Soliton curvatures of surfaces and spaces

Journal of Mathematical Physics

Volumn 38, Issue 1, 1997, Pages 434-457

  Konopelchenko, B G ^a,b  

^a BUDKER INSTITUTE OF NUCLEAR PHYSICS   (Russian Federation)

^b UNIVERSITY OF SALENTO   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031282820     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531826     Document Type: Article

References (89)

1. J. S. Langer, Rev. Mod. Phys. 52, 1 (1980).
2. A. R. Bishop, L. J. Cambell, and P. J. Channels, in Fronts, Interfaces and Patterns (North-Holland, New York, 1984).
3. P. Pelce, in Dynamics of Curved Fronts (Academic, New York, 1986).
4. R. C. Brower, D. A. Kessler, J. Koplik, and H. Levine, Phys. Rev. A 29, 1335 (1984).
5. A. M. Polyakov, Phys. Lett. B 103, 207,211 (1981).
6. D. Nelson, T. Piran, and S. Weinberg, in Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, 1989).
7. M. J. Duff, C. N. Pope, and E. Sergin, in Supermembranes and Physics in 2+1-Dimensions (World Scientific, Singapore, 1990).
8. D. J Gross, T. Piran, and S. Weinberg, in Two Dimensional Quantum Gravity and Random Surfaces (World Scientific, Singapore, 1992).
9. C. W. Mizner, K. S. Thone, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1970).
10. A. Z. Petrov, Einstein Spaces (Pergamon, Oxford, 1969).
11. G. Darboux, Leçons Sur La Théorie Des Surfaces Et Les Applications Geometriques Du Calcul Infinitesimal Gauthier-Villars, Paris, 1977-1896), t1-4.
12. L. Bianchi, Lezioni Di Geometria Differentiale, 2nd ed. (Piza, Spoerri, 1902).
13. L. P. Eisenhart, A Treatise On The Differential Geometry Of Curves And Surfaces (Dover, New York, 1909).
14. A. R. Forsyth, Lectures on the Differential Geometry of Curves and Surfaces (Cambridge University, Cambridge, 1920).
15. J. Liouville, J. Math. Pure Appl. 18, 71 (1853).
16. A. V. Bäcklund, Lunds Univ. Arsskr. Avd. 19, 7 (1883).
17. G. Tzitzeica, C. R. Acad. Sci. Paris, 150, 995, 1227 (1910).
18. E. P. Lane, Projective Differential Geometry of Curves and Surfaces (University of Chicago, Chicago, 1932).
19. S.-T. Yau, in Seminar on Differential Geometry (Princeton University, Princeton, 1982).
20. T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampere Equations (Springer, New York, 1982).
21. S.-T. Yau, "Survey on partial differential equations in differential geometry," in Ref. 19, p. 185.
22. J. I. Kazdan, "Gaussian and scalar curvature, an update," in Ref. 19, p. 185.
23. G. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1015 (1967).
24. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevsky, Theory of Solitons (Nauka, Moscow, 1980) (Consultants Bureau, New York, 1984).
25. M. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform (SIAM, Philadelphia, 1981).
26. F. Calogero and A. Degasperis, Spectral Transform and Solitons (North-Holland, Amsterdam, 1982), Vol. 1.
27. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Waves (Academic, New York, 1982).
28. A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985).
29. M. J. Ablowitz and P. A. Clarkson, Solitons Nonlinear Evolution Equations and Inverse Scattering (Cambridge University, Cambridge, 1991).
30. B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations (Plenum, New York, 1992).
31. B. G. Konopelchenko, Solitons in Multidimensions (World Scientific, Singapore, 1993).
32. F. Lund, "Solitons and geometry," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut (Reidel, Dordrecht, 1978).
33. R. Sasaki, Nucl. Phys. B 154, 434 (1979).
34. B. G. Konopelchenko, Phys. Lett. A 72, 101 (1979).
35. S. S. Chern and K. Tenenblat, J. Diff. Geom. 16, 347 (1981).
36. S. S. Chern and K. Tenenblat, Stud. Appl. Math. 74, 55 (1986).
37. L. P. Jorde and K. Tenenblat, Stud. Appl. Math. 77, 103 (1987).
38. R. Beals, M. Rabelo, and K. Tenenblat, Stud. Appl. Math. 81, 125 (1989).
39. K. Tenenblat and C.-L. Terng, Ann. Math. 111, 477 (1980).
40. C.-L. Terng, Ann. Math. 111, 491 (1980).
41. M. Ablowitz, R. Beals, and K. Tenenblat, Stud. Appl. Math. 74, 177 (1986).
42. M. V. Savelliev, Teor. Mat. Fyz. 69, 411 (1986).
43. V. S. Dryuma, Math. Stud. 124, 56 (1992).
44. V. S. Dryuma, Teor. Mat. Fyz. 99, 241 (1994).
45. B. G. Konopelchenko and W. K. Schief, "Lame and Zakharov-Manakov systems: Combescure, Darboux and Bäcklund transformations," preprint AM 93/9, Sydney, 1993.
46. A. Sym, "Soliton surfaces and their applications," in Lecture Notes in Physics (Springer, New York, 1985), Vol. 239, p. 154.
47. A. I. Bobenko, Math. Ann. 290, 207 (1990).
48. B. G. Konopelchenko, "Induced surfaces and their integrable dynamics," preprint, Budker INP 93-114, Novosibirsk, 1993; Stud. Appl. Math. 96, 9 (1996).
49. B. G. Konopelchenko, "Induced surfaces and their integrable dynamics," preprint, Budker INP 93-114, Novosibirsk, 1993; Stud. Appl. Math. 96, 9 (1996).
50. B. G. Konopelchenko, Phys. Lett. A 183, 153 (1993).
51. L. P. Eisenhart, Riemann Geometry (Princeton University, Princeton, 1926).
52. E. Cotton, Ann. Fac. Sci. Toulousse Ser. 2 1, 385 (1899).
53. K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
54. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Nauka, Moscow, 1974).
55. B. G. Konopelchenko, Rev. Math. Phys. 2, 399 (1990).
56. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991).
57. G. L. Lamb, Math. Phys. 18, 1654 (1977).
58. R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett. 67, 3203 (1991).
59. J. Langer and R. Perline, Nonlinear Sci. 1, 71 (1991).
60. K. Makayama, H. Segur, and M. Wadati, Phys. Rev. Lett. 60, 2603 (1992).
61. F. Calogero, Lett. Nuovo Cim. 14, 443 (1975).
62. V. E. Zakharov and S. V. Manakov, Teor. Mat. Fyz. 27, 283 (1976).
63. O. I. Bogoyavlensky, Breaking Solitons (Nauka, Moscow, 1991).
64. B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, DAN SSSR 229, 15 (1976).
65. A. P. Veselov and S. P. Novikov, DAN SSSR 279, 20 (1984).
66. M. C. Cheney, J. Math. Phys. 25, 94 (1984).
67. S. P. Novikov and A. P. Veselov, Physica D 18, 267 (1986).
68. P. G. Grinevich and R. G. Novikov, Funct. Anal. Appl. 19, 32 (1985).
69. P. G. Grinevich and S. V. Manakov, Funct. Anal. Appl. 22, 14 (1986).
70. M. Boiti, J. J.-P. Leon, M. Manna, and F. Pempinelli, Inverse Prob. 2, 27 (1986).
71. P. G. Grinevich, Teor. Mat. Fyz. 69, 307 (1986).
72. P. G. Grinevich and R. G. Novikov, Funct. Anal. Appl. 22, 23 (1988).
73. S. M. Natanson, Funct. Anal. Pliloz. 23, 41 (1989).
74. I. A. Taimanov, Mat. Sb. 181, 939 (1990).
75. R. G. Novikov, J. Funct. Anal. 103, 409 (1992).
76. T. Y. Tsai, Inverse Prob. 9, 763 (1993).
77. L. P. Nizhnik, Inverse Scattering Problems For Hyperbolic Equations (Naukova Dumka, Kiev, 1991).
78. M. Boiti, J. P. Leon, M. Manna, and F. Pempinelli, Inverse Prob. 2, 271 (1986).
79. C. Athorne and J. J. C. Nimmo, Inverse Prob. 7, 809 (1991).
80. V. G. Dubrovsky and B. G. Konopelchenko, Inverse Prob. 9, 391 (1993).
81. L. P. Nizhnik, DAN SSSR 254, 332 (1980).
82. L. Martinez Alonso, E. Medina Reus, and E. Olmedilla Moreno, Phys. Lett. A 159, 384 (1991).
83. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
84. E. Hopf, Commun. Pure Appl. Math. 3, 201 (1950).
85. J. D. Cole, Quart. Appl. Math. 9, 225 (1951).
86. M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ. 19, 943 (1983).
87. H. Harada, J. Phys. Soc. Jpn. 54, 9507 (1985); 56, 3847 (1987).
88. H. Harada, J. Phys. Soc. Jpn. 54, 9507 (1985); 56, 3847 (1987).
89. V. E. Zakharov, in Inverse Problems in Action, edited by P. C. Sabatier (Springer, Berlin, 1990), p. 602.