Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 062, 14 pages      math-ph/0612048
Contribution to the Vadim Kuznetsov Memorial Issue

Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

Artur Sergyeyev
Mathematical Institute, Silesian University in Opava, Na Rybnícku 1, 746 01 Opava, Czech Republic

Received December 15, 2006, in final form April 23, 2007; Published online April 26, 2007

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure J can be written as the Lie derivative of J −1 along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamiltonian structure of zero or first order, including Hamiltonian operators of the Dubrovin-Novikov type.

Key words: weakly nonlocal Hamiltonian structure; symplectic structure; Lie derivative.

pdf (288 kb)   ps (182 kb)   tex (18 kb)


  1. Blaszak M., Multi-Hamiltonian theory of dynamical systems, Springer, Heidelberg, 1998.
  2. Bocharov A.V. et al., Symmetries and conservation laws for differential equations of mathematical physics, American Mathematical Society, Providence, RI, 1999.
  3. Cooke D.B., Compatibility conditions for Hamiltonian pairs, J. Math. Phys. 32 (1991), no. 11, 3071-3076.
  4. Degiovanni L., Magri F., Sciacca V., On deformation of Poisson manifolds of hydrodynamic type, Comm. Math. Phys. 253 (2005), 1-24, nlin.SI/0103052.
  5. Dorfman I., Dirac structures and integrability of nonlinear evolution equations, John Wiley & Sons, Chichester, 1993.
  6. Dubrovin B.A., Novikov S.P., Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method, Soviet Math. Dokl. 27 (1983), 665-669.
  7. Dubrovin B.A., Novikov S.P., On Poisson brackets of hydrodynamic type, Soviet Math. Dokl. 30 (1984), 651-654.
  8. Ferapontov E.V., Compatible Poisson brackets of hydrodynamic type, J. Phys. A: Math. Gen. 34 (2001), 2377-2388, math.DG/0005221.
  9. Ferapontov E.V., Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funct. Anal. Appl. 25 (1991), 195-204.
  10. Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type, differential geometry and applications, Am. Math. Soc. Trans. 170 (1995), 33-58.
  11. Finkel F., Fokas A.S., On the construction of evolution equations admitting a master symmetry, Phys. Lett. A 293 (2002), 36-44, nlin.SI/0112002.
  12. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47-66.
  13. Lichnerowicz A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253-300.
  14. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
  15. Maltsev A.Ya., Weakly nonlocal symplectic structures, Whitham method and weakly nonlocal symplectic structures of hydrodynamic type, J. Phys. A: Math. Gen. 38 (2005), 637-682, nlin.SI/0405060.
  16. Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly non-local Poisson brackets, Phys. D 156 (2001), no. 1-2, 53-80, nlin.SI/0006030.
  17. Mikhailov A.V., Shabat A.B., Sokolov V.V., The symmetry approach to classification of integrable equations, in What is Integrability?, Editor V.E. Zakharov, Springer, New York, 1991, 115-184.
  18. Mikhailov A.V., Shabat A.B., Yamilov R.I., The symmetry approach to classification of nonlinear equations. Complete lists of integrable systems, Russ. Math. Surv. 42 (1987), no. 4, 1-63.
  19. Mikhailov A.V., Yamilov R.I., Towards classification of (2+1)-dimensional integrable equations. Integrability conditions. I, J. Phys. A: Math. Gen. 31 (1998), 6707-6715.
  20. Mokhov O.I., Symplectic and Poisson geometry on loop sapces of manifolds in nonlinear equations, in Topics in Topology and Mathematical Physics, Editor S.P. Novikov, AMS, Providence, RI, 1995, 121-151, hep-th/9503076.
  21. Mokhov O.I., Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type, Theoret. and Math. Phys. 133 (2002), no. 2, 1557-1564, math.DG/0201281.
  22. Mokhov O.I., Compatible nonlocal Poisson brackets of hydrodynamic type and related integrable hierarchies, Theoret. and Math. Phys. 132 (2002), no. 1, 942-954, math.DG/0201242.
  23. Oevel W., Rekursionmechanismen für Symmetrien und Erhaltungssätze in Integrablen Systemen, Ph.D. Thesis, University of Paderborn, Paderborn, 1984.
  24. Olver P.J., Applications of Lie groups to differential equations, Springer, New York, 1993.
  25. Sergyeyev A., On recursion operators and nonlocal symmetries of evolution equations, in Proc. Sem. Diff. Geom., Editor D. Krupka, Silesian University in Opava, Opava, 2000, 159-173, nlin.SI/0012011.
  26. Sergyeyev A., A simple way to make a Hamiltonian system into bi-Hamiltonian one, Acta Appl. Math. 83 (2004), 183-197, nlin.SI/0310012.
  27. Sergyeyev A., Why nonlocal recursion operators produce local symmetries: new results and applications, J. Phys. A: Math. Gen. 38 (2005), no. 15, 3397-3407, nlin.SI/0410049.
  28. Smirnov R.G., Bi-Hamiltonian formalism: a constructive approach, Lett. Math. Phys. 41 (1997), 333-347.
  29. Sokolov V.V., On symmetries of evolution equations, Russ. Math. Surv. 43 (1988), no. 5, 165-204.
  30. Vaisman I., Lectures on the geometry of Poisson manifolds, Birkhäuser, Basel, 1994.
  31. Wang J.P., Symmetries and conservation laws of evolution equations, Ph.D. Thesis, Vrije Universiteit van Amsterdam, Amsterdam, 1998.
  32. Wang J.P., A list of 1+1 dimensional integrable equations and their properties, J. Nonlinear Math. Phys. 9 (2002), suppl. 1, 213-233.

Previous article   Next article   Contents of Volume 3 (2007)