Soliton equations and differential geometry

Journal of Differential Geometry

Volumn 45, Issue 2, 1997, Pages 407-445

  Terng, Chuu Lian ^a  

^a NORTHEASTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031065010     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1214459804     Document Type: Article

References (41)

1. M. J. Ablowitz, R. Beals &K. Tenenblat, On the solution of the generalized wave and generalized sine-Gordon eguations, Stud. Appl. Math. 74 (1987) 177-203.
2. M. J. Ablowitz &P. A. Clarkson, Solitons, non-linear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991.
3. M. J. Ablowitz, D. J. Kaup, A. C. Newell &H. Segur, Nonlinear evolution equations of physical significance, Phy. Rev. Lett. 31 (1973) 125-127.
4. M. J. Ablowitz, D. J. Kaup, A. C. Newell &H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249-315.
5. M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math. 50 (1979) 219-248.
6. M. Adler &P. van Moerbeke, Completely inteqrable systems, Euclidean Lie algebras and curves, Adv. Math. 38 (1980) 267-317.
7. R. Beals &R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984) 39-90.
8. R. Beals &R. R. Coifman, Inverse scattering and evolution equations, Comm. Pure Appl. Math. 38 (1985) 29-42.
9. R. Beals &R. R. Coifman, Linear spectral problems, non-linear equations and the d-method, Inverse Probl. 5 (1989) 87-130.
10. R. Beals &D. H. Sattinger. On the complete integrability of complete inteqrable systems, Comm. Math. Phys. 138 (1991) 409-436.
11. R. Beals &K. Tenenblat, Inverse scattering and the Bdcklund transformation for the generalized wave and generalized sine-Gordon equations, Stud. Appl. Math. 78 (1988) 227-256.
12. B. Budagov, Completely inteqrable model of classical field theory with nontrivial particle interaction in two-dimensional space-time, Dokl. Akad. Nauk SSSR 235 (1977) 24-56(in Russian).
13. E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919) 132-208.
14. P. Chernoff &J. Marsden, Properties of infinite dimensional Hamiltonian system, Lecture Notes in Math., Vol. 425, Springer, Berlin, 1974.
15. V. G. Drinfel’d &V. V. Sokolov, Equations of Korteweg-de Vries type and simple Lie algebras, Dokl. Akad. Nauk SSSR 258 (1981) 11-16, (Trans, as Soviet Math. Dokl. 23, 457-462.)
16. L. D. Faddeev &L. A. Ta, Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987.
17. H. Flaschka, A. C. Newell &T. Ratiu, Kac-Moody Lie algebras and soliton equations, II. Lax equations associated with A^\ Phys. D9 (1983) 303-323.
18. H. Flaschka, A. C. Newell &T. Ratiu, Kac-Moody Lie algebras and soliton equations, IV. Lax equations associated with A^\ Phys. D9 (1983) 333-345.
19. I. M. Gel’fand &L. A. Dikii, Fractional powers of operators and Hamiltonian systems, Funkcional Anal, i Prilozen. 10 (1976).
20. I. M. Gel’fand &I. Ya Dorfman, Hamiltonian operators and algebraic structures related to them, Functional Anal. Appl. 13 (1979) 248-261.
21. S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1978.
22. V. G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge, 1985.
23. B. Kostant, The solution to a generalized Toda lattice and repjresentation theory, Adv. Math. 34 (1979) 195-338.
24. B. A. Kuperschmidt &G. Wilson, Modifying Lax equations and the second Hamiltonian structure, Invent. Math. 62 (1981) 403-436.
25. F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156-1162.
26. S. V. Manakov, An example of a completely integrable nonlinear wave field with nontrivial dynamics (Lee model) Teoret. Math, and Phys. 28 (1976) 172-179.
27. J. D. Moore, Isometric immersions of space forms in space forms, Pacific J. Math. 40 (1972) 157-166.
28. A. C. Newell, Solitons in mathematics and physics, SIAM, CBMS-NSF Vol. 48, 1985.
29. S. Novikov, S. Manakov, L. B. Pitaevskii &V. E. Zakharov, Theory of solitons, Plenum, New York, 1984.
30. R. S. Palais &C. L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Math., Vol. 1353, Springer, Berlin, 1988.
31. A. Pressley, &G. B. Segal, Loop groups, Oxford Science Publ., Clarendon Press, Oxford, 1986.
32. A. G. Reyman &Semonov Tian-Shansky, Current algebras and non-linear partial differential equations, Soviet Math. Dokl. 21 (1980) 630-634.
33. D. H. Sattinger, Hamiltonian hierarchies on semi-simple Lie algebras, Stud. Appl. Math. 72 (1984) 65-86.
34. D. H. Sattinger, Hamiltonian hierarchies on semi-simple Lie algebras, Stud. Appl. Math. 72 (1984) 65-86.
35. D. H. Sattinger &V. D. Zurkowski, Gauge theory of Bdcklund transformations.II, Phys. D26 (1987) 225-250.
36. K. Tenenblat, Bdcklund’s theorem for submanifolds of space forms and a generalized wave equation, Bol. Soc. Brasil. Mat. 16 (1985) 67-92.
37. 2n-\ Ann. Math. Ill (1980) 477-490.
38. C. L. Terng, A higher dimensional generalization of the sine-Gordon equation and its soliton theory, Ann. Math. Ill (1980) 491-510.
39. K. Uhlenbeck. Harmonic maps into Lie group (classical solutions of the Chiral model), J. Differential Geom. 30 (1989) 1-50.
40. K. Uhlenbeck, On the connection between harmonic maps and the self-dual Yang-Mills and the sine-Gordon equations, J. Geom. Phys. 8 (1992) 283-316.
41. G. Wilson, The modified Lax equations and two dimensional Toda lattice equations associated with simple Lie algebras, Ergodic Theory Dynamical Systems. I, 30 (1981) 361-380