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

SIGMA 9 (2013), 022, 21 pages      arXiv:1303.1259
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Integrable Flows for Starlike Curves in Centroaffine Space

Annalisa Calini a, b, Thomas Ivey a and Gloria Marí Beffa c
a) College of Charleston, Charleston SC, USA
b) National Science Foundation, Arlington VA, USA
c) University of Wisconsin, Madison WI, USA

Received September 07, 2012, in final form February 27, 2013; Published online March 06, 2013

We construct integrable hierarchies of flows for curves in centroaffine R3 through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and prove that the restricted hierarchy of flows on curves that project to conics in RP2 induces the Kaup-Kuperschmidt hierarchy at the curvature level.

Key words: integrable curve evolutions; centroaffine geometry; Boussinesq hierarchy; bi-Hamiltonian systems.

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  1. Arnold V.I., Khesin B.A., Topological methods in hydrodynamics, Applied Mathematical Sciences, Vol. 125, Springer-Verlag, New York, 1998.
  2. Brylinski J.-L., Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, Vol. 107, Birkhäuser Boston Inc., Boston, MA, 1993.
  3. Calini A., Ivey T., Marí Beffa G., Remarks on KdV-type flows on star-shaped curves, Phys. D 238 (2009), 788-797, arXiv:0808.3593.
  4. Chou K.S., Qu C., Integrable motions of space curves in affine geometry, Chaos Solitons Fractals 14 (2002), 29-44.
  5. Dickson R., Gesztesy F., Unterkofler K., Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys. 11 (1999), 823-879, solv-int/9809004.
  6. Doliwa A., Santini P.M., An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A 185 (1994), 373-384.
  7. Dorfman I., Dirac structures and integrability of nonlinear evolution equations, Nonlinear Science: Theory and Applications, John Wiley & Sons Ltd., Chichester, 1993.
  8. Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1985), 1975-2036.
  9. Hasimoto R., A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485.
  10. Huang R., Singer D.A., A new flow on starlike curves in R3, Proc. Amer. Math. Soc. 130 (2002), 2725-2735.
  11. Lamb G.L., Solitons on moving space curves, J. Math. Phys. 18 (1977), 1654-1661.
  12. Langer J., Recursion in curve geometry, New York J. Math. 5 (1999), 25-51.
  13. Langer J., Perline R., Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994), 1732-1737, solv-int/9401001.
  14. Langer J., Perline R., Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93.
  15. Marí Beffa G., Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds, Pacific J. Math. 247 (2010), 163-188.
  16. Marí Beffa G., Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc. 357 (2005), 2799-2827.
  17. Marí Beffa G., The theory of differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999), 363-391.
  18. Marsden J., Weinstein A., Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), 305-323.
  19. Mokhov O., Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations, in Topics in Topology and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 170, Amer. Math. Soc., Providence, RI, 1995, 121-151, hep-th/9503076.
  20. Musso E., Motions of curves in the projective plane inducing the Kaup-Kupershmidt hierarchy, SIGMA 8 (2012), 030, 20 pages, arXiv:1205.5329.
  21. Nakayama K., Motion of curves in hyperboloid in the Minkowski space, J. Phys. Soc. Japan 67 (1998), 3031-3037.
  22. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  23. Olver P.J., Moving frames and differential invariants in centro-affine geometry, Lobachevskii J. Math. 31 (2010), 77-89.
  24. Ovsienko V., Schwartz R., Tabachnikov S., The pentagram map: A discrete integrable system, Comm. Math. Phys. 299 (2010), 409-446, arXiv:0810.5605.
  25. Ovsienko V., Tabachnikov S., Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Mathematics, Vol. 165, Cambridge University Press, Cambridge, 2005.
  26. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
  27. Sanders J.A., Wang J.P., Integrable systems and their recursion operators, Nonlinear Anal. 47 (2001), 5213-5240.
  28. Schwartz R.E., Tabachnikov S., Elementary surprises in projective geometry, Math. Intelligencer 32 (2010), 31-34, arXiv:0910.1952.
  29. Wang J.P., A list of 1+1 dimensional integrable equations and their properties, J. Nonlinear Math. Phys. 9 (2002), suppl. 1, 213-233.
  30. Wilczynski E.J., Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig, 1906.

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