cited by 34 publications

References 21 publications

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra gl∞(n)

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