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Uspekhi Mat. Nauk, 1998, Volume 53, Issue 3(321), Pages 85–192 (Mi umn19)  

This article is cited in 48 scientific papers (total in 48 papers)

Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies


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English version:
Russian Mathematical Surveys, 1998, 53:3, 515–622

Bibliographic databases:

UDC: 517.9+514.7+515.16
MSC: 53D05, 53D17, 37K10
Received: 10.02.1998

Citation: O. I. Mokhov, “Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems”, Uspekhi Mat. Nauk, 53:3(321) (1998), 85–192; Russian Math. Surveys, 53:3 (1998), 515–622

Citation in format AMSBIB
\by O.~I.~Mokhov
\paper Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems
\jour Uspekhi Mat. Nauk
\yr 1998
\vol 53
\issue 3(321)
\pages 85--192
\jour Russian Math. Surveys
\yr 1998
\vol 53
\issue 3
\pages 515--622

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    This publication is cited in the following articles:
    1. O. I. Mokhov, “Compatible Poisson Structures of Hydrodynamic Type and Associativity Equations”, Proc. Steklov Inst. Math., 225 (1999), 269–284  mathnet  mathscinet  zmath
    2. Kholodenko, AL, “Boundary conformal field theories, limit sets of Kleinian groups and holography”, Journal of Geometry and Physics, 35:2–3 (2000), 193  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. O. I. Mokhov, “Compatible and Almost Compatible Pseudo-Riemannian Metrics”, Funct. Anal. Appl., 35:2 (2001), 100–110  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. V. Balandin, G. V. Potëmin, “On non-degenerate differential-geometric Poisson brackets of third order”, Russian Math. Surveys, 56:5 (2001), 976–977  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. O. I. Mokhov, “Compatible Dubrovin–Novikov Hamiltonian operators and the Lie derivative”, Russian Math. Surveys, 56:6 (2001), 1175–1176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Maltsev A.Y., Novikov S.P., “On the local systems Hamiltonian in the weakly non-local Poisson brackets”, Physica D, 156:1–2 (2001), 53–80  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. O. I. Mokhov, “Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics”, Theoret. and Math. Phys., 130:2 (2002), 198–212  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. O. I. Mokhov, “Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them”, Theoret. and Math. Phys., 132:1 (2002), 942–954  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. O. I. Mokhov, “Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type”, Theoret. and Math. Phys., 133:2 (2002), 1557–1564  mathnet  crossref  crossref  mathscinet  isi  elib
    10. O. I. Mokhov, “The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies”, Funct. Anal. Appl., 37:2 (2003), 103–113  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. O. I. Mokhov, “Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature”, Theoret. and Math. Phys., 136:1 (2003), 908–916  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. Lorenzoni, P, “A bi-Hamiltonian approach to the sine-Gordon and Liouville hierarchies”, Letters in Mathematical Physics, 67:2 (2004), 83  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    13. A Ya Maltsev, “Weakly nonlocal symplectic structures, Whitham method and weakly nonlocal symplectic structures of hydrodynamic type”, J Phys A Math Gen, 38:3 (2005), 637  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    14. Jen-Hsu Chang, “On the waterbag model of the dispersionless KP hierarchy (II)”, J Phys A Math Theor, 40:43 (2007), 12973  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. O. Bogoyavlenskij, P. Reynolds, “Form-invariant Poisson brackets of hydrodynamic type with several spatial variables”, J Math Phys (N Y ), 49:5 (2008), 053520  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    16. O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    17. Victor D. Gershun, “Integrable String Models in Terms of Chiral Invariants of $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{SP}(n)$ Groups”, SIGMA, 4 (2008), 041, 16 pp.  mathnet  crossref  mathscinet  zmath
    18. J. T. Ferguson, “Flat pencils of symplectic connections and Hamiltonian operators of degree 2”, Journal of Geometry and Physics, 58:4 (2008), 468  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    19. O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234  mathnet  crossref  mathscinet  zmath  isi  elib
    20. Ferguson, JT, “SECOND-ORDER DEFORMATIONS OF HYDRODYNAMIC-TYPE Poisson BRACKETS”, Glasgow Mathematical Journal, 51A (2009), 75  crossref  mathscinet  zmath  isi  scopus  scopus
    21. Bogoyavlenskij O.I., Reynolds A.P., “Criteria for Existence of a Hamiltonian Structure”, Regular & Chaotic Dynamics, 15:4–5 (2010), 431–439  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    22. O. I. Mokhov, “Deformations of Poisson Structures by Closed $3$-Forms”, Math. Notes, 89:6 (2011), 899–902  mathnet  crossref  crossref  mathscinet  isi
    23. A.P. Reynolds, O.I. Bogoyavlenskij, “Lie algebra structures for four-component Hamiltonian hydrodynamic type systems”, Journal of Geometry and Physics, 61:12 (2011), 2400  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    24. O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    25. E. Parodi, “On classification of discrete, scalar-valued Poisson brackets”, Journal of Geometry and Physics, 62:10 (2012), 2059  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    26. Vitagliano L., “Partial Differential Hamiltonian Systems”, Can. J. Math.-J. Can. Math., 65:5 (2013), 1164–1200  crossref  mathscinet  zmath  isi  scopus  scopus
    27. E.V. Ferapontov, M.V. Pavlov, R.F. Vitolo, “Projective-geometric aspects of homogeneous third-order Hamiltonian operators”, Journal of Geometry and Physics, 2014  crossref  mathscinet  isi  scopus  scopus
    28. D. J. Cirilo-Lombardo, V. D. Gershun, “Integrable hydrodynamic equations for initial chiral currents and infinite hydrodynamic chains from WZNW model and string model of WZNW type with SU(2), SO(3), SP(2), SU(∞), SO(∞), SP(∞) constant torsions”, Int. J. Mod. Phys. A, 29:24 (2014), 1450134  crossref  zmath  isi  scopus  scopus
    29. E.V.. Ferapontov, Paolo Lorenzoni, Andrea Savoldi, “Hamiltonian Operators of Dubrovin-Novikov Type in 2D”, Lett Math Phys, 2014  crossref  mathscinet  isi  scopus  scopus
    30. Andrea Savoldi, “Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D”, J. Phys. A: Math. Theor, 48:26 (2015), 265202  crossref  mathscinet  zmath  isi  scopus  scopus
    31. M.V.. Pavlov, R.F.. Vitolo, “On the Bi-Hamiltonian Geometry of WDVV Equations”, Lett Math Phys, 2015  crossref  mathscinet  isi  scopus  scopus
    32. Mikhail B. Sheftel, Devrim Yazici, “Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański”, SIGMA, 12 (2016), 091, 17 pp.  mathnet  crossref
    33. Savoldi A., “On deformations of one-dimensional Poisson structures of hydrodynamic type with degenerate metric”, J. Geom. Phys., 104 (2016), 246–276  crossref  mathscinet  zmath  isi  elib  scopus
    34. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182  mathnet
    35. Pavlov M.V. Vitolo R.F., “Remarks on the Lagrangian representation of bi-Hamiltonian equations”, J. Geom. Phys., 113 (2017), 239–249  crossref  mathscinet  zmath  isi  scopus
    36. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    37. Dynnikov I.A., Glutsyuk A.A., Mironov A.E., Taimanov I.A., Vesnin A.Yu., “The Conference “Dynamics in Siberia”, Novosibirsk, February 26 - March 4, 2017”, Sib. Electron. Math. Rep., 14 (2017), A7–A30  mathnet  crossref  mathscinet  isi
    38. O. I. Mokhov, N. A. Strizhova, “Classification of the associativity equations possessing a Hamiltonian structure of Dubrovin–Novikov type”, Russian Math. Surveys, 73:1 (2018), 175–177  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    39. Lorenzoni P. Savoldi A. Vitolo R., “Bi-Hamiltonian Structures of KdV Type”, J. Phys. A-Math. Theor., 51:4 (2018), 045202  crossref  mathscinet  zmath  isi  scopus  scopus
    40. Ferapontov E.V. Pavlov M.V. Vitolo R.F., “Systems of Conservation Laws With Third-Order Hamiltonian Structures”, Lett. Math. Phys., 108:6 (2018), 1525–1550  crossref  mathscinet  zmath  isi  scopus  scopus
    41. O. I. Mokhov, N. A. Pavlenko, “Classification of the associativity equations with a first-order Hamiltonian operator”, Theoret. and Math. Phys., 197:1 (2018), 1501–1513  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    42. O. I. Mokhov, N. A. Strizhova, “Liouville integrability of the reduction of the associativity equations on the set of stationary points of an integral in the case of three primary fields”, Russian Math. Surveys, 74:2 (2019), 369–371  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. Casati M. Ferapontov E.V. Pavlov M.V. Vitolo R.F., “On a Class of Third-Order Nonlocal Hamiltonian Operators”, J. Geom. Phys., 138 (2019), 285–296  crossref  mathscinet  zmath  isi  scopus
    44. Odesskii A., “Poisson Structures on Loop Spaces of Cpn and An R-Matrix Associated With the Universal Elliptic Curve”, J. Geom. Phys., 140 (2019), 152–160  crossref  mathscinet  isi  scopus
    45. Ontani R., “Some Remarks on the Operator Formalism For Nonlocal Poisson Brackets”, J. Math. Phys., 61:9 (2020), 093506  crossref  isi
    46. Meleshko S.V., “Complete Group Classification of the Two -Dimensional Shallow Water Equations With Constant Coriolis Parameter in Lagrangian Coordinates”, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105293  crossref  isi
    47. Nakpim W., Meleshko S.V., “Conservation Laws of the One-Dimensional Equations of Relativistic Gas Dynamics in Lagrangian Coordinates”, Int. J. Non-Linear Mech., 124 (2020), 103496  crossref  isi
    48. V. M. Buchstaber, A. V. Mikhailov, “Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves”, Russian Math. Surveys, 76:4 (2021), 587–652  mathnet  crossref  crossref  isi
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