Differential geometry of the vortex filament equation

Journal of Geometry and Physics

Volumn 28, Issue 1-2, 1998, Pages 195-207

  Yasui, Yukinori ^a   Sasaki, Norihito ^a  

^a Osaka Prefecture University   (Japan)

Author keywords

Hamiltonian pair; Hereditary operator; Integrable Hamiltonian system; Master symmetries; Vortex filament equation

Indexed keywords

EID: 0032209457     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(98)00024-2     Document Type: Article

References (17)

1. [1] L.S. Da Rois, Sul Moto d'un liquido indefinito con un filetto vorticoso di forma qualunque, Rend. Circ. Mat. Palermo 22 (1906) 117-135.
2. [2] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1977) 477-485.
3. [3] L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987.
4. [4] J. Marsden, A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7 (1983) 305-323.
5. [5] J. Langer, R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991) 71-93.
6. [6] P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977) 1212-1215; P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
7. [6] P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977) 1212-1215; P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
8. [7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156-1162; F. Magri, A geometrical approach to the nonlinear solvable equations, in: Nonlinear Evolution Equations and Dynamical Systems, Lecture Notes in Physics, Springer, Berlin, 1980, pp. 233-263.
9. [7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156-1162; F. Magri, A geometrical approach to the nonlinear solvable equations, in: Nonlinear Evolution Equations and Dynamical Systems, Lecture Notes in Physics, Springer, Berlin, 1980, pp. 233-263.
10. [8] I.M. Gel'fand, I.Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979) 248-262; I.M. Gel'fand, I.Ya. Dorfman, The Schouten bracket and Hamiltonian operators, Funct. Anal. Appl. 14 (1980) 223-226.
11. [8] I.M. Gel'fand, I.Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979) 248-262; I.M. Gel'fand, I.Ya. Dorfman, The Schouten bracket and Hamiltonian operators, Funct. Anal. Appl. 14 (1980) 223-226.
12. [9] B. Fuchssteiner, A.S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4 (1981) 47-66.
13. [10] B. Fuchssteiner, Mastersymmetries, Higher order time-dependent symmetries and conseved densities of nonlinear evolution equations, Prog. Theor. Phys. 70 (1983) 1508-1522.
14. [11] V.E. Zakharov, B.G. Konopelchenko, On the theory of recursion operator, Commun. Math. Phys. 94 (1984) 483-509.
15. [12] A.S. Fokas, Symmetries and integrability, Stud. Appl. Math. 77 (1987) 253-299.
16. [13] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester, 1993.
17. [14] N. Sasaki, Differential geometry and integrability of the Hamiltonian system of a closed vortex filament, Lett. Math. Phys. 39 (1997) 229-241.