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Journal articles on the topic 'Bihamiltonian'

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Author: Grafiati

Published: 10 March 2023

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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 (March 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 lay out its connection to Gelfand–Zakharevich bihamiltonian geometry.

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2

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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3

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 (December 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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4

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

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5

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

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6

Carlet, Guido, Hessel Posthuma, and Sergey Shadrin. "Bihamiltonian Cohomology of KdV Brackets." Communications in Mathematical Physics 341, no. 3 (January 2, 2016): 805–19. http://dx.doi.org/10.1007/s00220-015-2540-4.

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7

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

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8

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

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9

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

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10

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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11

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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12

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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13

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

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14

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

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15

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

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16

MOROSI, CARLO, and LIVIO PIZZOCCHERO. "ON THE BIHAMILTONIAN INTERPRETATION OF THE LAX FORMALISM." Reviews in Mathematical Physics 07, no. 03 (April 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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17

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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18

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

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19

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

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20

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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21

Liu, Si-Qi, and Youjin Zhang. "Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case." Communications in Mathematical Physics 324, no. 3 (October 16, 2013): 897–935. http://dx.doi.org/10.1007/s00220-013-1822-y.

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22

Dubrovin, Boris, Si-Qi Liu, and Youjin Zhang. "Bihamiltonian Cohomologies and Integrable Hierarchies II: The Tau Structures." Communications in Mathematical Physics 361, no. 2 (June 14, 2018): 467–524. http://dx.doi.org/10.1007/s00220-018-3176-y.

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23

Marshall, I. D. "The Kowalevski Top: its r-Matrix Interpretation and Bihamiltonian Formulation." Communications in Mathematical Physics 191, no. 3 (February 1, 1998): 723–34. http://dx.doi.org/10.1007/s002200050285.

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24

Falqui, Gregorio, Franco Magri, and Marco Pedroni. "Bihamiltonian Geometry, Darboux Coverings,¶and Linearization of the KP Hierarchy." Communications in Mathematical Physics 197, no. 2 (October 1, 1998): 303–24. http://dx.doi.org/10.1007/s002200050452.

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25

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 (August 1991): 298–302. http://dx.doi.org/10.1016/0370-2693(91)91043-u.

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26

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

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27

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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28

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

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29

Dubrovin, Boris, and Youjin Zhang. "Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation." Communications in Mathematical Physics 198, no. 2 (November 1, 1998): 311–61. http://dx.doi.org/10.1007/s002200050480.

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30

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 (October 2008): 780–837. http://dx.doi.org/10.1016/j.aim.2008.06.009.

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31

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 (June 2010): 063504. http://dx.doi.org/10.1063/1.3431971.

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32

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

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33

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 (August 12, 2004): 581–612. http://dx.doi.org/10.1007/s00220-004-1150-3.

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34

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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35

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

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36

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 (November 1993): 267–88. http://dx.doi.org/10.1007/bf02108075.

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37

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 (June 1992): 89–101. http://dx.doi.org/10.1007/bf00398305.

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38

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 (July 1999): 150–58. http://dx.doi.org/10.1016/s0370-2693(99)00633-4.

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39

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 (January 18, 2006): 1099–114. http://dx.doi.org/10.1088/0305-4470/39/5/006.

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40

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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41

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

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42

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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43

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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44

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

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45

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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