Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

Sugny, Dominique; Mardešić, Pavao; Pelletier, Michèle; Jébrane, Ahmed; Jauslin, H. R.

doi:10.1063/1.2863614

Cited by 26 publications

(44 citation statements)

References 36 publications

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“…(2). The class of Hamiltonian functions that we consider is defined in §3.2 and it includes as special cases all systems considered in [12,19,22,[27][28][29][30]32].…”

Section: 4mentioning

confidence: 99%

“…For the specific choices of H used in [27,32] the integral map F = (J, H ) has one curve of critical values C − that ends at a point c * when n 1 = 1, or two curves of critical values, C − and C + that join at a point c * when n 1 ≥ 2, see Fig. 3.…”

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confidence: 99%

“…3. Then there is a basis (n 1 n 2 a, b) of an index-n 1 n 2 subgroup H( j, h) of the first homology group H 1 (F −1 ( j, h)) which after parallel transport along Γ comes back to (n 1 n 2 a − b, b) [27,32]. Thus in the basis (n 1 n 2 a, b) the monodromy matrix reads 1 0 1 1 , and formally in the basis (a, b) of H 1 (F −1 ( j, h)) the matrix becomes 1 0…”

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confidence: 99%

“…Fractional monodromy was soon thereafter studied in n 1 : (−n 2 ) resonant systems [27,31,32] for n 1 , n 2 coprime natural numbers with n 1 < n 2 . In such systems the Hamiltonian H has, in suitable coordinates, the integral…”

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confidence: 99%

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Uncovering Fractional Monodromy

Efstathiou

Broer

2013

Commun. Math. Phys.

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Abstract:The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n 1 :(−n 2 ) resonant Hamiltonian systems with n 1 , n 2 coprime natural numbers. We consider, in particular, systems that for n 1 , n 2 > 1 contain one-parameter families of singular fibres which are 'curled tori'. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n 1 , n 2 . Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.

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“…(2). The class of Hamiltonian functions that we consider is defined in §3.2 and it includes as special cases all systems considered in [12,19,22,[27][28][29][30]32].…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Uncovering Fractional Monodromy

Efstathiou

Broer

2013

Commun. Math. Phys.

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“…The aim of the present article is to render these new mathematical tools accessible to a broad audience in the context of nonlinear optics. In this respect, this paper can also be viewed as a pedagogical introduction to this geometric approach of Hamiltonian singularities, which may subsequently h2elp the interested reader to enter into a more specialized literature of Hamiltonian dynamics [28,29,33].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian tools for the analysis of optical polarization control

et al. 2012

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The study of the polarization dynamics of two counterpropagating beams in optical fibers has recently been the subject of a growing renewed interest, from both the theoretical and experimental points of view. This system exhibits a phenomenon of polarization attraction, which can be used to achieve a complete polarization of an initially unpolarized signal beam, almost without any loss of energy. Along the same way, an arbitrary polarization state of the signal beam can be controlled and converted into any other desired state of polarization, by adjusting the polarization state of the counterpropagating pump beam. These properties have been demonstrated in various different types of optical fibers, i.e., isotropic fibers, spun fibers, and telecommunication optical fibers. This article is aimed at providing a rather complete understanding of this phenomenon of polarization attraction on the basis of new mathematical techniques recently developed for the study of Hamiltonian singularities. In particular, we show the essential role that play the peculiar topological properties of singular tori in the process of polarization attraction. We provide here a pedagogical introduction to this geometric approach of Hamiltonian singularities and give a unified description of the polarization attraction phenomenon in various types of optical fiber systems.

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