Advertisement

Letters in Mathematical Physics

, Volume 105, Issue 8, pp 1135–1163 | Cite as

On the Bi-Hamiltonian Geometry of WDVV Equations

  • Maxim V. Pavlov
  • Raffaele F. VitoloEmail author
Article

Abstract

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).

Mathematics Subject Classification

37K05 37K10 37K20 37K25 

Keywords

Hamiltonian operator Jacobi identity Monge metric hydrodynamic-type system WDVV equations Casimirs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balandin A.V., Potemin G.V.: On non-degenerate differential-geometric Poisson brackets of third order. Russ. Math. Surv. 56(5), 976–977 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baran, H., Marvan, M.: Jets: Differential calculus on jet spaces and diffieties for Maple 10+. (2010) http://jets.math.slu.cz/
  3. 3.
    Dolgachev, I.V.: Classical algebraic geometry. A modern view, pp. 639. Cambridge University Press, Cambridge (2012)Google Scholar
  4. 4.
    Dorfman I.: Dirac structures and integrability of nonlinear evolution equations. Wiley, England (1993)Google Scholar
  5. 5.
    Doyle P.W.: Differential geometric Poisson bivectors in one space variable. J. Math. Phys. 34(4), 1314–1338 (1993)MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Dubrovin, B.A.: Geometry of 2D topological field theories. Lecture Notes in Math, vol. 1620, pp. 120–348. Springer-Verlag (1996)Google Scholar
  7. 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), 665–669 (1983)zbMATHGoogle Scholar
  8. 8.
    Dubrovin B.A., Novikov S.P.: Poisson brackets of hydrodynamic type. Sov. Math. Dokl. 30(3), 651–2654 (1984)zbMATHGoogle Scholar
  9. 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, 50–70 (1993)MathSciNetADSCrossRefzbMATHGoogle Scholar
  10. 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, 391–397 (1993)MathSciNetADSCrossRefGoogle Scholar
  11. 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), 257–262 (1994)MathSciNetGoogle Scholar
  12. 12.
    Ferapontov E.V.: Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants. Diff. Geom. Appl. 5, 121–152 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ferapontov E.V.: Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems. Diff. Geometry and its Appl. 5, 335–369 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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, 649–669 (1997)MathSciNetADSCrossRefGoogle Scholar
  15. 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), 195–203 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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. 75–91. Birkhäuser. Boston (1996)Google Scholar
  17. 17.
    Ferapontov, E.V., Pavlov, M.V., Vitolo, R.F.: Projective-geometric aspects of homogeneous third-order Hamiltonian operators. J. Geom. Phys. 85, 16–28 (2014). doi: 10.1016/j.geomphys.2014.05.027
  18. 18.
    Ferapontov E.V., Sharipov R.A.: On first-order conservation laws for systems of hydrodynamic type equations. Theor. Math. Phys. 108(1), 937–952 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Getzler E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke J. Math. 111, 535–560 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kalayci J., Nutku Y.: Bi-Hamiltonian structure of a WDVV equation in 2d topological field theory. Phys. Lett. A 227, 177–182 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Kalayci J., Nutku Y.: Alternative bi-Hamiltonian structures for WDVV equations of associativity. J. Phys. A Math. Gen. 31, 723–734 (1998)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Kersten, P., Krasil’shchik, I., Verbovetsky, A.: Hamiltonian operators and \({\ell ^{\ast}}\) -coverings. J. Geom. Phys. 50, 273–302 (2004) arXiv:math/0304245
  23. 23.
    Kersten P., Krasil’shchik I., Verbovetsky A., Vitolo R.: On integrable structures for a generalized Monge-Ampere equation. Theor. Math. Phys. 128(2), 600–615 (2012)MathSciNetGoogle Scholar
  24. 24.
    Magri F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Mokhov O.I.: Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Russ. Math. Surv. 53(3), 515–622 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nutku Y., Pavlov M.V.: Multi Lagrangians for Integrable Systems. J. Math. Phys. 43(3), 1441–1460 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Pavlov M.V., Tsarev S.P.: Tri-hamiltonian structures of egorov systems of hydrodynamic type. Funkts. Anal. Prilozh. 37(1), 38–54 (2003)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Potemin G.V.: On Poisson brackets of differential-geometric type. Sov. Math. Dokl. 33, 30–33 (1986)zbMATHGoogle Scholar
  29. 29.
    Potemin G.V.: On third-order Poisson brackets of differential geometry. Russ. Math. Surv. 52, 617–618 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Potemin, G.V.: Some aspects of differential geometry and algebraic geometry in the theory of solitons. PhD Thesis, Moscow, Moscow State University, pp. 99 (1991)Google Scholar
  31. 31.
    Vitolo, R.F.: CDE: a Reduce package for computations in the geometry of differential equations, software, user guide and examples freely available at http://gdeq.or. See also the Reduce website http://reduce-algebra.sourceforge.net/
  32. 32.
    Vitolo, R.: On the Lagrangian representation of WDVV equations (2015, submitted)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematical PhysicsLebedev Physical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Department of Applied MathematicsNational Research Nuclear University MEPHIMoscowRussia
  3. 3.Department of Mathematics and Physics “E. De Giorgi”University of SalentoLecceItaly

Personalised recommendations