Reduction of abelian functions and algebraically integrable systems. I

Journal of Mathematical Sciences

Volumn 106, Issue 6, 2001, Pages 3395-3486

  Belokolos, E D ^a   Enolskii, V Z ^a  

^a UNIVERSITY OF REGINA   (Canada)

Author keywords

[No Author keywords available]

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EID: 23944449678     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1023/A:1011983313249     Document Type: Article

References (89)

1. R. D. M. Accola, Riemann Surfaces, Theta-Functions, and Abelian Automorphisms Groups, Lect. Notes Math., Vol. 485, Springer, Berlin (1975).
2. A. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, London-New York (1965).
3. A. A. Albert, "A note on the Poincaré theorem on impure Riemann matrices," Ann. Math., 36, 151-156 (1935).
4. M. B. Babich, "Effectivization of the formulas of finite-zone integration of the sine-Gordon equation for a curve of genus 3," Funkts. Anal. Prilozh., 19, No. 3, 75-76 (1985).
5. M. B. Babich, A. I. Bobenko, and V. B. Matveev, "Reduction of Riemann theta-functions of genus g to theta-functions of lower genus and symmetries of algebraic curves," Dokl. Akad. Nauk SSSR, 272, 13-17 (1983).
6. M. B. Babich, A. I. Bobenko, and V. B. Matveev, "Solutions of nonlinear equations, integrable by the inverse scattering method in theta-functions, and symmetry of algebraic curves," Izv. Akad. Nauk SSSR, Ser. Mat., 49, 511-529 (1985).
7. H. F. Baker, Abel's Theorem and the Allied Theory, Including the Theory of Theta-Functions, Cambridge Univ. Press, Cambridge (1897).
8. H. F. Baker, "On the hyperelliptic sigma functions," Amer. J. Math., 20, 301-384 (1898).
9. H. F. Baker, Multiply Periodic Functions, Cambridge Univ. Press, Cambridge (1907).
10. H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1955).
11. E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-Geometrical Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).
12. E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, and V. B. Matveev, "Algebraic-geometric principles of superposition of finite-zone solutions of integrable nonlinear equations," Russ. Math. Surveys, 41, 1-49 (1986).
13. C. Birkenhake and H. Lange, "Cubic theta relations," J. Reine Angew. Math., 407, 167-177 (1990).
14. A. I. Bobenko, "On the periodic solution of the sine-Gordon equation," Punkts. Anal. Prilozh., 18, No. 3, 6-19 (1985).
15. A. I. Bobenko, "All constant mean curvature tori in R3, S3, H3 in terms of theta functions," Math. Ann., 290, 209-245 (1991).
16. A. I. Bobenko, "Surfaces in terms of 2 × 2 matrices. Old and new integrable cases," In: Harmonic Maps and Integrable Systems. Aspects Math., Vol. 23 (Leeds) (A. P. Fordy and J. C. Wood, Eds.), Vieweg (1993).
17. O. Bolza, "Über die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades," Math. Ann., 28, 47-496 (1887).
18. M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford (1964).
19. V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, "Kleinian functions, hyperelliptic Jacobians, and applications," In: Reviews in Mathematics and Mathematical Physics (S. P. Novikov and I. M. Krichever, Eds.), Vol. 10:2, Gordon and Breach (1997), pp. 1-125.
20. V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, "Recursive family of polynomials generated by Sylvester's identity and the addition theorem for hyperelliptic Kleinian functions," Punkts. Anal. Prilozh., 31, No. 4, 19-32 (1997).
21. V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, "Rational analogues of the abelian functions," Funkts. Anal. Prilozh., 33, No. 2 (1999).
22. H. Burkhardt, "Beiträge zur Theorie der hyperelliptische Sigmafunktionen," Math. Ann., 32, 381-442 (1888).
23. F. Conforto, Abelsche Funktionen und algebraische Geometry, Springer, Berlin-Göttingen-Heiserberg (1956).
24. B. A. Dubrovin, "Theta functions and nonlinear equations," Russ. Math. Surveys, 36, 11-80 (1981).
25. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, "Nonlinear equations of the KdV type, finite gap linear operators and abelian varieties," Usp. Mat. Nauk, 31, No. 1, 55-135 (1976).
26. J. C. Eilbeck, V. Z. Enolskii, and D. V. Leykin, "On the Kleinian construction of Abelian functions of canonical algebraic curves," In: Proc. Conf. SIDE III: Symmetries of Integrable Difference Equations, Saubadia, May 1998, 1999, in press, pp. 1-22.
27. H. M. Farkas and I. Kra, Riemann Surfaces, Springer, New York (1980).
28. J. D. Fay, Theta Functions on Riemann Surfaces, Lect. Notes Math. (Berlin), Vol. 352, Springer (1973).
29. J. D. Fay, "On the Riemann-Jacobi formula," In: Nachrichten der Akadedemie der Wissenschaften in Gottingen, II. Mathematisch-Physikalische Klasse, 5 (1979), pp. 61-73.
30. J. D. Fay, "On the even-order vanishing of Jacobian theta functions," Duke Math. J., 51, 109-132 (1984).
31. E. V. Ferapontov, Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry, Preprint (1998).
32. A. R. Forsyth, "Memoir on the theta-functions, particularly those of two variables," Phil. Trans., 173, 325-400 (1982).
33. P. Griffith and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1978).
34. R. Hirota, "Exact solutions of the Kortewegde Vries equation for multiple collisions of solitons," Phys. Rev. Lett., 27, 1192-1194 (1972).
35. R. Hirota, "Direct methods in soliton theory," In: Topics in Current Physics (R. Bullough and P. Caudrey, Eds.), Vol. 17, Springer (1980), pp. 157-175.
36. R. Hirota, Y. Ohta, and J. Satsuma, "Solutions of the Kadomtsev-Petviashvili equation and the two-dimensional Toda equations," J. Phys. Soc. Jpn., 57, No. 6, 1901-1904 (1988).
37. R. Horiuchi, "Normal coverings of hyperelliptic Riemann surfaces," J. Math. Kyoto Univ., 19, 487-524 (1979).
38. R. W. H. T. Hudson, Kummer's Quartic Surface, Cambridge Univ. Press, Cambridge (1990).
39. J. Igusa, "On Siegel modular forms of genus two, II," Amer. J. Math., 86, 392-412 (1964).
40. J. Igusa, "Problems on Abelian functions at the time of Poincaré and some at present," Bull. AMS, 6, 161-174 (1982).
41. A. R. Its and V. B. Matveev, "Hill's operators with a finite number of lacunae and multisoliton solutions of the Korteweg-de Vries equation," Teor. Mat. Fiz., 23, 51-67 (1975).
42. C. M. Jessop, A Treatise on the Line Complex, Cambridge Univ. Press, Cambridge (1916).
43. V. G. Kac, Infinite-Dimensional Lie Algebras, Cambridge Univ. Press, Cambridge (1990).
44. F. Klein, "Über hyperelliptische Sigmafunktionen," Math. Ann., 32, 351-380 (1888).
45. F. Klein, Gesammelte Mathematische Abhandlungen, Vol. III, Teubner, Berlin (1923).
46. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics, Interscience Publisher, New York (1964).
47. H. Knörrer, "Die Fresnelsche Wellenfläche," In: Mathematische Miniaturen, Vol. 3, Birkhäuser (1986), pp. 115-141.
48. S. Koizumi, "Theta relations and projective normality of Abelian varieties," Amer. J. Math., 198, 865-889 (1976).
49. S. Koizumi, "Remarks on Takase's paper 'A generalization of Rosenhain's normal form with applications," Proc. Jpn. Acad. 73, 12-13 (1997).
50. L. Königsberger, "Über die Transformation der Abelschen Funktionen erster Ordnung," J. Reine Angew. Math., 64, 17-42 (1865).
51. S. Kowalewski, "Über die Reduction einer bestimmten Klasse Abelscher Integrale 3ten Ranges auf elliptische Integrale," Acta Math., 4, 393-414 (1884).
52. xx = sin u," Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 878-881 (1976).
53. M. Krause, Theorie der doppeltperidischen Funktionen einer veränderlichen Grösse, Vol. I, Teubner, Leipzig (1895).
54. M. Krause, Theorie der doppeltperidischen Funktionen einer veränderlichen Grösse, Vol. II, Teubner, Leipzig (1897).
55. A. Krazer, Lehrbuch der Thetafunktionen, Teubner, Leipzig (1903).
56. A. Krazer and W. Wirtinger, Abelsche Funktionen und allgemeine Thetafunktionen, Teubner (1915).
57. A. Kuribayashi and Komiya, "On Weierstrass points and automorphism of curves of genus three," Lect. Notes Math., Vol. 732, Springer (1978), pp. 252-299.
58. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982).
59. H. Lange and C. Birkenhake, Complex Abelian Varieties, Springer, Berlin (1992).
60. D. V. Leykin, "On Weierstrass cubics for hyperelliptic functions," Usp. Mat. Nauk, 50(6), 191-192 (1995).
61. D. V. Leykin, Dynamical systems integrable in hyperelliptic functions, Ph.D. dissertation, Kiev (1998).
62. S. Lu, "Explicit equations for KP and MKP hierarchies," In: Theta Functions, Bowdoin 1987. Proc. Symp. Pure Math. (Providence, Rhode Island) (L. Ehrenpreis and R. Gunning, Eds.), Vol. 49:1, Amer. Math. Soc. (1989), pp. 101-105.
63. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1979).
64. A. I. Markushevich, Introduction to the Classical Theory of Abelian Functions [in Russian], Nauka, Moscow (1979).
65. H. H. Martens, "A footnote to the Poincaré complete reducibility theorem," Publ. Mat., 36, 111-129 (1992).
66. V. B. Matveev, Abelian functions and solitons, Preprint No. 373, Wroslaw University (1976).
67. V. B. Matveev and A. O. Smirnov, "On the riemann theta functions of trigonal curves and solutions of the Bousinnesq and KP equations," Lett. Math. Phys., 14, 25-31 (1987).
68. V. B. Matveev and A. O. Smirnov, "Some comments on the solvable chiral Potts model," Mat. Zametki, 46(3), 31-39 (1989).
69. T. Muir, A Treatise on the Theory of Determinants, Dover Publ., New York (1928).
70. D. Mumford, Curves and Their Jacobians, Michigan Univ. Press, Ann Arbor (1975).
71. D. Mumford, Tata Lectures on Theta, Vols. 1 and 2, Birkhäuser, Boston (1983, 1984).
72. S. P. Novikov, "Periodic problem for the Korteweg de Vries equation," Funkts. Anal. Prilozh., 74, 54-66 (1974).
73. H. Poincaré, "Sur la reduction des integrales abéliennes," Bull. Soc. Math. France, 12, 124-143 (1884).
74. H. Poincaré, "Sur les fonctions abéliennes," J. Amer. Math., 8, 289-342 (1886).
75. G. Rosenhain, "Abhandlung über die Funktionen zweier Variabler mit vier Perioden," Mem. Pres. l'Acad de Sci. de France des savants, IX, 361-455 (1851).
76. R. Sasaki, "Modular forms vanishing at the reducible points of the upper-half space," J. Reine Angew. Math., 345, 111-121 (1983).
77. M. Sato, "Soliton equations as dynamical systems on infinite dimension Grasmanian manifold," RIMS Kokyuroku, 439, 30-46. (1981)
78. C. L. Siegel, Analytic Functions of Several Variables, Academic Press, New York (1949).
79. C. L. Siegel, Topics in Complex Function Theory. Vol. III. Abelian Functions and Modular Functions of Several Variables, Wiley Interscience, New York-London-Sydney-Toronto (1973).
80. A. O. Smirnov, "A matrix analogue of the Appell theorem and reduction of multidimensional Riemann theta-functions," Mat. Sb., 61, 379-388 (1988).
81. A. O. Smirnov, "Finite-gap elliptic solutions of the KdV equation," Acta Appl. Math., 36, 125-166 (1994).
82. I. A. Taimanov, "Secants of Abelian varieties, theta-functions, and soliton equations," Usp. Mat. Nauk, 52, No. 1, 149-224 (1997).
83. K. Takase, "A generalization of Rosenhain's normal form for hyperelliptic curves with applications," Proc. Jpn. Acad., 72, 162-165 (1996).
84. J. Thomae, "Beitrag zur bestimmung von θ(0,… ,0) durch die Klassenmoduln algebraischer Funktionen," J. Reine Angew. Math., 71, 201 (1869).
85. M. A. Tikhomandritzky, Eléments de la théorie des integrales abéliennes, Nouv. Ed., St. Petersbourgh (1911).
86. G. van der Geer, "On the geometry of a Siegel modular threefold," Math. Ann., 260, 317-350 (1982).
87. H. Weber, "Anwendung der Thetafunktionen zweiter Veranderlicher auf die Theorie der Bewegung Kerpers in einer Flussigkeit," Math. Ann., 14, 173-206 (1879).
88. K. Weierstrass, "Beitrag zur Theorie der Abel'schen Integrale," In: Jahreber. Kónigl. Katolischen Gymnasium zu Braunsberg in dem Schuljahre 1848/49 (1849), pp. 3-23.
89. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1973).