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Funktsional. Anal. i Prilozhen., 1977, Volume 11, Issue 1, Pages 15–31 (Mi faa2041)  

This article is cited in 98 scientific papers (total in 98 papers)

Integration of nonlinear equations by the methods of algebraic geometry

I. M. Krichever


Full text: PDF file (1957 kB)
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English version:
Functional Analysis and Its Applications, 1977, 11:1, 12–26

Bibliographic databases:

UDC: 513.015.7+517.944
Received: 20.08.1976

Citation: I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry”, Funktsional. Anal. i Prilozhen., 11:1 (1977), 15–31; Funct. Anal. Appl., 11:1 (1977), 12–26

Citation in format AMSBIB
\Bibitem{Kri77}
\by I.~M.~Krichever
\paper Integration of nonlinear equations by the methods of algebraic geometry
\jour Funktsional. Anal. i Prilozhen.
\yr 1977
\vol 11
\issue 1
\pages 15--31
\mathnet{http://mi.mathnet.ru/faa2041}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=494262}
\zmath{https://zbmath.org/?q=an:0346.35028|0368.35022}
\transl
\jour Funct. Anal. Appl.
\yr 1977
\vol 11
\issue 1
\pages 12--26
\crossref{https://doi.org/10.1007/BF01135528}


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    This publication is cited in the following articles:
    1. I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations”, Russian Math. Surveys, 32:6 (1977), 185–213  mathnet  crossref  mathscinet  zmath
    2. I. M. Gel'fand, L. A. Dikii, “The resolvent and Hamiltonian systems”, Funct. Anal. Appl., 11:2 (1977), 93–105  mathnet  crossref  mathscinet  zmath
    3. B. A. Dubrovin, “Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties”, Funct. Anal. Appl., 11:4 (1977), 265–277  mathnet  crossref  mathscinet  zmath
    4. I. M. Krichever, “Algebraic curves and non-linear difference equations”, Russian Math. Surveys, 33:4 (1978), 255–256  mathnet  crossref  mathscinet  zmath
    5. I. M. Krichever, “Commutative rings of ordinary linear differential operators”, Funct. Anal. Appl., 12:3 (1978), 175–185  mathnet  crossref  mathscinet  zmath
    6. I. V. Cherednik, “Differential equations for the Baker–Akhiezer functions of algebraic curves”, Funct. Anal. Appl., 12:3 (1978), 195–203  mathnet  crossref  mathscinet  zmath
    7. I. M. Krichever, S. P. Novikov, “Holomorphic bundles over Riemann surfaces and the Kadomtsev–Petviashvili equation. I”, Funct. Anal. Appl., 12:4 (1978), 276–286  mathnet  crossref  mathscinet  zmath
    8. Yu. I. Manin, “Matrix solitons and bundles over curves with singularities”, Funct. Anal. Appl., 12:4 (1978), 286–295  mathnet  crossref  mathscinet  zmath
    9. I. V. Cherednik, “Conservation laws and elements of scattering theory for principal chiral fields ($d=1$)”, Theoret. and Math. Phys., 41:2 (1979), 997–1002  mathnet  crossref  mathscinet  isi
    10. A. P. Veselov, “Hamiltonian formalism for the Novikov–Krichever equations for the commutativity of two operators”, Funct. Anal. Appl., 13:1 (1979), 1–6  mathnet  crossref  mathscinet  zmath
    11. I. V. Cherednik, “One generalization of the KdV and sin-Gordon differential equations”, Funct. Anal. Appl., 13:1 (1979), 68–69  mathnet  crossref  mathscinet  zmath
    12. I. M. Krichever, S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations”, Russian Math. Surveys, 35:6 (1980), 53–79  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. G. B. Shabat, “A system of equations of S. P. Novikov”, Funct. Anal. Appl., 14:2 (1980), 158–160  mathnet  crossref  mathscinet  zmath
    14. I. M. Krichever, “Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles”, Funct. Anal. Appl., 14:4 (1980), 282–290  mathnet  crossref  mathscinet  zmath  isi
    15. B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. V. P. Maslov, G. A. Omel'yanov, “Asymptotic soliton-form solutions of equations with small dispersion”, Russian Math. Surveys, 36:3 (1981), 73–149  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. B. A. Dubrovin, “The Kadomtsev–Petviashvili equation and the relations between the periods of holomorphic differentials on Riemann surfaces”, Math. USSR-Izv., 19:2 (1982), 285–296  mathnet  crossref  mathscinet  zmath
    18. I. M. Krichever, “Baxter's equations and algebraic geometry”, Funct. Anal. Appl., 15:2 (1981), 92–103  mathnet  crossref  mathscinet  zmath  isi
    19. I. V. Cherednik, “Solutions of algebraic type of asymmetric differential equations”, Funct. Anal. Appl., 15:3 (1981), 231–232  mathnet  crossref  mathscinet  zmath  isi
    20. I. M. Krichever, “The Peierls model”, Funct. Anal. Appl., 16:4 (1982), 248–263  mathnet  crossref  mathscinet  isi
    21. I. V. Cherednik, “On the group-theoretical interpretation of Baker functions and $\tau$-functions”, Russian Math. Surveys, 38:6 (1983), 113–114  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    22. I. M. Gel'fand, I. V. Cherednik, “The abstract Hamiltonian formalism for the classical Yang–Baxter bundles”, Russian Math. Surveys, 38:3 (1983), 1–22  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    23. I. V. Cherednik, “Integrable differential equations and coverings of elliptic curves”, Math. USSR-Izv., 22:2 (1984), 357–377  mathnet  crossref  mathscinet  zmath
    24. I. M. Krichever, “The laplace method, algebraic curves, and nonlinear equations”, Funct. Anal. Appl., 18:3 (1984), 210–223  mathnet  crossref  mathscinet  zmath  isi
    25. P. G. Grinevich, “Vector rank of commuting matrix differential operators. Proof of S. P. Novikov's criterion”, Math. USSR-Izv., 28:3 (1987), 445–465  mathnet  crossref  mathscinet  zmath
    26. E. D. Belokolos, A. I. Bobenko, V. B. Matveev, V. Z. Ènol'skii, “Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations”, Russian Math. Surveys, 41:2 (1986), 1–49  mathnet  crossref  mathscinet  zmath  isi
    27. I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model”, Funct. Anal. Appl., 20:3 (1986), 203–214  mathnet  crossref  mathscinet  zmath  isi
    28. R. A. Sharipov, “Multiplet solutions of the Kadomtsev–Petviashvili equation against a finite-zone background”, Russian Math. Surveys, 42:5 (1987), 177–178  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    29. I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations”, Funct. Anal. Appl., 22:3 (1988), 200–213  mathnet  crossref  mathscinet  zmath  isi
    30. Yu. A. Neretin, “Holomorphic extensions of representations of the group of the diffeomorphisms of the circle”, Math. USSR-Sb., 67:1 (1990), 75–97  mathnet  crossref  mathscinet  zmath  isi
    31. B. A. Dubrovin, S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory”, Russian Math. Surveys, 44:6 (1989), 35–124  mathnet  crossref  mathscinet  zmath  adsnasa
    32. O. I. Mokhov, “Commuting differential operators of rank 3, and nonlinear differential equations”, Math. USSR-Izv., 35:3 (1990), 629–655  mathnet  crossref  mathscinet  zmath
    33. A. I. Zhivkov, “Isospectral classes of matrix-valued finite-gap operators with symmetries”, Russian Math. Surveys, 44:1 (1989), 269–270  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    34. I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Russian Math. Surveys, 44:2 (1989), 145–225  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    35. A. V. Zabrodin, “Fermions on a Riemann surface and the Kadomtsev–Petviashvili equation”, Theoret. and Math. Phys., 78:2 (1989), 167–177  mathnet  crossref  mathscinet  zmath  isi
    36. V. I. Dragovich, “Solutions of the Yang equation with rational irreducible spectral curves”, Russian Acad. Sci. Izv. Math., 42:1 (1994), 51–65  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    37. V. M. Buchstaber, I. M. Krichever, “Vector addition theorems and Baker–Akhiezer functions”, Theoret. and Math. Phys., 94:2 (1993), 142–149  mathnet  crossref  mathscinet  zmath  isi
    38. A. V. Marshakov, “On integrable systems and supersymmetric gauge theories”, Theoret. and Math. Phys., 112:1 (1997), 791–826  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    39. I. M. Krichever, “Algebraic-Geometric $n$-Orthogonal Curvilinear Coordinate Systems and Solutions of the Associativity Equations”, Funct. Anal. Appl., 31:1 (1997), 25–39  mathnet  crossref  crossref  mathscinet  zmath  isi
    40. I. A. Taimanov, “Secants of Abelian varieties, theta functions, and soliton equations”, Russian Math. Surveys, 52:1 (1997), 147–218  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    41. A. D. Mironov, “Group theory approach to the $\tau$-function and its quantization”, Theoret. and Math. Phys., 114:2 (1998), 127–183  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    42. A. V. Marshakov, “Strings, SUSY gauge theories, and integrable systems”, Theoret. and Math. Phys., 121:2 (1999), 1409–1461  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    43. D. P. Novikov, “Algebraic-geometric solutions of the Krichever–Novikov equation”, Theoret. and Math. Phys., 121:3 (1999), 1567–1573  mathnet  crossref  crossref  mathscinet  zmath  isi
    44. Krichever, I, “Periodic and almost-periodic potentials in inverse problems”, Inverse Problems, 15:6 (1999), R117  isi
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    46. Treibich, A, “Difference analogs of elliptic KdV solitons and Schrodinger operators”, International Mathematics Research Notices, 2003, no. 6, 313  isi
    47. S. V. Oblezin, “Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations”, Funct. Anal. Appl., 38:2 (2004), 111–124  mathnet  crossref  crossref  mathscinet  zmath  isi
    48. A. E. Mironov, “A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2”, Sb. Math., 195:5 (2004), 711–722  mathnet  crossref  crossref  mathscinet  zmath  isi
    49. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    50. A. V. Domrin, “The Riemann Problem and Matrix-Valued Potentials with a Convergent Baker–Akhiezer Function”, Theoret. and Math. Phys., 144:3 (2005), 1264–1278  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    51. V. M. Buchstaber, D. V. Leikin, “Addition Laws on Jacobian Varieties of Plane Algebraic Curves”, Proc. Steklov Inst. Math., 251 (2005), 49–120  mathnet  mathscinet  zmath
    52. A. E. Mironov, “Commuting Rank 2 Differential Operators Corresponding to a Curve of Genus 2”, Funct. Anal. Appl., 39:3 (2005), 240–243  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    53. S. V. Oblezin, “Isomonodromic deformations of $\mathfrak{sl}(2)$ Fuchsian systems on the Riemann sphere”, Mosc. Math. J., 5:2 (2005), 415–441  mathnet  mathscinet  zmath
    54. V. M. Buchstaber, I. M. Krichever, “Integrable equations, addition theorems, and the Riemann–Schottky problem”, Russian Math. Surveys, 61:1 (2006), 19–78  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    55. Fledrich, P, “Hyperelliptic osculating covers and KdV solutions periodic in tau”, International Mathematics Research Notices, 2006, 73476  isi
    56. E. Previato, V. Z. Ènol'skii, “Ultra-elliptic solitons”, Russian Math. Surveys, 62:4 (2007), 796–798  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    58. E. Yu. Bunkova, V. M. Buchstaber, “Heat Equations and Families of Two-Dimensional Sigma Functions”, Proc. Steklov Inst. Math., 266 (2009), 1–28  mathnet  crossref  mathscinet  zmath  isi  elib
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    60. Treibich A., “Nonlinear evolution equations and hyperelliptic covers of elliptic curves”, Regular & Chaotic Dynamics, 16:3–4 (2011), 290–310  crossref  isi
    61. Badanin A., Korotyaev E.L., “Even Order Periodic Operators on the Real Line”, Int Math Res Not, 2012, no. 5, 1143–1194  isi
    62. G. Aminov, “Limit relation between Toda chains and the elliptic $SL(N,\mathbb C)$ top”, Theoret. and Math. Phys., 171:2 (2012), 575–588  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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    65. A. V. Domrin, “On holomorphic solutions of equations of Korteweg–de Vries type”, Trans. Moscow Math. Soc., 73 (2012), 193–206  mathnet  crossref  mathscinet  zmath  elib
    66. Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.  mathnet  crossref  mathscinet
    67. V. N. Davletshina, “O samosopryazhennykh kommutiruyuschikh differentsialnykh operatorakh ranga dva”, Sib. elektron. matem. izv., 10 (2013), 109–112  mathnet
    68. Konopelchenko B.G. Ortenzi G., “Birkhoff Strata of Sato Grassmannian and Algebraic Curves”, J. Nonlinear Math. Phys., 20:3 (2013), 309–347  crossref  isi
    69. V. N. Davletshina, E. I. Shamaev, “On commuting differential operators of rank $2$”, Siberian Math. J., 55:4 (2014), 606–610  mathnet  crossref  mathscinet  isi
    70. P. G. Grinevich, S. P. Novikov, “Spectrally meromorphic operators and non-linear systems”, Russian Math. Surveys, 69:5 (2014), 924–926  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    71. Romano S., “4-Dimensional Frobenius Manifolds and Painlevé' VI”, Math. Ann., 360:3-4 (2014), 715–751  crossref  isi
    72. Geng X. Zhai Yu. Dai H.H., “Algebro-Geometric Solutions of the Coupled Modified Korteweg-de Vries Hierarchy”, Adv. Math., 263 (2014), 123–153  crossref  isi
    73. Mironov A.E., “Self-Adjoint Commuting Ordinary Differential Operators”, Invent. Math., 197:2 (2014), 417–431  crossref  isi
    74. Treibich A., “Hyperelliptic D-Osculating Covers and Rational Surfaces”, Bull. Soc. Math. Fr., 142:3 (2014), 379–409  isi
    75. V. S. Oganesyan, “Commuting differential operators of rank 2 and arbitrary genus $g$ with polynomial coefficients”, Russian Math. Surveys, 70:1 (2015), 165–167  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    77. V. N. Davletshina, “Commuting differential operators of rank $2$ with trigonometric coefficients”, Siberian Math. J., 56:3 (2015), 405–410  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    78. Mironov A.E. Saparbayeva B.T., “on the Eigenfunctions of the One-Dimensional Schrodinger Operator With a Polynomial Potential”, Dokl. Math., 91:2 (2015), 171–172  crossref  isi
    79. Romano S., “Frobenius Structures on Double Hurwitz Spaces”, Int. Math. Res. Notices, 2015, no. 2, 538–577  crossref  isi
    80. V. S. Oganesyan, “Common Eigenfunctions of Commuting Differential Operators of Rank $2$”, Math. Notes, 99:2 (2016), 308–311  mathnet  crossref  crossref  mathscinet  isi  elib
    81. V. S. Oganesyan, “Commuting Differential Operators of Rank 2 with Polynomial Coefficients”, Funct. Anal. Appl., 50:1 (2016), 54–61  mathnet  crossref  crossref  mathscinet  isi  elib
    82. V. S. Oganesyan, “On operators of the form $\partial_x^4+u(x)$ from a pair of commuting differential operators of rank 2 and genus $g$”, Russian Math. Surveys, 71:3 (2016), 591–593  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    83. A. Treibich, “Tangential Polynomials and Matrix KdV Elliptic Solitons”, Funct. Anal. Appl., 50:4 (2016), 308–318  mathnet  crossref  crossref  mathscinet  isi  elib
    84. B. T. Saparbayeva, “On the Schrödinger operator connected with a family of Hamiltonian-minimal Lagrangian surfaces in $\mathbb CP^2$”, Siberian Math. J., 57:6 (2016), 1077–1081  mathnet  crossref  crossref  isi  elib
    85. Mironov A.E. Zheglov A.B., “Commuting Ordinary Differential Operators with Polynomial Coefficients and Automorphisms of the First Weyl Algebra”, Int. Math. Res. Notices, 2016, no. 10, 2974–2993  crossref  mathscinet  isi  elib  scopus
    86. Xianguo Geng, Xin Zeng, “Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy”, Theoret. and Math. Phys., 190:1 (2017), 18–42  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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    88. Oganesyan V., “Explicit Characterization of Some Commuting Differential Operators of Rank 2”, Int. Math. Res. Notices, 2017, no. 6, 1623–1640  crossref  isi
    89. V. S. Oganesyan, “Commuting Differential Operators of Rank 2 with Rational Coefficients”, Funct. Anal. Appl., 52:3 (2018), 203–213  mathnet  crossref  crossref  isi  elib
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  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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