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


SIGMA 8 (2012), 030, 20 pages      arXiv:1205.5329      http://dx.doi.org/10.3842/SIGMA.2012.030
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy

Emilio Musso
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy

Received February 08, 2012, in final form May 11, 2012; Published online May 24, 2012

Abstract
The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation are described.

Key words: local motion of curves; integrable evolution equations; Kaup-Kupershmidt hierarchy; geometric variational problems; projective differential geometry.

pdf (737 kb)   tex (271 kb)

References

  1. Anderson T.C., Marí Beffa G., A completely integrable flow of star-shaped curves on the light cone in Lorentzian R4, J. Phys. A: Math. Theor. 44 (2011), 445203, 21 pages.
  2. 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.
  3. Cartan E., Sur un problème du Calcul des variations en Géométrie projective plane, in Oeuvres Complètes, Partie III, Vol. 2, Gauthier Villars, Paris, 1955, 1105-1119.
  4. Chou K.S., Qu C., Integrable equations and motions of plane curves, in Proceedinds of Fourth International Conference "Symmetry in Nonlinear Mathematical Physics" (July 9-15, 2001, Kyiv), Proceedings of Institute of Mathematics, Kyiv, Vol. 43, Part 1, Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych, Institute of Mathematics, Kyiv, 2002, 281-290.
  5. Chou K.S., Qu C., Integrable equations arising from motions of plane curves, Phys. D 162 (2002), 9-33.
  6. Chou K.S., Qu C., Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), 487-517.
  7. Chou K.S., Qu C., Integrable motions of space curves in affine geometry, Chaos Solitons Fractals 14 (2002), 29-44.
  8. Chou K.S., Qu C., Motions of curves in similarity geometries and Burgers-mKdV hierarchies, Chaos Solitons Fractals 19 (2004), 47-53.
  9. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  10. Fordy A.P., Gibbons J., Some remarkable nonlinear transformations, Phys. Lett. A 75 (1980), 325.
  11. Fuchssteiner B., Oevel W., The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covaria, J. Math. Phys. 23 (1982), 358-363.
  12. Goldstein R.E., Petrich D.M., Solitons, Euler's equation, and vortex patch dynamics, Phys. Rev. Lett. 69 (1992), 555-558.
  13. Goldstein R.E., Petrich D.M., The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203-3206.
  14. Halphen G.H., Sur les invariants differentiels, Gauthier Villars, Paris, 1878.
  15. Huang R., Singer D.A., A new flow on starlike curves in R3, Proc. Amer. Math. Soc. 130 (2002), 2725-2735.
  16. Ivey T.A., Integrable geometric evolution equations for curves, in The Geometrical Study of Differential Equations (Washington, DC, 2000), Contemp. Math., Vol. 285, Amer. Math. Soc., Providence, RI, 2001, 71-84.
  17. Kaup D.J., On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx+6Qψx+6Rψ=λψ, Stud. Appl. Math. 62 (1980), 189-216.
  18. Kudryashov N.A., Two hierarchies of ordinary differential equations and their properties, Phys. Lett. A 252 (1999), 173-179.
  19. Langer J., Perline R., Curve motion inducing modified Korteweg-de Vries systems, Phys. Lett. A 239 (1998), 36-40.
  20. Lawden D.F., Elliptic functions and applications, Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York, 1989.
  21. Li Y., Qu C., Shu S., Integrable motions of curves in projective geometries, J. Geom. Phys. 60 (2010), 972-985.
  22. Marí Beffa G., Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc. 357 (2005), 2799-2827.
  23. Marí Beffa G., Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds, Ann. Inst. Fourier (Grenoble) 58 (2008), 1295-1335.
  24. Musette M., Verhoeven C., Nonlinear superposition formula for the Kaup-Kupershmidt partial differential equation, Phys. D 144 (2000), 211-220.
  25. Musso E., Congruence curves of the Goldstein-Petrich flows, in Harmonic Maps and Differential Geometry, Contemp. Math., Vol. 542, Amer. Math. Soc., Providence, RI, 2011, 99-113.
  26. Musso E., Variational problems for plane curves in centro-affine geometry, J. Phys. A: Math. Theor. 43 (2010), 305206, 24 pages.
  27. Musso E., Nicolodi L., Hamiltonian flows on null curves, Nonlinearity 23 (2010), 2117-2129, arXiv:0911.4467.
  28. Musso E., Nicolodi L., Reduction for the projective arclength functional, Forum Math. 17 (2005), 569-590.
  29. Nakayama K., Segur H., Wadati M., Integrability and the motion of curves, Phys. Rev. Lett. 69 (1992), 2603-2606.
  30. 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.
  31. Parker A., On soliton solutions of the Kaup-Kupershmidt equation. I. Direct bilinearisation and solitary wave, Phys. D 137 (2000), 25-33.
  32. Pinkall U., Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332.
  33. Qu C., Si Y., Liu R., On affine Sawada-Kotera equation, Chaos Solitons Fractals 15 (2003), 131-139.
  34. Rogers C., Carillo S., On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Phys. Scripta 36 (1987), 865-869.
  35. Thorbergsson G., Umehara M., Sextactic points on a simple closed curve, Nagoya Math. J. 167 (2002), 55-94, math.DG/0008137.
  36. Umehara M., A simplification of the proof of Bol's conjecture on sextactic points, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 10-12.
  37. Verhoeven C., Musette M., Extended soliton solutions for the Kaup-Kupershmidt equation, J. Phys. A: Math. Gen. 34 (2001), 2515-2523.
  38. Wazwaz A.M., Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput. 182 (2006), 283-300.
  39. Weiss J., On classes of integrable systems and the Painlevé property, J. Math. Phys. 25 (1984), 13-24.
  40. Wilczynski E.J., Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig, 1906.
  41. Zait R.A., Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations, Chaos Solitons Fractals 15 (2003), 673-678.

Previous article  Next article   Contents of Volume 8 (2012)