Geometric realizations of Fordy-Kulish nonlinear Schrödinger systems

Pacific Journal of Mathematics

Volumn 195, Issue 1, 2000, Pages 157-178

  Langer, Joel ^a   Perline, Ron ^b  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b DREXEL UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0004456299     PISSN: 00308730     EISSN: None     Source Type: Journal    
DOI: 10.2140/pjm.2000.195.157     Document Type: Article

References (30)

1. V. Arnold, Mathematical Methods of Mechanics, Springer-Verlag, 1980.
2. C. Athorne and A. Fordy, Generalized KdV and mKdV equations associated with symmetric spaces, J. Phys. A, 20 (1987), 1377-1386.
3. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
4. R. Bishop, There is more than one way to frame a curve, American Mathematical Monthly, (March 1975), 246-251.
5. A. Calini and T. Ivey, Bäcklund transformations and knots of constant torsion, J. Knot Theory and its Ramifications, 7(6) (1998), 719-746.
6. A. Doliwa and P.M. Santini, An elementary geometric characterization of the integrable motions of a curve, Physics Letters A, 185 (1994), 373-384.
7. A.P. Fordy and P. Kulish, Nonlinear Schrödinger equations and simple Lie algebras, Communications in Mathematical Physics, 89 (1983), 427-443.
8. O. Garay and J. Langer, Taimanov's motion of surfaces and Bäcklund transformations for curves, J. Conformal Geometry and Dynamics, 3 (1999), 37-49.
9. C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-de Vries equation, Physical Review Letters, 19 (1967), 1095-1097.
10. R. Goldstein and D. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Physical Review Letters, 67(23) (1991), 3203-3206.
11. H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics, 51 (1972), 477-485.
12. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978.
13. J. Langer, Recursion in curve geometry, New York J. Math., 5 (1999), 25-51.
14. J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci., 1 (1991), 71-93.
15. _, The filament equation, the Heisenberg model, and the non-linear Schrödinger equation, in W. Shadwick, P. Krishnaprasad, T. Ratiu, eds., Fields Institute Communications, Mechanics Day, AMS, Providence, RI, 1996, 181-188.
16. _, The planar filament equation, in W. Shadwick, P. Krishnaprasad, T. Ratiu, eds., Fields Institute Communications, Mechanics Days, AMS, Providence, RI, 1996, 171-180.
17. _, Local geometric invariants of integrable evolution equations, J. Math. Phys., 35 (1994), 1732-1737.
18. 3 J. London Math. Soc., 30(2) (1984), 512-520.
19. _, Liouville integrability of geometric variational problems, Comm. Math. Helv., 69 (1994), 272-280.
20. _, Lagrangian aspects of the Kirchoff elastic rod, SIAM Review, 38 (1996), 605-618.
21. J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica, 7D (1983), 305-323.
22. R. Perline, Localized induction equation and pseudospherical surfaces, Journal of Physics A, 27 (1994), 5335-5344.
23. _, Localized induction hierarchy and Weingarten systems, Physics Letters A, 220 (1996), 70-74.
24. K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys., 46 (1976), 207-221.
25. R.L. Ricca, Rediscovery of the Da Rios equations, Nature, 352 (1991), 561-562.
26. N. Sasaki and Y. Yasui, Differential geometry of the vortex filament equation, J. Geom. Phys., 28(1-2) (1998), 195-207.
27. A. Shabat and V. Zakharov, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69.
28. A. Sym, Soliton surfaces and their applications, in 'Geometrical aspects of the Einstein equations and integrable systems', Lecture Notes in Physics, 239 (1985), 154-231.
29. C.L. Terng and K. Uhlenbeck, Poisson actions and scattering theory for integrable systems, preprint.
30. _, Schrödinger flows on Grassmanians, preprint math.dg/9901086.