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Dokl. Akad. Nauk SSSR, 1976, Volume 227, Number 2, Pages 291–294 (Mi dan40219)  

This article is cited in 34 scientific papers (total in 34 papers)


Algebraic-geometric construction of the Zakharov–Shabat equations and their periodic solutions

I. M. Krichever

Open Joint-Stock Company "Krzhizhanovsky Power Engineering institute", Moscow

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Bibliographic databases:
UDC: 513.836
Presented: А. Д. Александров
Received: 08.10.1975

Citation: I. M. Krichever, “Algebraic-geometric construction of the Zakharov–Shabat equations and their periodic solutions”, Dokl. Akad. Nauk SSSR, 227:2 (1976), 291–294

Citation in format AMSBIB
\by I.~M.~Krichever
\paper Algebraic-geometric construction of the Zakharov--Shabat equations and their periodic solutions
\jour Dokl. Akad. Nauk SSSR
\yr 1976
\vol 227
\issue 2
\pages 291--294

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    This publication is cited in the following articles:
    1. V. G. Drinfeld, “Commutative subrings of certain noncommutative rings”, Funct. Anal. Appl., 11:1 (1977), 9–12  mathnet  crossref  mathscinet  zmath
    2. I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry”, Funct. Anal. Appl., 11:1 (1977), 12–26  mathnet  crossref  mathscinet  zmath
    3. I. M. Gel'fand, L. A. Dikii, “The resolvent and Hamiltonian systems”, Funct. Anal. Appl., 11:2 (1977), 93–105  mathnet  crossref  mathscinet  zmath
    4. 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
    5. V. V. Sokolov, “Examples of commutative rings of differential operators”, Funct. Anal. Appl., 12:1 (1978), 65–66  mathnet  crossref  mathscinet  zmath
    6. I. M. Krichever, “Commutative rings of ordinary linear differential operators”, Funct. Anal. Appl., 12:3 (1978), 175–185  mathnet  crossref  mathscinet  zmath
    7. V. V. Sokolov, “Birationally isomorphic commutative rings of differential operators”, Funct. Anal. Appl., 12:3 (1978), 234–236  mathnet  crossref  mathscinet  zmath
    8. 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
    9. 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
    10. 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
    11. B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. I. M. Krichever, “The laplace method, algebraic curves, and nonlinear equations”, Funct. Anal. Appl., 18:3 (1984), 210–223  mathnet  crossref  mathscinet  zmath  isi
    13. 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
    14. 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
    15. 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
    16. I. M. Krichever, S. P. Novikov, “Algebras of virasoro type, riemann surfaces and structures of the theory of solitons”, Funct. Anal. Appl., 21:2 (1987), 126–142  mathnet  crossref  mathscinet  zmath  isi
    17. A. O. Smirnov, “A matrix analogue of Appell's theorem and reductions of multidimensional Riemann theta-functions”, Math. USSR-Sb., 61:2 (1988), 379–388  mathnet  crossref  mathscinet  zmath
    18. B. A. Dubrovin, S. M. Natanzon, “Real theta-function solutions of the Kadomtsev–Petviashvili equation”, Math. USSR-Izv., 32:2 (1989), 269–288  mathnet  crossref  mathscinet  zmath
    19. I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations”, Funct. Anal. Appl., 22:3 (1988), 200–213  mathnet  crossref  mathscinet  zmath  isi
    20. 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
    21. 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
    22. A. O. Smirnov, “On some set of elliptic solutions of the Boussinesq equation”, Theoret. and Math. Phys., 109:3 (1996), 1515–1522  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. S. P. Novikov, I. A. Dynnikov, “Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds”, Russian Math. Surveys, 52:5 (1997), 1057–1116  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    24. 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
    25. A. Treibich, “Hyperelliptic tangential covers and finite-gap potentials”, Russian Math. Surveys, 56:6 (2001), 1107–1151  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. A. A. Akhmetshin, Yu. S. Vol'vovskii, I. M. Krichever, “Elliptic Families of Solutions of the Kadomtsev–Petviashvili Equation and the Field Elliptic Calogero–Moser System”, Funct. Anal. Appl., 36:4 (2002), 253–266  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    27. S. M. Natanzon, “Witten Solution for the Gelfand–Dikii Hierarchy”, Funct. Anal. Appl., 37:1 (2003), 21–31  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. 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
    29. I. A. Taimanov, “Singular spectral curves in finite-gap integration”, Russian Math. Surveys, 66:1 (2011), 107–144  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    30. X. Zeng, X. Geng, “Quasiperiodic solutions of the discrete Chen–Lee–Liu hierarchy”, Theoret. and Math. Phys., 179:3 (2014), 649–678  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    31. Hui Wang, Xianguo Geng, “Algebro-geometric solutions to a new hierarchy of soliton equations”, Zhurn. matem. fiz., anal., geom., 11:4 (2015), 359–398  mathnet  crossref  mathscinet
    32. 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
    33. Vladimir P. Kotlyarov, “A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their Finite-Gap Solutions”, SIGMA, 14 (2018), 082, 27 pp.  mathnet  crossref
    34. Simonetta Abenda, Petr G. Grinevich, “Real soliton lattices of the Kadomtsev–Petviashvili II equation and desingularization of spectral curves: the $\mathrm {Gr^{ \scriptscriptstyle TP}}(2,4)$ case”, Proc. Steklov Inst. Math., 302 (2018), 1–15  mathnet  crossref  crossref  isi  elib
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