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

SIGMA 4 (2008), 034, 23 pages      arXiv:0803.3866      https://doi.org/10.3842/SIGMA.2008.034
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

Gloria Marí Beffa
Department of Mathematics, University of Wisconsin, Madison, WI 53705, USA

Received November 14, 2007, in final form March 13, 2008; Published online March 27, 2008

Abstract
In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.

Key words: invariant evolutions of curves; Hermitian symmetric spaces; Poisson brackets; differential invariants; projective differential invariants; equations of KdV type; completely integrable PDEs; moving frames; geometric realizations.

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References

  1. Anco S., Hamiltonian flows of curves in G/SO(N) and vector soliton equations of mKdV and sine-Gordon type, SIGMA 2 (2006), 044, 18 pages, nlin.SI/0512046.
  2. Anco S., Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations, J. Phys. A: Math. Gen. 39 (2006), 2043-2072, nlin.SI/0512051.
  3. Arms R.J., Hama F.R., Localized-induction concept on a curved vortex and motion of an elliptic vortex ring, Phys. Fluids 8 (1965), 553-559.
  4. Baston R.J., Almost Hermitian symmetric manifolds. I. Local twistor theory, Duke Math. J. 63 (1991), 81-112.
  5. Bailey T.N., Eastwood M.G., Conformal circles and parametrizations of curves in conformal manifolds, Proc. Amer. Math. Soc. 108 (1990), 215-222.
  6. Bailey T.N., Eastwood M.G., Complex paraconformal manifolds - their differential geometry and twistor theory, Forum Math. 3 (1991), 61-103.
  7. Calini A., Ivey T., Marí Beffa G., Remarks on KdV-type flows on star-shaped curves, in preparation.
  8. Cartan É., La Méthode du Repère Mobile, la Théorie des Groupes Continus, et les Espaces Généralisés, Exposés de Géométrie, no. 5, Hermann, Paris, 1935.
  9. Cartan É., La Théorie des Groupes Finis et Continus et la Géométrie Différentielle Traitées par la Méthode du Repère Mobile, Cahiers Scientifiques, Vol. 18, Gauthier-Villars, Paris, 1937.
  10. Cartan É., Les espaces à connexion conforme, Oeuvres Complètes, III.1, Gauthier-Villars, Paris, 1955, 747-797.
  11. Doliwa A., Santini P.M., An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A 185 (1994), 373-384.
  12. Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, in Current Problems in Mathematics, Itogi Nauki i Tekhniki, Vol. 24, VINITI, Moscow, 1984, 81-180 (in Russian).
  13. Ferapontov E.V., Isoparametric hypersurfaces in spheres, integrable non-diagonalizable systems of hydrodynamic type, and N-wave systems, Differential Geom. Appl. 5 (1995), 335-369.
  14. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  15. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  16. Fialkov A., The conformal theory of curves, Trans. Amer. Math. Soc. 51 (1942), 435-501.
  17. Gay-Balmaz F., Ratiu T.S., Group actions on chains of Banach manifolds and applications to fluid dynamics, Ann. Global Anal. Geom., to appear.
  18. González-López A., Hernández Heredero R., Marí Beffa G., Invariant differential equations and the Adler-Gel'fand-Dikii bracket, J. Math. Phys. 38 (1997), 5720-5738, hep-th/9603199.
  19. Griffiths P.A., On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775-814.
  20. Green M.L., The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Math. J. 45 (1978), 735-779.
  21. Hasimoto R., A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485.
  22. Hitchin N.J., Segal G.B., Ward R.S., Integrable systems: twistors, loop groups and riemann surfaces, Oxford Graduate Texts in Mathematics, Clarendon Press, Oxford, 1999.
  23. Hubert E., Generation properties of differential invariants in the moving frame methods, Preprint.
  24. Ivey T.A., Integrable geometric evolution equations for curves, Contemp. Math. 285 (2001), 71-84.
  25. Chou K.-S., Qu C., Integrable equations arising from motions of plane curves, Phys. D 162 (2002), 9-33.
  26. Chou K.-S., Qu C., Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), 487-517.
  27. Kobayashi S., Nagano T., On filtered Lie algebras and geometric structures. I, J. Math. Mech. 13 (1964), 875-907.
  28. Langer J., Perline R., Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93.
  29. Langer J., Perline R., Geometric realizations of Fordy-Kulish nonlinear Schrödinger systems, Pacific J. Math. 195 (2000), 157-178.
  30. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
  31. Marí Beffa G., Hamiltonian structures on the space of differential invariants of curves in flat semisimple homogenous manifolds, Asian J. Math., to appear.
  32. Marí Beffa G., Poisson geometry of differential invariants of curves in some nonsemisimple homogenous spaces, Proc. Amer. Math. Soc. 134 (2006), 779-791.
  33. Marí Beffa G., Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Inst. Fourier (Grenoble), to appear.
  34. Marí Beffa G., On completely integrable geometric evolutions of curves of Lagrangian planes, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 111-131.
  35. Marí Beffa G., Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc. 357 (2005), 2799-2827.
  36. Marí Beffa G., The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999), 363-391.
  37. Marí Beffa G., Completely integrable curve flows in O(2n+1,2n+1)/H, in preparation.
  38. Marí Beffa G., Sanders J., Wang J.P., Integrable systems in three-dimensional Riemannian geometry, J. Nonlinear Sci. 12 (2002), 143-167.
  39. Marsden J.E., Weinstein A., Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), 305-323.
  40. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Springer-Verlag, 1986.
  41. Olver P., Invariant variational problems and integrable curve flows, Cocoyoc, Mexico, 2005, presentation.
  42. Ochiai T., Geometry associated with semisimple flat homogeneous spaces, Trans. Amer. Math. Soc. 152 (1970), 159-193.
  43. Ovsienko V., Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. Fac. Sci. Toulouse Math. (6) 6 (1993), 73-96.
  44. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
  45. Pressley A., Segal G., Loop groups, Graduate Texts in Mathematics, Springer, 1997.
  46. Sanders J., Wang J.P., Integrable systems in n-dimensional Riemannian geometry, Moscow Math. J. 3 (2004), 1369-1393, math.AP/0301212.
  47. Terng C.L., Thorbergsson G., Completely integrable flows on adjoint orbits, Results Math. 40 (2001), 286-309, math.DG/0108154.
  48. Terng C.L., Uhlenbeck K., Schrödinger flows on Grassmannians, in Integrable Systems, Geometry and Topology, AMS/IP Stud. Adv. Math., Vol. 36, Amer. Math. Soc., Providence, 2006, 235-256, math.DG/9901086.
  49. Terng C.L., Uhlenbeck K., Bäcklund transformations and loop group actions, Comm. Pure Appl. Math. 53 (2000), 1-75, math.DG/9805074.
  50. Wilczynski E.J., Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig, 1906.
  51. Yasui Y., Sasaki N., Differential geometry of the vortex filament equation, J. Geom. Phys. 28 (1998), 195-207, hep-th/9611073.

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