Algebraic Extensions of Gaudin Models

Musso, Fabio; Petrera, Matteo; Ragnisco, Orlando

doi:10.2991/jnmp.2005.12.s1.39

Cited by 17 publications

(49 citation statements)

References 11 publications

(22 reference statements)

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“…The remarkable feature of the above procedure is that the contracted models inherit the linear r-matrix algebra (2.6) of the ancestor system. The following proposition holds [14,17,18].…”

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

“…In particular we shall describe them in terms of their (linear) r-matrix formulation, providing their Lax matrices and r-matrices. For further details we remand at the references [5,6,8,10,11,17,20,22,24,25,26].…”

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

“…As shown in [17,19,20] one can construct integrable many-body systems starting with the onebody Lax matrices given in equations (3.4), (3.5) and (3.6). Such systems describe completely integrable (long-range) chains of interacting tops on the Lie-Poisson manifold associated with ⊕ M e * (3), being M the number of tops appearing in the chain.…”

Section: Integrable Chains Of Interacting Topsmentioning

confidence: 99%

“…the matrices in equations (2.3), (2.4) and (2.5) with N = 2. To do this a second ingredient is needed: as shown in [14,17,18,20] we have to consider the pole coalescence λ i = ε ν i , i = 1, 2. This fusion procedure can be considered as the analytical counterpart of the algebraic contraction given by the map in equation (3.1).…”

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

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From su(2) Gaudin Models to Integrable Tops

Petrera

Ragnisco

2007

SIGMA

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Abstract. In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax representation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.

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“…The remarkable feature of the above procedure is that the contracted models inherit the linear r-matrix algebra (2.6) of the ancestor system. The following proposition holds [14,17,18].…”

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

Section: Integrable Chains Of Interacting Topsmentioning

confidence: 99%

Section: A Short Review Of Su(2) Gaudin Modelsmentioning

confidence: 99%

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From su(2) Gaudin Models to Integrable Tops

Petrera

Ragnisco

2007

SIGMA

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“…The main results are derived in [5,6]. To this aim, let us briefly review some relevant features of the trigonometric Gaudin model.…”

Section: The Kirchhof F Top By a Contraction Of Trigonometric Gaudin mentioning

confidence: 99%

Bäcklund Transformations for the Kirchhoff Top

Ragnisco¹

2011

SIGMA

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Abstract. We construct Bäcklund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the sl(2) trigonometric Gaudin model. Our BTs are integrable maps providing an exact timediscretization of the system, inasmuch as they preserve both its Poisson structure and its invariants. Moreover, in some special cases we are able to show that these maps can be explicitly integrated in terms of the initial conditions and of the "iteration time" n. Encouraged by these partial results we make the conjecture that the maps are interpolated by a specific one-parameter family of hamiltonian flows, and present the corresponding solution. We enclose a few pictures where the orbits of the continuous and of the discrete flow are depicted.

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An Integrable Discretization of the Rational $${\mathfrak su(2)}$$ Gaudin Model and Related Systems

Petrera

Suris

2008

Commun. Math. Phys.

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The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational su(2) Gaudin model through certain contraction procedures. In the second part, we derive an explicit integrable Poisson map discretizing a particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper. † petrera@ma.tum.de. ⋄ suris@ma.tum.de. 1 H (2) 1 = p, z 1 + = 1 2 z 1 , z 1 . The map (6) for N = 2 coincides with the integrable discretization of the Lagrange top found in [6]: z 0 = z 0 + ε[ p, z 1 ] , z 1 = (1 + ε z 0 ) z 1 (1 + ε z 0 ) −1 ,

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Algebraic Extensions of Gaudin Models

Cited by 17 publications

References 11 publications

From su(2) Gaudin Models to Integrable Tops

From su(2) Gaudin Models to Integrable Tops

Bäcklund Transformations for the Kirchhoff Top

An Integrable Discretization of the Rational $${\mathfrak su(2)}$$ Gaudin Model and Related Systems

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