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Journal articles on the topic 'Bihamiltonian'

Author: Grafiati

Published: 10 March 2023

Last updated: 31 July 2025

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1

GUHA, PARTHA. "BIDIFFERENTIAL CALCULI, BICOMPLEX STRUCTURE AND ITS APPLICATION TO BIHAMILTONIAN SYSTEMS." International Journal of Geometric Methods in Modern Physics 03, no. 02 (2006): 209–32. http://dx.doi.org/10.1142/s0219887806001120.

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In this exposition, we study the relationship between the bihamiltonian formalism of completely integrable systems using the bidifferential calculi introduced by Dimakis and Müller-Hoissen in [1] and the bihamiltonian formulation of integrable systems with a finite number of degrees of freedom via the Frölicher–Nijenhuis geometry. This pair of bidifferetial operators are used to construct alternative Lie algebroids as shown by Camacaro and Carinena. We find its connection to Finsler geometry. We also find the dispersionless integrable hierarchies using the bidifferential ideals. Finally, we la

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2

FIGUEROA-O'FARRILL, JOSÉ M., EDUARDO RAMOS, and JAVIER MAS. "INTEGRABILITY AND BIHAMILTONIAN STRUCTURE OF THE EVEN ORDER SKDV HIERARCHIES." Reviews in Mathematical Physics 03, no. 04 (1991): 479–501. http://dx.doi.org/10.1142/s0129055x91000175.

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We study reductions of the even order SKP hierarchy. We prove that these systems are integrable and bihamiltonian. We derive an infinite set of independent polynomial conservation laws, prove their nontriviality, and derive Lenard relations between them. A further reduction of the simplest such hierarchy is identified with the supersymmetric KdV hierarchy of Manin and Radul. We prove that it inherits all the bihamiltonian and integrability properties from the unreduced hierarchy.

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3

Odesskii, A. "Bihamiltonian Elliptic Structures." Moscow Mathematical Journal 4, no. 4 (2004): 941–46. http://dx.doi.org/10.17323/1609-4514-2004-4-4-941-946.

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4

Dinar, Yassir. "Low-dimensional bihamiltonian structures of topological type." Journal of Mathematical Physics 64, no. 3 (2023): 033502. http://dx.doi.org/10.1063/5.0130899.

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We construct local bihamiltonian structures from classical W-algebras associated with non-regular nilpotent elements of regular semisimple type in Lie algebras of types A2 and A3. They form exact Poisson pencils and admit a dispersionless limit, and their leading terms define logarithmic or trivial Dubrovin–Frobenius manifolds. We calculate the corresponding central invariants, which are expected to be constants. In particular, we get Dubrovin–Frobenius manifolds associated with the focused Schrödinger equation and Hurwitz space M0;1,0 and the corresponding bihamiltonian structures of topologi

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5

Casati, Paolo, Gregorio Falqui, Franco Magri та Marco Pedroni. "Bihamiltonian reductions and ωn-algebras". Journal of Geometry and Physics 26, № 3-4 (1998): 291–310. http://dx.doi.org/10.1016/s0393-0440(97)00060-0.

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6

Ibort, A., F. Magri, and G. Marmo. "Bihamiltonian structures and Stäckel separability." Journal of Geometry and Physics 33, no. 3-4 (2000): 210–28. http://dx.doi.org/10.1016/s0393-0440(99)00051-0.

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7

Carlet, Guido, Hessel Posthuma, and Sergey Shadrin. "Bihamiltonian Cohomology of KdV Brackets." Communications in Mathematical Physics 341, no. 3 (2016): 805–19. http://dx.doi.org/10.1007/s00220-015-2540-4.

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8

Jodeit, Max, and Peter J. Olver. "On the equation grad f = M grad g." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 116, no. 3-4 (1990): 341–58. http://dx.doi.org/10.1017/s0308210500031541.

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SynopsisThe system of differential equations ∇f = M∇g, where M is a given square matrix, arises in many contexts. A complete solution to this problem in the case when M is a constant matrix is presented here. Applications to continuum mechanics and biHamiltonian systems are indicated.

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9

Casati, Paolo, and Giovanni Ortenzi. "Bihamiltonian Equations on Polynomial Virasoro Algebras." Journal of Nonlinear Mathematical Physics 13, no. 3 (2006): 352–64. http://dx.doi.org/10.2991/jnmp.2006.13.3.3.

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10

Marmo, G., A. Simoni, and F. Ventriglia. "BiHamiltonian quantum systems and Weyl quantization." Reports on Mathematical Physics 48, no. 1-2 (2001): 149–57. http://dx.doi.org/10.1016/s0034-4877(01)80074-4.

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11

Gelfand, Israel M., and Ilya Zakharevich. "Webs, Veronese curves, and bihamiltonian systems." Journal of Functional Analysis 99, no. 1 (1991): 150–78. http://dx.doi.org/10.1016/0022-1236(91)90057-c.

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12

Liu, Si-Qi, Zhe Wang, and Youjin Zhang. "Super tau-covers of bihamiltonian integrable hierarchies." Journal of Geometry and Physics 170 (December 2021): 104351. http://dx.doi.org/10.1016/j.geomphys.2021.104351.

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13

Turiel, Francisco. "Classification of (1,1) tensor fields and bihamiltonian structures." Banach Center Publications 33, no. 1 (1996): 449–58. http://dx.doi.org/10.4064/-33-1-449-458.

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14

Liu, Si-Qi, and Youjin Zhang. "Deformations of semisimple bihamiltonian structures of hydrodynamic type." Journal of Geometry and Physics 54, no. 4 (2005): 427–53. http://dx.doi.org/10.1016/j.geomphys.2004.11.003.

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15

Izosimov, Anton. "Stability in bihamiltonian systems and multidimensional rigid body." Journal of Geometry and Physics 62, no. 12 (2012): 2414–23. http://dx.doi.org/10.1016/j.geomphys.2012.09.006.

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16

Abellanas, L., and A. Galindo. "A Harry Dym class of bihamiltonian evolution equations." Physics Letters A 107, no. 4 (1985): 159–60. http://dx.doi.org/10.1016/0375-9601(85)90831-x.

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17

MOROSI, CARLO, and LIVIO PIZZOCCHERO. "ON THE BIHAMILTONIAN INTERPRETATION OF THE LAX FORMALISM." Reviews in Mathematical Physics 07, no. 03 (1995): 389–430. http://dx.doi.org/10.1142/s0129055x95000177.

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We propose a general framework for constructing systematically the Lax formulation of the soliton equations using the bi-Hamiltonian formalism. The method is applied to several examples, both classical and supersymmetric.

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18

Rastelli, Giovanni, and Manuele Santoprete. "Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures." Journal of Geometric Mechanics 7, no. 4 (2015): 483–515. http://dx.doi.org/10.3934/jgm.2015.7.483.

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19

Xue, Ting, and Youjin Zhang. "Bihamiltonian Systems of Hydrodynamic Type and Reciprocal Transformations." Letters in Mathematical Physics 75, no. 1 (2006): 79–92. http://dx.doi.org/10.1007/s11005-005-0031-9.

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20

Hua, Zheng, and Alexander Polishchuk. "Elliptic bihamiltonian structures from relative shifted Poisson structures." Journal of Topology 16, no. 4 (2023): 1389–422. http://dx.doi.org/10.1112/topo.12315.

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AbstractIn this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf.

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21

Falqui, Gregorio, Franco Magri, and Marco Pedroni. "Bihamiltonian Geometry and Separation of Variables for Toda Lattices." Journal of Nonlinear Mathematical Physics 8, sup1 (2001): 118–27. http://dx.doi.org/10.2991/jnmp.2001.8.s.21.

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22

FALQUI, Gregorio, Franco MAGRI, and Marco PEDRONI. "Bihamiltonian Geometry and Separation of Variables for Toda Lattices." Journal of Non-linear Mathematical Physics 8, Supplement (2001): 118. http://dx.doi.org/10.2991/jnmp.2001.8.supplement.21.

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23

Liu, Si-Qi, and Youjin Zhang. "Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case." Communications in Mathematical Physics 324, no. 3 (2013): 897–935. http://dx.doi.org/10.1007/s00220-013-1822-y.

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24

Dubrovin, Boris, Si-Qi Liu, and Youjin Zhang. "Bihamiltonian Cohomologies and Integrable Hierarchies II: The Tau Structures." Communications in Mathematical Physics 361, no. 2 (2018): 467–524. http://dx.doi.org/10.1007/s00220-018-3176-y.

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25

Marshall, I. D. "The Kowalevski Top: its r-Matrix Interpretation and Bihamiltonian Formulation." Communications in Mathematical Physics 191, no. 3 (1998): 723–34. http://dx.doi.org/10.1007/s002200050285.

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26

Falqui, Gregorio, Franco Magri, and Marco Pedroni. "Bihamiltonian Geometry, Darboux Coverings,¶and Linearization of the KP Hierarchy." Communications in Mathematical Physics 197, no. 2 (1998): 303–24. http://dx.doi.org/10.1007/s002200050452.

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27

Figueroa-O'Farrill, JoséM, Javier Mas, and Eduardo Ramos. "Bihamiltonian structure of the KP hierarchy and the WKP algebra." Physics Letters B 266, no. 3-4 (1991): 298–302. http://dx.doi.org/10.1016/0370-2693(91)91043-u.

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28

ZHANG, Ling, and Dafeng ZUO. "The super-bihamiltonian reduction on C∞(1, OSP(1|2))." Acta Mathematica Scientia 34, no. 2 (2014): 537–45. http://dx.doi.org/10.1016/s0252-9602(14)60026-6.

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29

Mikhailov, Andrei. "Bihamiltonian structure of the classical superstring in $AdS_5 × S^5$." Advances in Theoretical and Mathematical Physics 14, no. 6 (2010): 1585–620. http://dx.doi.org/10.4310/atmp.2010.v14.n6.a1.

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30

Zakharevich, Ilya. "Kronecker webs, bihamiltonian structures, and the method of argument translation." Transformation Groups 6, no. 3 (2001): 267–300. http://dx.doi.org/10.1007/bf01263093.

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31

Dubrovin, Boris, and Youjin Zhang. "Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation." Communications in Mathematical Physics 198, no. 2 (1998): 311–61. http://dx.doi.org/10.1007/s002200050480.

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32

Dubrovin, Boris, Si-Qi Liu, and Youjin Zhang. "Frobenius manifolds and central invariants for the Drinfeld–Sokolov bihamiltonian structures." Advances in Mathematics 219, no. 3 (2008): 780–837. http://dx.doi.org/10.1016/j.aim.2008.06.009.

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33

Wu, Chao-Zhong, and Dingdian Xu. "Bihamiltonian structure of the two-component Kadomtsev–Petviashvili hierarchy of type B." Journal of Mathematical Physics 51, no. 6 (2010): 063504. http://dx.doi.org/10.1063/1.3431971.

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34

Zhang, Youjin. "Deformations of the Bihamiltonian Structures on the Loop Space of Frobenius Manifolds." Journal of Nonlinear Mathematical Physics 9, sup1 (2002): 243–57. http://dx.doi.org/10.2991/jnmp.2002.9.s1.20.

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35

Khesin, B., A. Levin, and M. Olshanetsky. "Bihamiltonian Structures and Quadratic Algebras in Hydrodynamics and on Non-Commutative Torus." Communications in Mathematical Physics 250, no. 3 (2004): 581–612. http://dx.doi.org/10.1007/s00220-004-1150-3.

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36

Dinar, Yassir Ibrahim. "W-algebras and the equivalence of bihamiltonian, Drinfeld–Sokolov and Dirac reductions." Journal of Geometry and Physics 84 (October 2014): 30–42. http://dx.doi.org/10.1016/j.geomphys.2014.06.003.

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37

Barakat, Aliaa. "On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type." Advances in Mathematics 219, no. 2 (2008): 604–32. http://dx.doi.org/10.1016/j.aim.2008.05.010.

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38

Morosi, Carlo, and Livio Pizzocchero. "On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach." Communications in Mathematical Physics 158, no. 2 (1993): 267–88. http://dx.doi.org/10.1007/bf02108075.

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39

Casati, Paolo, and Marco Pedroni. "Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view." Letters in Mathematical Physics 25, no. 2 (1992): 89–101. http://dx.doi.org/10.1007/bf00398305.

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40

Popowicz, Z. "Odd bihamiltonian structure of new supersymmetric N=2,4 Korteweg de Vries equation and odd SUSY Virasoro-like algebra." Physics Letters B 459, no. 1-3 (1999): 150–58. http://dx.doi.org/10.1016/s0370-2693(99)00633-4.

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41

Aratyn, H., J. F. Gomes, and A. H. Zimerman. "On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type." Journal of Physics A: Mathematical and General 39, no. 5 (2006): 1099–114. http://dx.doi.org/10.1088/0305-4470/39/5/006.

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42

Brouzet, Robert, Pierre Molino, and Francisco Javier Turiel. "Géométrie des systémes bihamiltoniens." Indagationes Mathematicae 4, no. 3 (1993): 269–96. http://dx.doi.org/10.1016/0019-3577(93)90002-g.

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43

RIGAL, M. "Systèmes bihamiltoniens en dimension impaire." Annales Scientifiques de l’École Normale Supérieure 31, no. 3 (1998): 345–59. http://dx.doi.org/10.1016/s0012-9593(98)80138-9.

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44

Gershun, V. D. "Bihamiltonity as the origin of T-duality of the closed string model." Nuclear Physics B - Proceedings Supplements 102-103 (September 2001): 71–76. http://dx.doi.org/10.1016/s0920-5632(01)01538-9.

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45

Liu, Si-Qi, Zhe Wang, and Youjin Zhang. "Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations." Communications in Mathematical Physics, February 18, 2023. http://dx.doi.org/10.1007/s00220-023-04658-0.

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46

Buryak, Alexandr, Paolo Rossi, and Sergey Shadrin. "Towards a bihamiltonian structure for the double ramification hierarchy." Letters in Mathematical Physics 111, no. 1 (2021). http://dx.doi.org/10.1007/s11005-020-01341-6.

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47

Liu, Si-Qi, Zhe Wang, and Youjin Zhang. "Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro Symmetries." Communications in Mathematical Physics, July 18, 2022. http://dx.doi.org/10.1007/s00220-022-04433-7.

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48

Liu, Si-Qi, Zhe Wang, and Youjin Zhang. "Variational Bihamiltonian Cohomologies and Integrable Hierarchies III: Linear Reciprocal Transformations." Communications in Mathematical Physics, August 11, 2023. http://dx.doi.org/10.1007/s00220-023-04817-3.

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