×

Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach. (English) Zbl 0909.34073

Summary: The authors provide an overview on elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. The treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L05 General spectral theory of ordinary differential operators
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991. · Zbl 0762.35001
[2] Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249 – 315. · Zbl 0408.35068
[3] Milton Abramowitz and Irene A. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. · Zbl 0171.38503
[4] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), no. 1, 1 – 30. · Zbl 0428.35067
[5] H. Airault, H. P. McKean, and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), no. 1, 95 – 148. · Zbl 0338.35024
[6] N. I. Ahiezer, On the spectral theory of Lamé’s equation, Istor.-Mat. Issled. 23 (1978), 77 – 86, 357 (Russian).
[7] N. I. Akhiezer, Elements of the theory of elliptic functions, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. · Zbl 0694.33001
[8] G. L. Alfimov, A. R. It\cdot s, and N. E. Kulagin, Modulation instability of solutions of the nonlinear Schrödinger equation, Teoret. Mat. Fiz. 84 (1990), no. 2, 163 – 172 (Russian, with English summary); English transl., Theoret. and Math. Phys. 84 (1990), no. 2, 787 – 793 (1991). · Zbl 0718.35088
[9] P. É. Appell, Sur la transformation des équations différentielles linéaires, Comptes Rendus 91 (1880), 211-214. · JFM 12.0249.03
[10] F. M. Arscott, Periodic differential equations. An introduction to Mathieu, Lamé, and allied functions, International Series of Monographs in Pure and Applied Mathematics, Vol. 66. A Pergamon Press Book, The Macmillan Co., New York, 1964. · Zbl 0121.29903
[11] N. Asano and Y. Kato, Algebraic and spectral methods for nonlinear wave equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 49, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. · Zbl 0728.35001
[12] O. Babelon and M. Talon, The symplectic structure of the spin Calogero model, Phys. Lett. A 236 (1997), 462-468. CMP 98:06 · Zbl 0969.37525
[13] M. V. Babich, A. I. Bobenko, and V. B. Matveev, Reductions of Riemann theta functions of genus \? to theta functions of lesser genus, and symmetries of algebraic curves, Dokl. Akad. Nauk SSSR 272 (1983), no. 1, 13 – 17 (Russian). · Zbl 0556.35114
[14] M. V. Babich, A. I. Bobenko, and V. B. Matveev, Solution of nonlinear equations, integrable by the inverse problem method, in Jacobi theta-functions and the symmetry of algebraic curves, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 3, 511 – 529, 672 (Russian). · Zbl 0583.35012
[15] H. F. Baker, Note on the foregoing paper, “Commutative ordinary differential operators,” by J. L. Burchnall and J. W. Chaundy, Proc. Roy. Soc. London A 118 (1928), 584-593. · JFM 54.0439.02
[16] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994. · Zbl 0809.35001
[17] E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Ènol\(^{\prime}\)skiĭ, Algebro-geometric principles of superposition of finite-zone solutions of integrable nonlinear equations, Uspekhi Mat. Nauk 41 (1986), no. 2(248), 3 – 42 (Russian).
[18] E. D. Belokolos and V. Z. Ènol\(^{\prime}\)skiĭ, Verdier’s elliptic solitons and the Weierstrass reduction theory, Funktsional. Anal. i Prilozhen. 23 (1989), no. 1, 57 – 58 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 1, 46 – 47. · Zbl 0695.35171
[19] E. D. Belokolos and V. Z. Ènol\(^{\prime}\)skiĭ, Isospectral deformations of elliptic potentials, Uspekhi Mat. Nauk 44 (1989), no. 5(269), 155 – 156 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 5, 191 – 193. · Zbl 0701.35129
[20] E. D. Belokolos and V. Z. Ènol\(^{\prime}\)skiĭ, Reduction of theta functions and elliptic finite-gap potentials, Acta Appl. Math. 36 (1994), no. 1-2, 87 – 117. · Zbl 0827.35113
[21] Daniel Bennequin, Hommage à Jean-Louis Verdier: au jardin des systèmes intégrables, Integrable systems (Luminy, 1991) Progr. Math., vol. 115, Birkhäuser Boston, Boston, MA, 1993, pp. 1 – 36 (French). · Zbl 1151.90059
[22] G. D. Birkhoff, Existence and oscillation theorem for a certain boundary value problem, Trans. Amer. Math. Soc. 10 (1909), 259-270. · JFM 40.0375.01
[23] Björn Birnir, Complex Hill’s equation and the complex periodic Korteweg-de Vries equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 1 – 49. · Zbl 0592.47004
[24] Björn Birnir, Singularities of the complex Korteweg-de Vries flows, Comm. Pure Appl. Math. 39 (1986), no. 3, 283 – 305. · Zbl 0605.35076
[25] Björn Birnir, An example of blow-up, for the complex KdV equation and existence beyond the blow-up, SIAM J. Appl. Math. 47 (1987), no. 4, 710 – 725. · Zbl 0647.35076
[26] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math. 78 (1946), 1-96. · Zbl 0063.00523
[27] V. M. Buchstaber, V. Z. Enolskiĭ, and D. V. Leĭkin, Hyperelliptic Kleinian functions and applications, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, pp. 1 – 33. · Zbl 0911.14020
[28] -, Kleinian functions, hyperelliptic Jacobians and applications, to appear in Revs. in Mathematics and Mathematical Physics, Vol. 10, S. Novikov and I. Krichever , Gordon & Breach, pp. 1-115.
[29] J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. Ser. 2 21 (1923), 420-440. · JFM 49.0311.03
[30] -, Commutative ordinary differential operators, Proc. Roy. Soc. London A 118 (1928), 557-583. · JFM 54.0439.01
[31] -, Commutative ordinary differential operators. II.-The identity \(P^n=Q^m,\) Proc. Roy. Soc. London A134 (1932), 471-485.
[32] H. Burkhardt, Elliptische Functionen, 2nd ed., Verlag von Veit, Leipzig, 1906. · JFM 30.0396.01
[33] Mutiara Buys and Allan Finkel, The inverse periodic problem for Hill’s equation with a finite-gap potential, J. Differential Equations 55 (1984), no. 2, 257 – 275. · Zbl 0508.34013
[34] F. Calogero, Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cimento (2) 13 (1975), no. 11, 411 – 416.
[35] F. Calogero, Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related ”solvable” many-body problems, Nuovo Cimento B (11) 43 (1978), no. 2, 177 – 241 (English, with Italian and Russian summaries).
[36] R. C. Carlson and K. R. Goodearl, Commutants of ordinary differential operators, J. Differential Equations 35 (1980), no. 3, 339 – 365. · Zbl 0406.34020
[37] K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. · Zbl 0575.33001
[38] D. V. Čudnovs\(^{\prime}\)kiĭ and G. V. Choodnovsky, Pole expansions of nonlinear partial differential equations, Nuovo Cimento B (11) 40 (1977), no. 2, 339 – 353 (English, with Italian and Russian summaries).
[39] P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, and N. A. Kostov, Quasi-periodic solutions of the coupled nonlinear Schrödinger equations, Proc. Roy. Soc. London Ser. A 451 (1995), no. 1943, 685 – 700. · Zbl 0866.35110
[40] D. V. Chudnovsky, Meromorphic solutions of nonlinear partial differential equations and many-particle completely integrable systems, J. Math. Phys. 20 (1979), no. 12, 2416 – 2422. · Zbl 0455.35096
[41] David V. Chudnovsky and Gregory V. Chudnovsky, Travaux de J. Drach (1919), Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl. Math., vol. 92, Dekker, New York, 1984, pp. 445 – 453.
[42] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985. · Zbl 0064.33002
[43] E. Colombo, G. P. Pirola, and E. Previato, Density of elliptic solitons, J. Reine Angew. Math. 451 (1994), 161 – 169. · Zbl 0810.14011
[44] L. A. Dickey, Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, vol. 12, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. · Zbl 0753.35075
[45] Roger K. Dodd, J. Chris Eilbeck, John D. Gibbon, and Hedley C. Morris, Solitons and nonlinear wave equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982.
[46] Ron Donagi and Eyal Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, Integrable systems and quantum groups (Montecatini Terme, 1993) Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 1 – 119. · Zbl 0853.35100
[47] Ron Donagi and Edward Witten, Supersymmetric Yang-Mills theory and integrable systems, Nuclear Phys. B 460 (1996), no. 2, 299 – 334. · Zbl 0996.37507
[48] J. Drach, Sur les groupes complexes de rationalité et sur l’intégration par quadratures, C. R. Acad. Sci. Paris 167 (1918), 743-746. · JFM 46.0679.01
[49] -, Détermination des cas de réduction de’léquation différentielle \(d^2 y/dx^2=[\phi(x)+h]y\), C. R. Acad. Sci. Paris 168 (1919), 47-50. · JFM 47.0411.03
[50] -, Sur l’intégration par quadratures de’léquation \(d^2 y/dx^2=[\phi(x)+h]y\) , C. R. Acad. Sci. Paris 168 (1919), 337-340. · JFM 47.0412.01
[51] P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. · Zbl 0661.35001
[52] B. A. Dubrovin, A periodic problem for the Korteweg-de Vries equation in a class of short-range potentials, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 41 – 51 (Russian). · Zbl 0338.35022
[53] B. A. Dubrovin, Completely integrable Hamiltonian systems that are associated with matrix operators, and Abelian varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 28 – 41, 96 (Russian).
[54] -, Theta functions and non-linear equations, Russ. Math. Surv. 36:2 (1981), 11-92. · Zbl 0549.58038
[55] B. A. Dubrovin, Matrix finite-gap operators, Current problems in mathematics, Vol. 23, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 33 – 78 (Russian).
[56] B. A. Dubrovin and S. P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 6, 2131 – 2144 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 6, 1058 – 1063.
[57] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh and London, 1973. · Zbl 0287.34016
[58] J. C. Eilbeck and V. Z. Ènol\(^{\prime}\)skiĭ, Elliptic Baker-Akhiezer functions and an application to an integrable dynamical system, J. Math. Phys. 35 (1994), no. 3, 1192 – 1201. · Zbl 0978.33501
[59] J. C. Eilbeck and V. Z. Ènol\(^{\prime}\)skiĭ, Elliptic solutions and blow-up in an integrable Hénon-Heiles system, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 6, 1151 – 1164. · Zbl 0814.35118
[60] V. Z. Ènol\(^{\prime}\)skiĭ, On the solutions in elliptic functions of integrable nonlinear equations, Phys. Lett. A 96 (1983), no. 7, 327 – 330.
[61] V. Z. Ènol\(^{\prime}\)skiĭ, On the two-gap Lamé potentials and elliptic solutions of the Kovalevskaja problem connected with them, Phys. Lett. A 100 (1984), no. 9, 463 – 466.
[62] V. Z. Ènol\(^{\prime}\)skiĭ, Solutions in elliptic functions of integrable nonlinear equations connected with two-zone Lamé potentials, Dokl. Akad. Nauk SSSR 278 (1984), no. 2, 305 – 308 (Russian).
[63] V. Z. Ènol\(^{\prime}\)skiĭ and J. C. Eilbeck, On the two-gap locus for the elliptic Calogero-Moser model, J. Phys. A 28 (1995), no. 4, 1069 – 1088. · Zbl 0854.35097
[64] V. Z. Ènol\(^{\prime}\)skiĭ and N. A. Kostov, On the geometry of elliptic solitons, Acta Appl. Math. 36 (1994), no. 1-2, 57 – 86. · Zbl 0810.35104
[65] A. Erdélyi, On Lamé functions, Phil. Mag. (7) 31 (1941), 123-1130. · JFM 67.0236.02
[66] E. Fermi, J. Pasta, and S. M. Ulam, Studies in nonlinear problems, Technical Report LA-1940, Los Alamos Sci. Lab. Also in: Collected Papers of Enrico Fermi, Vol II, 978-988, University of Chicago Press, 1965.
[67] Allan Finkel, Eli Isaacson, and Eugene Trubowitz, An explicit solution of the inverse periodic problem for Hill’s equation, SIAM J. Math. Anal. 18 (1987), no. 1, 46 – 53. · Zbl 0622.34021
[68] H. Flaschka, On the inverse problem for Hill’s operator, Arch. Rational Mech. Anal. 59 (1975), no. 4, 293 – 309. · Zbl 0376.34016
[69] G. Floquet, Sur la théorie des équations différentielles linéaires, Ann. Sci. École Norm. Sup. 8 (1879), suppl., 1-132. · JFM 11.0239.01
[70] -, Sur les équations différentielles linéaires à coefficients périodiques, C. R. Acad. Sci. Paris 91 (1880), 880-882. · JFM 12.0259.01
[71] -, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. 12 (1883), 47-88. · JFM 15.0279.01
[72] -, Sur les équations différentielles linéaires à coefficients doublement périodiques, C. R. Acad. Sci. Paris 98 (1884), 38-39, 82-85. · JFM 16.0279.01
[73] -, Sur les équations différentielles linéaires à coefficients doublement périodiques, Ann. Sci. Ecole Norm. Sup. 1 (1884), 181-238.
[74] -, Addition a un mémorie sur les équations différentielles linéaires, Ann. Sci. Ecole Norm. Sup. 1 (1884), 405-408. · JFM 16.0279.02
[75] Andrew Russell Forsyth, Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Six volumes bound as three, Dover Publications, Inc., New York, 1959. · Zbl 0088.05802
[76] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097. · Zbl 1061.35520
[77] Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97 – 133. · Zbl 0291.35012
[78] C. S. Gardner and G. K. Morikawa, Similarity in the asymptotic behavior of collision free hydromagnetic waves and water waves, Research Report NYO-9082, Courant Institute, 1960.
[79] M. G. Gasymov, Spectral analysis of a class of second-order nonselfadjoint differential operators, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 14 – 19, 96 (Russian).
[80] M. G. Gasymov, Spectral analysis of a class of ordinary differential operators with periodic coefficients, Dokl. Akad. Nauk SSSR 252 (1980), no. 2, 277 – 280 (Russian). · Zbl 0527.34023
[81] Letterio Gatto and Silvio Greco, Algebraic curves and differential equations: an introduction, The Curves Seminar at Queen’s, Vol. VIII (Kingston, ON, 1990/1991) Queen’s Papers in Pure and Appl. Math., vol. 88, Queen’s Univ., Kingston, ON, 1991, pp. Exp. B, 69. · Zbl 0758.14012
[82] I. M. Gel\(^{\prime}\)fand and L. A. Dikiĭ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk 30 (1975), no. 5(185), 67 – 100 (Russian).
[83] I. M. Gel\(^{\prime}\)fand and L. A. Dikiĭ, Fractional powers of operators, and Hamiltonian systems, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 13 – 29 (Russian). · Zbl 0346.35085
[84] I. M. Gel\(^{\prime}\)fand and L. A. Dikiĭ, Integrable nonlinear equations and the Liouville theorem, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 8 – 20, 96 (Russian).
[85] F. Gesztesy and H. Holden, Darboux-type transformations and hyperelliptic curves, in preparation. · Zbl 1102.14301
[86] -, Hierarchies of Soliton Equations and their Algebro-Geometric Solutions, monograph in preparation.
[87] F. Gesztesy and R. Ratneseelan, An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev. Math. Phys. 10 (1998), 345-391. CMP 98:14
[88] Fritz Gesztesy and Barry Simon, The xi function, Acta Math. 176 (1996), no. 1, 49 – 71. · Zbl 0885.34070
[89] F. Gesztesy, B. Simon, and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators, J. d’Anal. Math. 70 (1996), 267-324. CMP 97:11 · Zbl 0951.34061
[90] F. Gesztesy and W. Sticka, On a theorem of Picard, Proc. Amer. Math. Soc. 126 (1998), 1089-1099. CMP 98:06 · Zbl 1125.34347
[91] F. Gesztesy and R. Weikard, Spectral deformations and soliton equations, Differential equations with applications to mathematical physics, Math. Sci. Engrg., vol. 192, Academic Press, Boston, MA, 1993, pp. 101 – 139. · Zbl 0795.35099
[92] -, Floquet theory revisited, Differential Equations and Mathematical Physics (ed. by I. Knowles), International Press, Boston, 1995, 67-84. · Zbl 0946.47031
[93] F. Gesztesy and R. Weikard, Lamé potentials and the stationary (m)KdV hierarchy, Math. Nachr. 176 (1995), 73 – 91. · Zbl 0843.35100
[94] F. Gesztesy and R. Weikard, Treibich-Verdier potentials and the stationary (m)KdV hierarchy, Math. Z. 219 (1995), no. 3, 451 – 476. · Zbl 0830.35119
[95] F. Gesztesy and R. Weikard, On Picard potentials, Differential Integral Equations 8 (1995), no. 6, 1453 – 1476. · Zbl 0846.35119
[96] Fritz Gesztesy and Rudi Weikard, A characterization of elliptic finite-gap potentials, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 7, 837 – 841 (English, with English and French summaries). · Zbl 0867.58035
[97] Fritz Gesztesy and Rudi Weikard, Picard potentials and Hill’s equation on a torus, Acta Math. 176 (1996), no. 1, 73 – 107. · Zbl 0927.37040
[98] -, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math. 181 (1998), to appear. · Zbl 0955.34073
[99] -, Toward a characterization of elliptic solutions of hierarchies of soliton equations, Contemp. Math., to appear.
[100] -, in preparation.
[101] M. Giertz, M. K. Kwong, and A. Zettl, Commuting linear differential expressions, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), no. 3-4, 331 – 347. · Zbl 0524.34011
[102] Jeremy Gray, Linear differential equations and group theory from Riemann to Poincaré, Birkhäuser Boston, Inc., Boston, MA, 1986. · Zbl 0596.01018
[103] Silvio Greco and Emma Previato, Spectral curves and ruled surfaces: projective models, The Curves Seminar at Queen’s, Vol. VIII (Kingston, ON, 1990/1991) Queen’s Papers in Pure and Appl. Math., vol. 88, Queen’s Univ., Kingston, ON, 1991, pp. Exp. F, 33. · Zbl 0758.14014
[104] P. G. Grinevich, Rational solutions of equations of commutation of differential operators, Funktsional. Anal. i Prilozhen. 16 (1982), no. 1, 19 – 24, 96 (Russian). · Zbl 0514.47034
[105] V. Guillemin and A. Uribe, Hardy functions and the inverse spectral method, Comm. Partial Differential Equations 8 (1983), no. 13, 1455 – 1474. · Zbl 0567.35073
[106] G.-H. Halphen, Memoire sur la reduction des equations differentielles lineaires aux formes integrales, Mem. pres. l’Acad. Sci., France 28 (1884), 1-300.
[107] -, Sur une nouvelle classe d’équations différentielles linéaires intégrables, C. R. Acad. Sci. Paris 101 (1885), 1238-1240. · JFM 17.0284.02
[108] -, Traité des Fonctions Elliptiques, tome 2, Gauthier-Villars, Paris, 1888.
[109] G. Hamel, Über die lineare Differentialgleichung zweiter Ordnung mit periodischen Koeffizienten, Math. Ann. 73 (1913), 371-412. · JFM 44.0375.02
[110] O. Haupt, Über lineare homogene Differentialgleichungen 2. Ordnung mit periodischen Koeffizienten, Math. Ann. 79 (1919), 278-285. · JFM 46.0666.03
[111] C. Hermite, Sur quelques applications des fonctions elliptiques, Comptes Rendus 85 (1877), 689-695, 728-732, 821-826. · JFM 09.0349.01
[112] -, Oeuvres, tome 3, Gauthier-Villars, Paris, 1912.
[113] G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math. 8 (1886), 1-36. Reprinted from a paper first published in 1877. · JFM 18.1106.01
[114] Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. · Zbl 0901.34001
[115] Harry Hochstadt, On the determination of a Hill’s equation from its spectrum, Arch. Rational Mech. Anal. 19 (1965), 353 – 362. · Zbl 0128.31201
[116] I. D. Iliev, E. Kh. Khristov, and K. P. Kirchev, Spectral methods in soliton equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 73, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. Appendix 3.B by A. Yanovski [A. B. Yanovskiĭ]. · Zbl 0874.35104
[117] E. L. Ince, Further investigations into the periodic Lamé functions, Proc. Roy. Soc. Edinburgh 60 (1940), 83-99. · Zbl 0027.21201
[118] -, Ordinary Differential Equations, Dover, New York, 1956.
[119] H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory — curves and periods, Nuclear Phys. B 477 (1996), no. 3, 855 – 877. · Zbl 0925.81362
[120] A. R. Its, Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestnik Leningrad. Univ. 7 Mat. Meh. Astronom. vyp. 2 (1976), 39 – 46, 162 (Russian, with English summary). · Zbl 0336.35025
[121] A. R. It\cdot s and V. Z. Ènol\(^{\prime}\)skiĭ, The dynamics of the Calogero-Moser system and reduction of hyperelliptic integrals to elliptic integrals, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 73 – 74 (Russian).
[122] A. R. Its and V. B. Matveev, Schrödinger operators with the finite-band spectrum and the \?-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz. 23 (1975), no. 1, 51 – 68 (Russian, with English summary).
[123] Katsunori Iwasaki, Inverse problem for Sturm-Liouville and Hill equations, Ann. Mat. Pura Appl. (4) 149 (1987), 185 – 206. · Zbl 0641.34012
[124] F. Klein, Über den Hermite’schen Fall der Lamé’schen Differentialgleichung, Math. Ann. 40 (1892), 125-129. · JFM 24.0308.01
[125] Q. Kong and A. Zettl, Dependence of eigenvalues of Sturm-Liouville problems on the boundary, J. Differential Equations 126 (1996), no. 2, 389 – 407. · Zbl 0856.34027
[126] Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations 131 (1996), no. 1, 1 – 19. · Zbl 0862.34020
[127] B. G. Konopel\(^{\prime}\)chenko, Elementary Bäcklund transformations, nonlinear superposition principle and solutions of the integrable equations, Phys. Lett. A 87 (1981/82), no. 9, 445 – 448.
[128] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422-443. · JFM 26.0881.02
[129] N. A. Kostov and V. Z. Ènol\(^{\prime}\)skiĭ, Spectral characteristics of elliptic solitons, Mat. Zametki 53 (1993), no. 3, 62 – 71 (Russian); English transl., Math. Notes 53 (1993), no. 3-4, 287 – 293. · Zbl 0818.35097
[130] S. Kotani, Generalized Floquet theory for stationary Schrödinger operators in one dimension, Chaos, Solitons and Fractals 8 (1997), 1817-1854. CMP 98:03 · Zbl 0936.34074
[131] M. Krause, Theorie der doppeltperiodischen Funktionen einer veränderlichen Grösse, Vol. 1, 1895, Vol. 2, 1897, Teubner, Leipzig. · JFM 26.0482.05
[132] I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977), 12-26. · Zbl 0368.35022
[133] -, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv. 32:6 (1977), 185-213.
[134] -, Rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of \(N\) particles on a line, Funct. Anal. Appl. 12 (1978), 59-61. · Zbl 0408.70010
[135] Igor Moiseevich Krichever, Elliptic solutions of the Kadomcev-Petviašvili equations, and integrable systems of particles, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 45 – 54, 95 (Russian).
[136] Igor Moiseevich Krichever, Nonlinear equations and elliptic curves, Current problems in mathematics, Vol. 23, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 79 – 136 (Russian).
[137] -, Rational solutions of the Zakharov-Shabat equations and completely integrable systems of \(N\) particles on a line, J. Sov. Math. 21, 335-345 (1983). · Zbl 0515.35005
[138] Igor Moiseevich Krichever, Elliptic solutions of nonlinear integrable equations and related topics, Acta Appl. Math. 36 (1994), no. 1-2, 7 – 25. · Zbl 0811.35123
[139] -, Elliptic solutions to difference non-linear equations and nested Bethe ansatz equations, preprint, solv-int/9804016.
[140] Igor Moiseevich Krichever, O. Babelon, E. Billey, and M. Talon, Spin generalization of the Calogero-Moser system and the matrix KP equation, Topics in topology and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 170, Amer. Math. Soc., Providence, RI, 1995, pp. 83 – 119. · Zbl 0843.58069
[141] I. M. Krichever and D. H. Phong, On the integrable geometry of soliton equations and \?=2 supersymmetric gauge theories, J. Differential Geom. 45 (1997), no. 2, 349 – 389. · Zbl 0889.58044
[142] I. Krichever, P. Wiegmann, and A. Zabrodin, Elliptic solutions to difference non-linear equations and related many-body problems, Commun. Math. Phys. 193 (1998), 373-396. CMP 98:13 · Zbl 0907.35124
[143] Igor Moiseevich Krichever and A. Zabrodin, Spin generalization of the Ruijsenaars-Schneider model, the nonabelian two-dimensionalized Toda lattice, and representations of the Sklyanin algebra, Uspekhi Mat. Nauk 50 (1995), no. 6(306), 3 – 56 (Russian); English transl., Russian Math. Surveys 50 (1995), no. 6, 1101 – 1150. · Zbl 0960.37030
[144] V. B. Kuznetsov, F. W. Nijhoff, and E. K. Sklyanin, Separation of variables for the Ruijsenaars system, Commun. Math. Phys. 189 (1997), 855-877. CMP 98:04 · Zbl 0892.58040
[145] Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467 – 490. · Zbl 0162.41103
[146] -, Outline of a theory of the KdV equation, Recent Mathematical Methods in Nonlinear Wave Propagation (ed. by T. Ruggeri), Lecture Notes in Mathematics 1640 (1996), Springer, Berlin, 70-102. CMP 98:07
[147] J. E. Lee and M. P. Tsui, The geometry and completeness of the two-phase solutions of the nonlinear Schrödinger equation, Nonlinear Evolution Equations and Dynamical Systems (ed. by S. Carillo and O. Ragnisco), Springer, Berlin, 1990, 94-97. CMP 91:02
[148] A. M. Levin and M. A. Olshanetsky, Hierarchies of isomonodromic deformations and Hitchin systems, preprint, hep-th/9709207. · Zbl 0938.37045
[149] A. Liapounoff, Sur une équation transcendante et les équations différentielles linéaires du second ordre à coefficients périodiques, Comptes Rendus 128 (1899), 1085-1088. · JFM 30.0303.01
[150] Wilhelm Magnus and Stanley Winkler, Hill’s equation, Dover Publications, Inc., New York, 1979. Corrected reprint of the 1966 edition. · Zbl 0158.09604
[151] A. V. Marshakov, On integrable systems and supersymmetric gauge theories, Teoret. Mat. Fiz. 112 (1997), no. 1, 3 – 46 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 112 (1997), no. 1, 791 – 826 (1998). · Zbl 0978.37503
[152] Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. · Zbl 0592.34011
[153] A. I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, Second English edition, Chelsea Publishing Co., New York, 1977. Translated and edited by Richard A. Silverman. · Zbl 0357.30002
[154] V. B. Matveev, Some comments on the rational solutions of the Zakharov-Schabat equations, Lett. Math. Phys. 3 (1979), no. 6, 503 – 512. · Zbl 0435.35074
[155] V. B. Matveev and A. O. Smirnov, Symmetric reductions of the Riemann \(\theta\)-function and some of their applications to the Schrödinger and Boussinesq equation, Amer. Math. Soc. Transl. (2) 157 (1993), 227-237. CMP 94:05 · Zbl 0807.35137
[156] David McGarvey, Operators commuting with translation by one. I. Representation theorems, J. Math. Anal. Appl. 4 (1962), 366 – 410. · Zbl 0115.33203
[157] D. C. McGarvey, Operators commuting with translation by one. II. Differential operators with periodic coefficients in \?_{\?}(-\infty ,\infty ), J. Math. Anal. Appl. 11 (1965), 564 – 596. , https://doi.org/10.1016/0022-247X(65)90105-8 D. C. McGarvey, Operators commuting with translation by one. III. Perturbation results for periodic differential operators, J. Math. Anal. Appl. 12 (1965), 187 – 234. · Zbl 0188.21201
[158] D. C. McGarvey, Operators commuting with translation by one. II. Differential operators with periodic coefficients in \?_{\?}(-\infty ,\infty ), J. Math. Anal. Appl. 11 (1965), 564 – 596. , https://doi.org/10.1016/0022-247X(65)90105-8 D. C. McGarvey, Operators commuting with translation by one. III. Perturbation results for periodic differential operators, J. Math. Anal. Appl. 12 (1965), 187 – 234. · Zbl 0188.21201
[159] H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217 – 274. · Zbl 0319.34024
[160] H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143 – 226. · Zbl 0339.34024
[161] J. Mertsching, Quasiperiodic solutions of the nonlinear Schrödinger equation, Fortschr. Phys. 35 (1987), no. 7, 519 – 536.
[162] G. Mittag-Leffler, Sur les équations différentielles linéaires à coefficients doublement périodiques, C. R. Acad. Sci. Paris, 90, 299-300 (1880). · JFM 12.0257.02
[163] Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202 – 1204. , https://doi.org/10.1063/1.1664700 Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys. 9 (1968), 1204 – 1209. · Zbl 0283.35019
[164] Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202 – 1204. , https://doi.org/10.1063/1.1664700 Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys. 9 (1968), 1204 – 1209. · Zbl 0283.35019
[165] J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197 – 220. · Zbl 0303.34019
[166] J. Moser, Integrable Hamiltonian systems and spectral theory, Lezioni Fermiane. [Fermi Lectures], Scuola Normale Superiore, Pisa, 1983.
[167] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related nonlinear equation, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 115 – 153.
[168] S. P. Novikov, A periodic problem for the Korteweg-de Vries equation. I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54 – 66 (Russian).
[169] S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. · Zbl 0598.35003
[170] M. A. Olshanetsky and A. M. Perelomov, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), no. 5, 313 – 400.
[171] A. R. Osborne and G. Boffetta, A summable multiscale expansion for the KdV equation, Nonlinear Evolution Equations: Integrability and Spectral Methods (ed. by A. Degasperis, A. P. Fordy, and M. Lakshmanan), Manchester Univ. Press, Manchester, 1990, 559-569. · Zbl 0726.35115
[172] Richard S. Palais, The symmetries of solitons, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 4, 339 – 403. · Zbl 0886.58040
[173] L. A. Pastur and V. A. Tkachenko, On the spectral theory of Schrödinger operators with periodic complex-valued potentials, Funktsional. Anal. i Prilozhen. 22 (1988), no. 2, 85 – 86 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 2, 156 – 158. · Zbl 0717.34096
[174] L. A. Pastur and V. A. Tkachenko, An inverse problem for a class of one-dimensional Schrödinger operators with complex periodic potential, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 6, 1252 – 1269 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 3, 611 – 629. · Zbl 0718.34015
[175] L. A. Pastur and V. A. Tkachenko, On the geometry of the spectrum of the one-dimensional Schrödinger operator with periodic complex-valued potential, Mat. Zametki 50 (1991), no. 4, 88 – 95, 159 (Russian); English transl., Math. Notes 50 (1991), no. 3-4, 1045 – 1050 (1992). · Zbl 0781.34054
[176] M. V. Pavlov, The nonlinear Schrödinger equation and the Bogolyubov-Whitham averaging method, Teoret. Mat. Fiz. 71 (1987), no. 3, 351 – 356 (Russian, with English summary).
[177] R. Pego, Origin of the KdV equation, Notices Amer. Math. Soc. 45 (1998), 358.
[178] Dmitry Pelinovsky, Rational solutions of the Kadomtsev-Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution, J. Math. Phys. 35 (1994), no. 11, 5820 – 5830. · Zbl 0817.35097
[179] E. Picard, Sur une généralisation des fonctions périodiques et sur certaines équations différentielles linéaires, C. R. Acad. Sci. Paris 89 (1879), 140-144. · JFM 11.0241.01
[180] -, Sur une classe d’équations différentielles linéaires, C. R. Acad. Sci. Paris 90 (1880), 128-131. · JFM 12.0257.01
[181] -, Sur les équations différentielles linéaires à coefficients doublement périodiques, J. reine angew. Math. 90 (1881), 281-302. · JFM 13.0255.01
[182] -, Leçons sur Quelques Équations Fonctionnelles, Gauthier Villars, Paris, 1928.
[183] Emma Previato, The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, Hamiltonian systems, transformation groups and spectral transform methods (Montreal, PQ, 1989) Univ. Montréal, Montreal, QC, 1990, pp. 57 – 67. · Zbl 0749.35047
[184] Emma Previato, Monodromy of Boussinesq elliptic operators, Acta Appl. Math. 36 (1994), no. 1-2, 49 – 55. · Zbl 0837.33012
[185] Emma Previato, Seventy years of spectral curves: 1923 – 1993, Integrable systems and quantum groups (Montecatini Terme, 1993) Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 419 – 481. · Zbl 0851.35120
[186] Emma Previato and Jean-Louis Verdier, Boussinesq elliptic solitons: the cyclic case, Proceedings of the Indo-French Conference on Geometry (Bombay, 1989) Hindustan Book Agency, Delhi, 1993, pp. 173 – 185. · Zbl 0844.14016
[187] F. S. Rofe-Beketov, On the spectrum of non-selfadjoint differential operators with periodic coefficients, Dokl. Akad. Nauk SSSR 152 (1963), 1312 – 1315 (Russian). · Zbl 0199.14002
[188] S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), no. 2, 191 – 213. · Zbl 0673.58024
[189] Jean-Jacques Sansuc and Vadim Tkachenko, Spectral properties of non-selfadjoint Hill’s operators with smooth potentials, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 371 – 385. · Zbl 0844.34087
[190] Jean-Jacques Sansuc and Vadim Tkachenko, Spectral parametrization of non-selfadjoint Hill’s operators, J. Differential Equations 125 (1996), no. 2, 366 – 384. · Zbl 0844.34088
[191] J.-J. Sansuc and V. Tkachenko, Characterization of the periodic and anti-periodic spectra of nonselfadjoint Hill’s operators, New results in operator theory and its applications, Oper. Theory Adv. Appl., vol. 98, Birkhäuser, Basel, 1997, pp. 216 – 224. · Zbl 0884.34090
[192] J. Schur, Über vertauschbare lineare Differentialausdrücke, Sitzungsber. der Berliner Math. Gesell. 4 (1905), 2-8. · JFM 36.0387.01
[193] Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5 – 65. · Zbl 0592.35112
[194] Takahiro Shiota, Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys. 35 (1994), no. 11, 5844 – 5849. · Zbl 0827.35118
[195] A. O. Smirnov, Elliptic solutions of the Korteweg-de Vries equation, Mat. Zametki 45 (1989), no. 6, 66 – 73, 111 (Russian); English transl., Math. Notes 45 (1989), no. 5-6, 476 – 481. · Zbl 0706.35122
[196] A. O. Smirnov, Real elliptic solutions of the sine-Gordon equation, Mat. Sb. 181 (1990), no. 6, 804 – 812 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 231 – 240. · Zbl 0705.35128
[197] A. O. Smirnov, Finite-gap elliptic solutions of the KdV equation, Acta Appl. Math. 36 (1994), no. 1-2, 125 – 166. · Zbl 0828.35119
[198] A. O. Smirnov, Solutions of the KdV equation that are elliptic in \?, Teoret. Mat. Fiz. 100 (1994), no. 2, 183 – 198 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 100 (1994), no. 2, 937 – 947 (1995). · Zbl 0875.35107
[199] A. O. Smirnov, The Dirac operator with elliptic potential, Mat. Sb. 186 (1995), no. 8, 133 – 141 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 8, 1213 – 1221. · Zbl 0863.35029
[200] A. O. Smirnov, Elliptic solutions of the nonlinear Schrödinger equation and a modified Korteweg-de Vries equation, Mat. Sb. 185 (1994), no. 8, 103 – 114 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 82 (1995), no. 2, 461 – 470. · Zbl 0854.35110
[201] -, On a class of elliptic solutions of the Boussinesq equations, Theoret. Math. Phys. 109 (1996), 1515-1522. CMP 98:01
[202] A. O. Smirnov, Solutions of the nonlinear Schrödinger equation that are elliptic in \?, Teoret. Mat. Fiz. 107 (1996), no. 2, 188 – 200 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 107 (1996), no. 2, 568 – 578. · Zbl 0927.35110
[203] A. O. Smirnov, On a class of elliptic potentials of the Dirac operator, Mat. Sb. 188 (1997), no. 1, 109 – 128 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 1, 115 – 135. · Zbl 0920.35122
[204] A. O. Smirnov, Real elliptic solutions of equations related to the sine-Gordon equation, Algebra i Analiz 8 (1996), no. 3, 196 – 211 (Russian); English transl., St. Petersburg Math. J. 8 (1997), no. 3, 513 – 524. · Zbl 0877.35114
[205] -, 3-elliptic solutions of the sine-Gordon equation, Math. Notes 62 (1997), 368-376. CMP 98:12
[206] V. V. Sokolov, Examples of commutative rings of differential operators, Funkcional. Anal. i Priložen. 12 (1978), no. 1, 82 – 83 (Russian). · Zbl 0408.34017
[207] I. A. Taĭmanov, Elliptic solutions of nonlinear equations, Teoret. Mat. Fiz. 84 (1990), no. 1, 38 – 45 (Russian, with English summary); English transl., Theoret. and Math. Phys. 84 (1990), no. 1, 700 – 706 (1991). · Zbl 0760.35021
[208] I. A. Taĭmanov, On the two-gap elliptic potentials, Acta Appl. Math. 36 (1994), no. 1-2, 119 – 124. · Zbl 0843.33011
[209] C.-L. Terng and K. Uhlenbeck, Poisson actions and scattering theory for integrable systems, preprint, dg-ga/9707004. · Zbl 0935.35163
[210] V. A. Tkachenko, Spectral analysis of the one-dimensional Schrödinger operator with periodic complex-valued potential, Sov. Math. Dokl. 5 (1964), 413-415. · Zbl 0188.46103
[211] V. A. Tkachenko, Spectral analysis of the nonselfadjoint Hill operator, Dokl. Akad. Nauk SSSR 322 (1992), no. 2, 248 – 252 (Russian); English transl., Soviet Math. Dokl. 45 (1992), no. 1, 78 – 82. · Zbl 0791.34061
[212] V. A. Tkachenko, Discriminants and generic spectra of nonselfadjoint Hill’s operators, Spectral operator theory and related topics, Adv. Soviet Math., vol. 19, Amer. Math. Soc., Providence, RI, 1994, pp. 41 – 71. · Zbl 0815.34018
[213] -, Spectral properties of periodic Dirac operator with skew-symmetric potential matrix, preprint, 1994.
[214] V. Tkachenko, Spectra of non-selfadjoint Hill’s operators and a class of Riemann surfaces, Ann. of Math. (2) 143 (1996), no. 2, 181 – 231. · Zbl 0856.34087
[215] -, Non-selfadjoint periodic Dirac operators, preprint, 1997.
[216] -, Non-selfadjoint periodic Dirac operators with finite-band spectrum, preprint, 1998.
[217] Armando Treibich, Tangential polynomials and elliptic solitons, Duke Math. J. 59 (1989), no. 3, 611 – 627. · Zbl 0698.14029
[218] Armando Treibich, Compactified Jacobians of tangential covers, Integrable systems (Luminy, 1991) Progr. Math., vol. 115, Birkhäuser Boston, Boston, MA, 1993, pp. 39 – 59. · Zbl 0833.14019
[219] Armando Treibich, Revêtements tangentiels et condition de Brill-Noether, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 8, 815 – 817 (French, with English and French summaries). · Zbl 0789.14023
[220] Armando Treibich, New elliptic potentials, Acta Appl. Math. 36 (1994), no. 1-2, 27 – 48. · Zbl 0824.14023
[221] -, Matrix elliptic solitons, Duke Math. J. 90 (1997), 523-547. CMP 98:04
[222] A. Treibich and J.-L. Verdier, Solitons elliptiques, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 437 – 480 (French). With an appendix by J. Oesterlé. · Zbl 0726.14024
[223] Armando Treibich and Jean-Louis Verdier, Revêtements tangentiels et sommes de 4 nombres triangulaires, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 1, 51 – 54 (French, with English summary). · Zbl 0712.33014
[224] Armando Treibich and Jean-Louis Verdier, Revêtements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J. 68 (1992), no. 2, 217 – 236 (French). · Zbl 0806.14013
[225] A. Treibich and J.-L. Verdier, Variétés de Kritchever des solitons elliptiques de KP, Proceedings of the Indo-French Conference on Geometry (Bombay, 1989) Hindustan Book Agency, Delhi, 1993, pp. 187 – 232 (French). · Zbl 0837.14011
[226] -, Au-delà des potentiels et rêvetements tangentiels hyperelliptiques exceptionnels, C. R. Acad. Sci. Paris 325 (1997), 1101-1106. CMP 98:10
[227] A. V. Turbiner, Lamé equation, \?\?(2) algebra and isospectral deformations, J. Phys. A 22 (1989), no. 1, L1 – L3. · Zbl 0662.34031
[228] K. L. Vaninsky, Trace formula for a system of particles with elliptic potential, preprint, solv-int/9707002. · Zbl 0921.35150
[229] J.-L. Verdier, New elliptic solitons, Algebraic analysis, Vol. II, Academic Press, Boston, MA, 1988, pp. 901 – 910.
[230] G. Wallenberg, Über die Vertauschbarkeit homogener linearer Differentialausdrücke, Arch. Math. Phys. 4 (1903), 252-268. · JFM 34.0350.01
[231] R. S. Ward, The Nahm equations, finite-gap potentials and Lamé functions, J. Phys. A 20 (1987), no. 10, 2679 – 2683. · Zbl 0639.34042
[232] R. Weikard, On Hill’s equation with a singular complex-valued potential, Proc. London Math. Soc. 76 (1998), 603-633. CMP 98:11 · Zbl 0905.34004
[233] -, On rational and periodic solutions of stationary KdV equations, preprint 1997.
[234] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. · JFM 53.0180.04
[235] George Wilson, Commuting flows and conservation laws for Lax equations, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 131 – 143. · Zbl 0427.35024
[236] George Wilson, Algebraic curves and soliton equations, Geometry today (Rome, 1984) Progr. Math., vol. 60, Birkhäuser Boston, Boston, MA, 1985, pp. 303 – 329. · Zbl 0581.35065
[237] A. Wintner, Stability and spectrum in the wave mechanics of lattices, Phys. Rev. 72 (1947), 81-82. · Zbl 0029.18601
[238] -, On the location of continuous spectra, Am. J. Math. 70 (1948), 22-30. · Zbl 0035.18003
[239] V. A. Yakubovich and V. M. Starzhinskii, Linear differential equations with periodic coefficients. 1, 2, Halsted Press [John Wiley & Sons] New York-Toronto, Ont.,; Israel Program for Scientific Translations, Jerusalem-London, 1975. Translated from Russian by D. Louvish.
[240] N. J. Zabusky and M. D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243. · Zbl 1201.35174
[241] V. E. Zakharov and L. D. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280-287. · Zbl 0257.35074
[242] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118 – 134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62 – 69.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.