Methods of algebraic geometry in the theory of non-linear equations

Russian Mathematical Surveys

Volumn 32, Issue 6, 1977, Pages 185-213

  Krichever, I M ^a  

^a NONE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0010989303     PISSN: 00360279     EISSN: None     Source Type: Journal    
DOI: 10.1070/RM1977v032n06ABEH003862     Document Type: Article

References (83)

1. Gardner C, J Green, M Kruskal and R Miura 1967 A method for solving the Korteweg-de Vries equation Phys. Rev. Lett. 19 1095-1098.
2. Lax P D 1968 Integrals of non-linear equations of evolution and solitary waves Comm. Pure Appl. Math. 21 467-490.
3. Lax P D 1969 Matematika 13 (5) 128-50.
4. Zakharov V E and L D Faddeev 1971 The Korteweg-de Vries equation is a completely integrable Hamiltonian system Funktsional. Anal. i Prilozhen. 5 (4) 18-27.
5. Zakharov V E and L D Faddeev 1971 Functional Anal. Appl. 5 280-287.
6. Gardner C 1971 Korteweg-de Vries equation and generalization. IV, The Korteweg-de Vries equation as a Hamiltonian system J. Math. Phys. 12 1548-1551.
7. Zakharov V E and A B Shabat 1971 An exact theory of two-dimensional self-focussing and of one-dimensional self-modulation of waves in non-linear media Zh. Eksp. Teor. Fiz. 61.
8. Zakharov V E and A B Shabat 1972 Soviet Phys. JETP 34 62-69.
9. Zakharov V E and A B Shabat 1973 On the interaction of solitons in a stable medium Zh. Eksp. Teor. Fiz. 64 1627-1639.
10. Zakharov V E and A B Shabat 1973 Soviet Phys. JETP 37 823-828.
11. Ablowitz M A, D J Kanp, A Cnewel and H Segur 1973 Method for solving the sine- Gordon equation Phys. Rev. Lett. 30 1262-1264.
12. Dryuma V S 1973 On the analytic solution of the two-dimensional Korteweg-de Vries equation Zh. Eksp. Teor. Fiz. Lett. 19 219-225.
13. Zakharov V E 1973 On the problem of the stochastization of one-dimensional chains of non-linear oscillators Zh. Eksp. Teor Fiz. 65 219-225.
14. Zakharov V E 1973 Soviet Phys. JETP 38 108-110.
15. Zakharov V E and S V Manakov 1975 The theory of resonance interaction of wave packets in non-linear media Zh. Eksp. Teor. Fiz. 69 1654-1673.
16. Zakharov V E and S V Manakov 1975 Soviet Phys. JETP 42 842-850.
17. Takhtadzhyan L A 1974 An exact theory of the propagation of ultra-short optical impulses in two-level media Zh. Eksp. Teor. Fiz. 66 476-489.
18. Takhtadzhyan L A 1975 Soviet Phys. JETP 39 228-233.
19. Zakharov V E, L A Takhtadzhyan and L D Faddeev 1974 A complete description of the solutions of the "sine-Gordon" equation Dokl. Akad. Nauk SSSR 219 1334-1337.
20. Zakharov V E, L A Takhtadzhyan and L D Faddeev 1974 Soviet Phys. Dokl. 19 824-826.
21. Toda M 1970 Waves in non-linear lattice Prog. Theor. Phys. Suppl. 45 174-200.
22. Jenon M 1974 Integrals of the Toda lattice Phys. Rev. B 9 (3) 1921-1923.
23. Flaschka H 1974 The Toda lattice. I Phys. Rev. B 9 (3) 1924-1925.
24. Flaschka H 1974 The Toda lattice. II Prog. Theor. Phys. 52 703-716.
25. Manakov S V 1974 The complete integrability and stochastization in discrete dynamic systems Zh. Eksp. Teor. Fiz. 67 543-555.
26. Manakov S V 1974 Soviet Phys. JETP 40 269-274.
27. Shabat A B 1973 On the Korteweg-de Vries equation Dokl. Akad. Nauk SSSR 211 1310-1313.
28. Shabat A B 1973 Soviet Math. Dokl. 14 1266-1270.
29. Zakharov V E and A B Shabat 1974 A scheme for the integration of non-linear equations of mathematical physics by the method of the inverse scattering problem. I Funktsional. Anal. i Prilozhen. 8 (3) 43-53.
30. Zakharov V E and A B Shabat 1974 Functional Anal. Appl. 8 226-235.
31. Novikov S P 1974 The periodic Korteweg-de Vries problem. I Funktsional. Anal. i Prilozhen. 8 (3) 54-56.
32. Novikov S P 1974 Functional Anal. Appl. 8 236-246.
33. Lax P D 1974 Periodic solutions of the K-dV equations Lect. Appl. Math. 15 85-96.
34. McKean H and Van Moerbeke 1975 The spectrum of Hill's equation Invent. Math. 30 217-274.
35. Marchenko V A 1974 The periodic Korteweg-de Vries problem Dokl. Akad. Nauk SSSR 217 276-279.
36. Marchenko V A 1974 Soviet Math. Dokl. 15 1052-1056.
37. Marchenko V A and I V Ostrovskii 1974 The periodic Korteweg-de Vries problem Mat. Sb. 95 331-356.
38. Marchenko V A and I V Ostrovskii 1974 Math. USSR-Sb. 24 319-344.
39. Marchenko V A 1977 Operator Sturm-Liuv illy a i ego prilozheniya, (The Sturm-liouville operator and its applications) (Naukova Dumka, Kiev).
40. Dubrovin V A, V B Matveev and S P Novikov 1976 Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds Uspekhi Mat. Nauk 31 (1) 55-136.
41. Dubrovin V A, V B Matveev and S P Novikov 1976 Russian Math. Surveys 31 (1) 59-146.
42. Kadomtsev B B and V I Petviasvili 1970 On the stability of solitary waves in weakly dispersing media Dokl. Akad. Nauk SSSR 192 753-756.
43. Kadomtsev B B and V I Petviasvili 1971 Soviet Phys. Dokl. 15 539-541.
44. Krichever I M 1976 An algebraic-geometrical construction of the Zakharov-Shabat equation and their periodic solutions Dokl. Akad. Nauk SSSR 227 291-294.
45. Krichever I M 1976 Soviet Math. Dokl. 17 394-397.
46. Krichever I M 1976 Algebraic curves and commuting matrix differential operators Funktsional. Anal i Prilozhen. 10 (2) 75-76.
47. Krichever I M 1976 Functional Anal. Appl. 10 144-146.
48. Krichever I M 1977 The integration of non-linear equations by methods of algebraic geometry Funktsional. Anal i Prilozhen. 11 (1) 15-31.
49. Krichever I M 1977 Functional Anal. Appl. 11 12-26.
50. Drinfeld V E 1977 On commutative subrings of some non-commutative rings Funktsional. Anal i Prilozhen. 11 (1) 11-14.
51. Drinfeld V E 1977 Functional Anal. Appl. 11 9-12.
52. Gel'fand I M and L A Dikii 1976 Fractional powers of operators and Hamiltonian systems Funktsional. Anal i Prilozhen. 10 (4) 13-29.
53. Gel'fand I M and L A Dikii 1976 Functional Anal. Appl. 10 259-273.
54. Gel'fand I M and L A Dikii 1977 Resolvents and Hamiltonian systems Funkts. Anal i Prilozhen. 11 (2) 11-27.
55. Gel'fand I M and L A Dikii 1977 Functional Anal. Appl. 11 93-105.
56. Dubrovin V A, I M Krichever and S P Novikov 1976 The Schrödinger equation in a periodic magnetic field, and Riemann surfaces Dokl. Akad. Nauk SSSR 229 15-18.
57. Dubrovin V A, I M Krichever and S P Novikov 1976 Soviet Math. Dokl. 17 947-951.
58. Akhiezer N I 1961 A continuous analogue of orthogonal polynomials on a system of intervals Dokl. Akad. Nauk SSSR 141 263-266.
59. Akhiezer N I 1961 Soviet Math. Dokl. 2 1409-1412.
60. Its A R and V B Matveev 1975 Schrödinger operators with a finite-zone spectrum and N-soliton solutions of the Korteweg-de Vries equation Teor. Mat. Fiz. 23 (1) 51-67.
61. Zverovich E I 1971 Boundary value problems of the theory of analytic functions Uspekhi Mat. Nauk 26 (1) 113-181.
62. Zverovich E I 1971 Russian Math. Surveys 26 (1) 117-192.
63. Springer G 1957 Introduction to Riemann surfaces (Addison-Wesley, Reading, Mass.).
64. Springer G 1961 Vvedenie v teoriyu riemanovykh poverkhnostei (Izdat. Inostr. Lit., Moscow) Translation.
65. Dubrovin B A 1976 Finite-zone linear operators and Abelian manifolds Uspekhi Mat. Nauk 31 (4) 259-260.
66. Dubrovin B A 1977 Completely integrable systems, associated with matrix operators Funktsional. Anal. i Prilozhen. 11 (4) 28-41.
67. Dubrovin B A 1977 Functional Anal. Appl. 11 265-277.
68. Manakov S V 1976 Note on the integration of the Euler equations for the dynamics of an n-dimensional rigid body Funktsional. Anal. i Prilozhen. 10 (4) 93-94.
69. Manakov S V 1976 Functional Anal. Appl. 10 328-329.
70. Arnold V I 1974 Matematicheskie metody klassicheskoi mekhaniki (Mathematical methods of classical mechanics) (Nauka, Moscow).
71. Dixmier G 1968 Sur les algebres de Weyl Bull. Soc. Math. France 96 209-242.
72. Dixmier G 1969 Matematika 13 (4) 27-40.
73. Dubrovin B A 1975 The inverse problem of scattering theory for periodic finite-zone potentials Funktsional. Anal. i Prilozhen. 9 (1) 65-66.
74. Dubrovin B A 1975 Functional Anal. Appl. 9 61-62.
75. Kruskal M and N Zabusky 1967 Interaction of 2 solitons in a collisionless plasma and the recurrence of initial states Phys. Rev. Lett. 19 1095-1098.
76. Bogoyavlenskii O I and S P Novikov 1976 On the connection of Hamiltonian formalisms of stationary and non-stationary problems Funktsional. Anal. i Prilozhen. 10 (1) 9-13.
77. Bogoyavlenskii O I and S P Novikov 1976 Functional Anal. Appl. 10 8-11.
78. Lax P D 1975 Periodic solution of the K-dV equation Comm. Pure Appl. Math. 38 (1) 141-188.
79. Gelfand I M and L A Dikii 1975 The asymptotic behaviour of the resolvent of Sturm-Liouville operators and an algebra of Korteweg-de Vries equations Uspekhi Mat. Nauk 30 (5) 67-100.
80. Gelfand I M and L A Dikii 1975 Russian Math. Surveys 30 (5) 77-113.
81. Alber S I 1976 The study of equations of the Korteweg-de Vries type by the method of recurrence relationsArXiv e-print (Inst. Khim. Fiz. Akad. Nauk SSSR, Chernogolovka).
82. Flaschka H and McLaflin 1976 Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions Prog. Theor. Phys. 53 (2) 438-451.
83. Airault H, H McKean and J Moser 1976 Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem Courant Inst. 8.