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Schrödinger flows, binormal motion for curves and the second AKNS-hierarchies. (English) Zbl 1048.37058
Summary: We present a unified geometric interpretation of the second AKNS-hierarchies via the geometric concept of Schrödinger flows in the category of symplectic manifolds and binormal motion for curves in the Minkowski 3-space.

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
Full Text: DOI
[1] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, London math soc lect note ser, vol. 149, (1992), Cambridge University Press
[2] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problem, Stud. appl. math., 53, 249-315, (1974) · Zbl 0408.35068
[3] Chang, N.H.; Shatah, J.; Uhlenbeck, K., Schrödinger maps, Commun. pure appl. math., 53, 590-620, (2000) · Zbl 1028.35134
[4] Chern, S.S.; Tenenblat, K., Pseudo-spherical surfaces and evolution equations, Stud. appl. math., 74, 55-83, (1986) · Zbl 0605.35080
[5] Ding, W.Y.; Wang, Y.D., Schrödinger flow of maps into symplectic manifolds, Sci. China, 47A, 746-755, (1998) · Zbl 0918.53017
[6] Ding, Q., A note on the NLS and Schrödinger flow of maps, Phys. lett. A, 248, 49-58, (1998)
[7] Ding, Q., The gauge equivalence of the NLS and the Schrödinger flow of maps in 2+1 dimensions, J. phys. A: math. gen., 32, 5087-5096, (1999) · Zbl 0941.35104
[8] Ding Q, Inoguchi J. Hasimoto surfaces derived from dark solitons, in preparation
[9] Ding, Q.; Zhu, Z., A geometric characterization of the nonlinear Schrödinger equation and its applications, Sci. China A, 45, 1225-1237, (2002) · Zbl 1099.37055
[10] Faddeev, L.D.; Takhatajan, L.A., Hamiltonian methods in the theory of solitons, (1987), Springer Verlag Berlin · Zbl 0632.58004
[11] (), English translation · Zbl 0112.11301
[12] Hasimoto, H., A soliton on a vortex filament, J. fluid mech., 51, 477-485, (1972) · Zbl 0237.76010
[13] Ivey, T.A., Helices, hasimoto surfaces and Bäcklund transformations, Can. math. bull., 43, 427-439, (2000) · Zbl 0992.53004
[14] Kobayashi, S., Transformation groups in differential geometry, (1972), Springer Verlag Berlin · Zbl 0246.53031
[15] Koiso, N., The vortex filament equation and a semi-linear Schrödinger equation in a Hermitian symmetric space, Osaka J. math., 34, 199-214, (1997) · Zbl 0926.53002
[16] Lakshmanan, M., Continuum spin system as an exactly solvable dynamical system, Phys. lett. A, 61, 53-54, (1979)
[17] Libermann, P., Sur le probléme d’equivalence de certaines structures infinitésimales, Ann. mat. pura appl., 36, 27-120, (1954) · Zbl 0056.15401
[18] Nahmod, A.; Stefanov, A.; Uhlenbeck, K., On Schrödinger maps, Commun. pure appl. math., 56, 114-151, (2003) · Zbl 1028.58018
[19] Rogers, C., Bäcklund transformations in soliton theory, (), 97-130
[20] Rogers, C.; Schief, W.K., Intrinsic geometry of the NLS equation and its auto-Bäcklund transformations, Stud. appl. math., 101, 267-287, (1998) · Zbl 1020.35100
[21] Sasaki, R., Soliton equation and pseudospherical surfaces, Nucl. phys. B, 154, 343-357, (1979)
[22] Scheif, W.K.; Rogers, C., Binormal motion of curves of constant curvature and torsion, Proc. royal soc. London A, 455, 3163-3188, (1999) · Zbl 0981.53061
[23] Terng CL, Uhlenbeck K. Schrödinger flows on Grassmannians, 1999, preprint math.DG/9901086
[24] Zakharov, V.E.; Takhtajan, L.A., Equivalence of a nonlinear Schrödinger equation and a Heisenberg ferromagnet equation, Theor. math. phys., 38, 26-35, (1979)
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