On the Bi-Hamiltonian Geometry of WDVV Equations

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• 作者：Maxim V. Pavlov ; Raffaele F. Vitolo
• 关键词：37K05 ; 37K10 ; 37K20 ; 37K25 ; Hamiltonian operator ; Jacobi identity ; Monge metric ; hydrodynamic ; type system ; WDVV equations ; Casimirs
• 刊名：Letters in Mathematical Physics
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：105
• 期：8
• 页码：1135-1163
• 全文大小：626 KB
• 参考文献：1.Balandin A.V., Potemin G.V.: On non-degenerate differential-geometric Poisson brackets of third order. Russ. Math. Surv. 56(5), 976-77 (2001)MathSciNet View Article MATH
  2.Baran, H., Marvan, M.: Jets: Differential calculus on jet spaces and diffieties for Maple 10+. (2010) http://?jets.?math.?slu.?cz/-/span>
  3.Dolgachev, I.V.: Classical algebraic geometry. A modern view, pp. 639. Cambridge University Press, Cambridge (2012)
  4.Dorfman I.: Dirac structures and integrability of nonlinear evolution equations. Wiley, England (1993)
  5.Doyle P.W.: Differential geometric Poisson bivectors in one space variable. J. Math. Phys. 34(4), 1314-338 (1993)MathSciNet ADS View Article MATH
  6.Dubrovin, B.A.: Geometry of 2D topological field theories. Lecture Notes in Math, vol. 1620, pp. 120-48. Springer-Verlag (1996)
  7.Dubrovin B.A., Novikov S.P.: Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method. Sov. Math. Dokl. 27(3), 665-69 (1983)MATH
  8.Dubrovin B.A., Novikov S.P.: Poisson brackets of hydrodynamic type. Sov. Math. Dokl. 30(3), 651-654 (1984)MATH
  9.Ferapontov E.V.: On integrability of \({ 3 \times 3}\) semi-Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants. Physica. D. 63, 50-0 (1993)MathSciNet ADS View Article MATH
  10.Ferapontov E.V.: On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants. Phys. lett. A. 179, 391-97 (1993)MathSciNet ADS View Article
  11.Ferapontov E.V.: Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants. Teor. i Mat. Fiz. 99(2), 257-62 (1994)MathSciNet
  12.Ferapontov E.V.: Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants. Diff. Geom. Appl. 5, 121-52 (1995)MathSciNet View Article MATH
  13.Ferapontov E.V.: Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems. Diff. Geometry and its Appl. 5, 335-69 (1995)MathSciNet View Article MATH
  14.Ferapontov E.V., Galvao C.A.P., Mokhov O.I., Nutku Y.: Bi-Hamiltonian structure of equations of associativity in 2-d topological field theory. Commun. Math. Phys. 186, 649-69 (1997)MathSciNet ADS View Article
  15.Ferapontov E.V., Mokhov O.I.: Equations of associativity of two-dimensional topological field theory as integrable Hamiltonian nondiagonalisable systems of hydrodynamic type. Funct. Anal. Appl. 30(3), 195-03 (1996)MathSciNet View Article MATH
  16.Ferapontov, E.V., Mokhov, O.I.: On the Hamiltonian representation of the associativity equations. In: Gelfand, I.M., Fokas, A.S.(eds.) Algebraic aspects of integrable systems: In memory of Irene Dorfman, pp. 75-1. Birkh?user. Boston (1996)
  17.Ferapontov, E.V., Pavlov, M.V., Vitolo, R.F.: Projective-geometric aspects of homogeneous third-order Hamiltonian operators. J. Geom. Phys. 85, 16-8 (2014). doi:10.-016/?j.?geomphys.-014.-5.-27
  18.Ferapontov E.V., Sharipov R.A.: On first-order conservation laws for systems of hydrodynamic type equations. Theor. Math. Phys. 108(1), 937-52 (1996)MathSciNet View Article MATH
  19.Getzler E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke J. Math. 111, 535-60 (2002)MathSciNet View Article
  20.Kalayci J., Nutku Y.: Bi-Hamiltonian structure of a WDVV equation in 2d topological field theory. Phys. Lett. A 227, 177-82 (1997)MathSciNet ADS View Article MATH
  21.Kalayci J., Nutku Y.: Alternative bi-Hamiltonian structures for WDVV equations of associativity. J. Phys. A Math. Gen. 31, 723-34 (1998)MathSciNet ADS View Article
  22.Kersten, P., Krasil’shchik, I., Verbovetsky, A.: Hamiltonian operators and \({\ell ^{\ast}}\) -coverings. J. Geom. Phys. 50, 273-02 (2004) arXiv:?math/-304245
  23.Kersten P., Krasil’shchik I., Verbovetsky A., Vitolo R.: On integrable structures for a generalized Monge-Ampere equation. Theor. Math. Phys. 128(2), 600-15 (2012)MathSciNet
  24.Magri F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156-162 (1978)MathSciNet ADS View Article
  25.Mokhov O.I.: Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Russ. Math. Surv. 53(3), 515-22 (1998)MathSciNet View Article MATH
  26.Nutku Y., Pavlov M.V.: Multi Lagrangians for Integrable Systems. J. Math. Phys. 43(3), 1441-460 (2002)MathSciNet ADS View Article MATH
  27.Pavlov M.V., Tsarev S.P.: Tri-hamiltonian structures of egorov systems of hydrodynamic type. Funkts. Anal. Prilozh. 37(1), 38-4 (2003)MathSciNet View Article
  28.Potemin G.V.: On Poisson brackets of differential-geometric type. Sov. Math. Dokl. 33, 30-3 (1986)MATH
  29.Potemin G.V.: On third-order Poisson brackets of differential geometry. Russ. Math. Surv.
• 作者单位：Maxim V. Pavlov (1) (2)
  Raffaele F. Vitolo (3)
  
  1. Department of Mathematical Physics, Lebedev Physical Institute of Russian Academy of Sciences, Leninskij Prospekt, 53, 119991, Moscow, Russia
  2. Department of Applied Mathematics, National Research Nuclear University MEPHI, Kashirskoe Shosse 31, 115409, Moscow, Russia
  3. Department of Mathematics and Physics “E. De Giorgi- University of Salento, Lecce, Italy
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Statistical Physics
  Geometry
  Group Theory and Generalizations
  
• 出版者：Springer Netherlands
• ISSN：1573-0530

文摘

We consider the WDVV associativity equations in the four-dimensional case. These nonlinear equations of third order can be written as a pair of six-component commuting two-dimensional non-diagonalizable hydrodynamic-type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin–Novikov type (of first and third order, respectively).