A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy

Acta Mathematica

Volumn 181, Issue 1, 1998, Pages 63-108

  Gesztesy, Fritz ^a   Weikard, Rudi ^b  

^a UNIVERSITY OF MISSOURI   (United States)

^b UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001804145     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02392748     Document Type: Article

References (105)

1. AIRAULT, H., MCKEAN, H. P. & MOSER, J., Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math., 30 (1977), 95-148.
2. AKHIEZER, N. I., On the spectral theory of Lamé's equation. Istor.-Mat. Issled., 23 (1978), 77-86, 357 (Russian).
3. _ Elements of the Theory of Elliptic Functions. Amer. Math. Soc., Providence, RI, 1990.
4. AL'BER, S. I., Investigation of equations of Korteweg-de Vries type by the method of recurrence relations. J. London Math. Soc., 19 (1979), 467-480 (Russian).
5. ALFIMOV, G. L., ITS, A. R. & KULAGIN, N. E., Modulation instability of solutions of the nonlinear Schrödinger equation. Theoret. and Math. Phys., 84 (1990), 787-793.
6. ARSCOTT, F. M., Periodic Differential Equations. MacMillan, New York, 1964.
7. BABICH, M. V., BOBENKO, A. I. & MATVEEV, V. B., Reductions of Riemann theta-functions of genus g to theta-functions of lower genus, and symmetries of algebraic curves. Soviet Math. Dokl., 28 (1983), 304-308.
8. _ Solutions of nonlinear equations integrable in Jacobi theta functions by the method of the inverse problem, and symmetries of algebraic curves. Math. USSR-Izv., 26 (1986), 479-496.
9. BELOKOLOS, E. D., BOBENKO, A. I., ENOL'SKII, V. Z., ITS, A. R. & MATVEEV, V. B., Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer-Verlag, Berlin, 1994.
10. BELOKOLOS, E. D., BOBENKO, A. I., MATVEEV, V. B. & ENOL'SKII, V. Z., Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations. Russian Math. Surveys, 41:2 (1986), 1-49.
11. BELOKOLOS, E. D. & ENOL'SKII, V. Z., Verdier elliptic solitons and the Weierstrass theory of reduction. Functional Anal. Appl., 23 (1989), 46-47.
12. _ Reduction of theta functions and elliptic finite-gap potentials. Acta Appl. Math., 36 (1994), 87-117.
13. BENNEQUIN, D., Hommage à Jean-Louis Verdier: au jardin des systèmes intégrables, in Integrable Systems: The Verdier Memorial Conference (Luminy, 1991), pp. 1-36. Birkhäuser, Boston, MA, 1993.
14. BUCHSTABER, V. M., ENOL'SKII, V. Z. & LEYKIN, D. V., Hyperelliptic Kleinian functions and applications, in Solitons, Geometry, and Topology: On the Crossroad, pp. 1-33. Amer. Math. Soc. Transl. Ser. 2, 179. Amer. Math. Soc., Providence, RI, 1997.
15. _ Kleinian functions, hyperelliptic Jacobians and applications. To appear in Revs. in Mathematics and Mathematical Physics, 10 (S. Novikov & I. Krichever, eds.), pp. 1-115. Gordon & Breach.
16. BURCHNALL, J. L. & CHAUNDY, T. W., Commutative ordinary differential operators. Proc. London Math. Soc. (2), 21 (1923), 420-440.
17. _ Commutative ordinary differential operators. Proc. Roy. Soc. London Ser. A, 118 (1928), 557-583.
18. BURKHARDT, H., Elliptische Funktionen, 2nd edition. Verlag von Veit, Leipzig, 1906.
19. CALOGERO, F., Exactly solvable one-dimensional many-body problems. Lett. Nuovo Cimento (2), 13 (1975), 411-416.
20. CHOODNOVSKY, D. V. & CHOODNOVSKY, G. V., Pole expansions of nonlinear partial differential equations. Nuovo Cimento B, 40 (1977), 339-353.
21. CHRISTIANSEN, P. L., EILBECK, J. C., ENOLSKII, V. Z. & KOSTOV, N. A., Quasi-periodic solutions of the coupled nonlinear Schrödinger equations. Proc. Roy. Soc. London Ser: A, 451 (1995), 685-700.
22. CHUDNOVSKY, D. V., Meromorphic solutions of nonlinear partial differential equations and many-particle completely integrable systems. J. Math. Phys., 20 (1979), 2416-2422.
23. CODDINGTON, E. A. & LEVINSON, N., Theory of Ordinary Differential Equations. Krieger, Malabar, 1984.
24. COLOMBO, E., PIROLA, G. P. & PREVIATO, E., Density of elliptic solitons. J. Reine Angew. Math., 451 (1994), 161-169.
25. DONAGI, R. & MARKMAN, E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), pp. 1-119. Lecture Notes in Math., 1620. Springer-Verlag, Berlin, 1996.
26. DUOROVIN, B. A. & NOVIKOV, S. P., Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation. Soviet Phys. JETP, 40 (1975), 1058-1063.
27. EHLERS, F. & KNÖRRER, H., An algebro-geometric interpretation of the Bäcklurid transformation for the Korteweg-de Vries equation. Comment. Math. Helv., 57 (1982), 1-10.
28. EILBECK, J. C. & ENOL'SKII, V. Z., Elliptic Baker-Akhiezer functions and an application to an integrable dynamical system. J. Math. Phys., 35 (1994), 1192-1201.
29. ENOL'SKII, V. Z., On the solutions in elliptic functions of integrable nonlinear equations. Phys. Lett. A, 96 (1983), 327-330.
30. _ On the two-gap Lamé potentials and elliptic solutions of the Kovalevskaja problem connected with them. Phys. Lett. A, 100 (1984), 463-466.
31. _ On solutions in elliptic functions of integrable nonlinear equations associated with two-zone Lamé potentials. Soviet Math. Dokl., 30 (1984), 394-397.
32. ENOL'SKII, V. Z. & EILBECK, J. C., On the two-gap locus for the elliptic Calogero-Moser model. J. Phys. A, 28 (1995), 1069-1088.
33. ENOL'SKII, V. Z. & KOSTOV, N. A., On the geometry of elliptic solitons. Acta Appl. Math., 36 (1994), 57-86.
34. FLOQUET, G., Sur les équations différentielles linéaires à coefficients doublement périodiques. C. R. Acad. Sci. Paris, 98 (1884), 38-39, 82-85.
35. _ Sur les équations différentielles linéaires à coefficients doublement périodiques. Ann. Sci. École Norm. Sup. (3), 1 (1884), 181-238.
36. _ Addition a un mémorie sur les équations différentielles linéaires. Ann. Sci. École Norm. Sup. (3), 1 (1884), 405-408.
37. GESZTESY, F. & HOLDEN, H., Darboux-type transformations and hyperelliptic curves. In preparation.
38. GESZTESY, F. & RATNASEELAN, R., An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys., 10 (1998), 345-391.
39. GESZTESY, F. & STICKA, W., On a theorem of Picard. Proc. Amer. Math. Soc., 126 (1998), 1089-1099.
40. GESZTESY, F. & WEIKARD, R., Lamé potentials and the stationary (m)KdV hierarchy. Math. Nachr., 176 (1995), 73-91.
41. _ Treibich-Verdier potentials and the stationary (m)KdV hierarchy. Math. Z., 219 (1995), 451-476.
42. _ On Picard potentials. Differential Integral Equations, 8 (1995), 1453-1476.
43. _ Floquet theory revisited, in Differential Equations and Mathematical Physics (I. Knowles, ed.), pp. 67-84. International Press, Boston, MA, 1995.
44. _ A characterization of elliptic finite-gap potentials. C. R. Acad. Sci. Paris Ser. I Math., 321 (1995), 837-841.
45. _ Picard potentials and Hill's equation on a torus. Acta Math., 176 (1996), 73-107.
46. _ Toward a characterization of elliptic solutions of hierarchies of soliton equations. To appear in Contemp. Math.
47. GREBERT, B. & KAPPELER, T., Gap estimates of the spectrum of the Zakharov Shabat system. Preprint, 1997.
48. GRECO, S. & PREVIATO, E., Spectral curves and ruled surfaces: projective models, in The Curves Seminar at Queen's, Vol. VIII (Kingston, ON, 1990/1991), Exp. F, 33 pp. Queen's Papers in Pure and Appl. Math., 88. Queen's Univ., Kingston, ON, 1991.
49. GRECO, S. & PREVIATO, E., Spectral curves and ruled surfaces: projective models, in The Curves Seminar at Queen's, Vol. VIII (Kingston, ON, 1990/1991), Exp. F, 33 pp. Queen's Papers in Pure and Appl. Math., 88. Queen's Univ., Kingston, ON, 1991.
50. HALPHEN, G.-H., Traité des fonctions elliptiques, tome 2. Gauthier-Villars, Paris, 1888.
51. HERMITE, C., Oeuvres, tome 3. Gauthier-Villars, Paris, 1912.
52. INCE, E. L., Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh Sect. A, 60 (1940), 83-99.
53. _ Ordinary Differential Equations. Dover, New York, 1956.
54. ITS, A. R., Inversion of hyperelliptic integrals and integration of nonlinear differential equations. Vestnik Leningrad Univ. Mat. Mekh. Astronom., 9 (1981), 121-129.
55. ITS, A. R. & ENOL'SKII, V. Z., Dynamics of the Calogero Moser system and the eduction of hyperelliptic integrals to elliptic integrals. Functional Anal. Appl., 20 (1986), 62-64.
56. KLEIN, F., Über den Hermite'schen Fall der Lamé'schen Differentialgleichung. Math. Ann., 40 (1892), 125-129.
57. KONOPELCHENKO, B. G., Elementary Bäcklund transformations, nonlinear superposition principle and solutions of the integrable equations. Phys. Lett. A, 87 (1982), 445-448.
58. KOSTOV, N. A. & ENOL'SKII, V. Z., Spectral characteristics of elliptic solitons. Math. Notes, 53 (1993), 287-293.
59. KRAUSE, M., Theorie der doppeltperiodischen Funktionen einer veränderlichen Grösse, Vol. 2. Teubner, Leipzig, 1897.
60. KRICHEVER, I. M., Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Functional Anal. Appl., 14 (1980), 282-290.
61. _ Nonlinear equations and elliptic curves. Revs. Sci. Technology, 23 (1983), 51-90.
62. Elliptic solutions of nonlinear integrable equations and related topics. Acta Appl. Math., 36 (1994), 7-25.
63. LEE, J. E. & TSUI, M. P., The geometry and completeness of the two-phase solutions of the nonlinear Schrödinger equation, in Nonlinear Evolution Equations and Dynamical Systems (Kolymbari, 1989), pp. 94-97. Springer-Verlag, Berlin, 1990.
64. MARCHENKO, V. A., Sturm-Liouville Operators and Applications. Birkhäuser, Basel, 1986.
65. MATVEEV, V. B. & SMIRNOV, A. O., Symmetric reductions of the Riemann θ-function and some of their applications to the Schrödinger and Boussinesq equation. Amer. Math. Soc. Transl. Ser. 2, 157 (1993), 227-237.
66. MERTSCHING, J., Quasiperiodic solutions of the nonlinear Schrödinger equation. Fortschr. Phys., 35 (1987), 519-536.
67. MITTAG-LEFFLER, G., Sur les équations différentielles linéaires à coefficients doublement périodiques. C. R. Acad. Sci. Paris, 90 (1880), 299-300.
68. MOKHOV, O. I., Commuting differential operators of rank 3, and nonlinear differential equations. Math. USSR-Izv., 35 (1990), 629-655.
69. MOSER, J., Three integrable Hamiltonian systems connected with isospectral deformations. Adv. in Math., 16 (1975), 197-220.
70. NEWELL, A. C., Solitons in Mathematics and Physics. SIAM, Philadelphia, 1985.
71. NOVIKOV, S. P., The periodic problem for the Korteweg de Vries equation. Functional Anal. Appl., 8 (1974), 236-246.
72. NOVIKOV, S., MANAKOV, S. V., PITAEVSKII, L. P. & ZAKHAROV, V. E., Theory of Solitons. Consultants Bureau, New York, 1984.
73. OHMIYA, M., KdV polynomials and Λ-operator. Osaka J. Math., 32 (1995), 409-430.
74. _ Spectrum of Darboux transformation of differential operator. Preprint, 1996.
75. OLSHANETSKY, M. A. & PERELOMOV, A. M., Classical integrable finite-dimensional systems related to Lie Algebras. Phys. Rep., 71 (1981), 313-400.
76. OSBORNE, A. R. & BOFFETTA, G., A summable multiscale expansion for the KdV equation, in Nonlinear Evolution Equations: Integrability and Spectral Methods (A. Degasperis, A. P. Fordy and M. Lakshmanan, eds.), pp. 559-569. Manchester Univ. Press, Manchester, 1990.
77. PAVLOV, M. V., Nonlinear Schrödinger equation and the Bogolyubov-Whitham method of averaging. Theoret. and Math. Phys., 71 (1987), 584-588.
78. PICARD, E., Sur une généralisation des fonctions périodiques et sur certaines équations différentielles linéaires. C. R. Acad. Sci. Paris, 89 (1879), 140-144.
79. Sur une classe d'équations différentielles linéaires. C. R. Acad. Sci. Paris, 90 (1880), 128-131.
80. _ Sur les équations différentielles linéaires à coefficients doublement périodiques. J. Reine Angew. Math., 90 (1881), 281-302.
81. PREVIATO, E., The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, in Hamiltonian Systems, Transformation Groups and Spectral Transform Methods (Montreal, PQ, 1989), pp. 57-67. Univ. Montréal, Montreal, PQ, 1990.
82. _ Monodromy of Boussinesq elliptic operators. Acta Appl. Math., 36 (1994), 49-55.
83. _ Seventy years of spectral curves: 1923-1993, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), pp. 419-481. Lecture Notes in Math., 1620. Springer-Verlag, Berlin, 1996.
84. SMIRNOV, A. O., Real finite-gap regular solutions of the Kaup-Boussinesq equation. Theoret. and Math. Phys., 66 (1986), 19-31.
85. _ A matrix analogue of Appell's theorem and reductions of multidimensional Riemann theta functions. Math. USSR-Sb., 61 (1988), 379-388.
86. Elliptic solutions of the Korteweg de Vries equation. Math. Notes, 45 (1989), 476-481.
87. Real elliptic solutions of the Sine-Gordon equation. Math. USSR-Sb., 70 (1990), 231-240.
88. _ Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math., 36 (1994), 125-166.
89. _ Solutions of the KdV equation elliptic in t. Theoret. and Math. Phys., 100 (1994), 937-947.
90. Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg-de Vries equation. Russian Acad. Sci. Sb. Math., 82 (1995), 461-470.
91. _ The Dirac operator with elliptic potential. Sb. Math., 186 (1995), 1213-1221.
92. TAIMANOV, I. A., Elliptic solutions of nonlinear equations. Theoret. and Math. Phys., 84 (1990), 700-706.
93. _ On the two-gap elliptic potentials. Acta Appl. Math., 36 (1994), 119-124.
94. TKACHENKO, V., Non-selfadjoint periodic Dirac operators. Preprint, 1997.
95. TREIBICH, A., Compactified Jacobians of tangential covers, in Integrable Systems: The Verdier Memorial Conference (Luminy, 1991), pp. 39-60. Birkhäuser, Boston, MA, 1993.
96. _ New elliptic potentials. Acta Appl. Math., 36 (1994), 27-48.
97. TREIBICH, A. & VERDIER, J.-L., Solitons elliptiques, in The Grothendieck Festschrift, Vol. III, pp. 437-480. Progr. Math., 115. Birkhäuser Boston, Boston, MA, 1990.
98. TREIBICH, A. & VERDIER, J.-L., Solitons elliptiques, in The Grothendieck Festschrift, Vol. III, pp. 437-480. Progr. Math., 115. Birkhäuser Boston, Boston, MA, 1990.
99. _ Revêtements tangentiels et sommes de 4 nombres triangulaires. C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 51-54.
100. _ Revêtements exceptionnels et sommes de 4 nombres triangulaires. Duke Math. J., 68 (1992), 217-236.
101. TURBINER, A. V., Lamé equation, sl(2) algebra and isospectral deformations. J. Phys. A, 22 (1989), L1-L3.
102. VERDIER, J.-L., New elliptic solitons, in Algebraic Analysis, Vol. II (M. Kashiwara and T. Kawai, eds.), pp. 901-910. Academic Press, Boston, MA, 1988.
103. WARD, R. S., The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A, 20 (1987), 2679-2683.
104. WEIKARD, R., On Hill's equation with a singular complex-valued potential. Proc. London Math. Soc. (3), 76 (1998), 603-633.
105. WHITTAKER, E. T. & WATSON, G. N., A Course of Modern Analysis. Cambridge Univ. Press, Cambridge, 1996.