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Funktsional. Anal. i Prilozhen., 1975, Volume 9, Issue 3, Pages 41–51 (Mi faa2261)  

This article is cited in 47 scientific papers (total in 47 papers)

Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials

B. A. Dubrovin

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English version:
Functional Analysis and Its Applications, 1975, 9:3, 215–223

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Received: 17.09.1974

Citation: B. A. Dubrovin, “Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials”, Funktsional. Anal. i Prilozhen., 9:3 (1975), 41–51; Funct. Anal. Appl., 9:3 (1975), 215–223

Citation in format AMSBIB
\by B.~A.~Dubrovin
\paper Periodic problems for the Korteweg--de Vries equation in the class of finite band potentials
\jour Funktsional. Anal. i Prilozhen.
\yr 1975
\vol 9
\issue 3
\pages 41--51
\jour Funct. Anal. Appl.
\yr 1975
\vol 9
\issue 3
\pages 215--223

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    This publication is cited in the following articles:
    1. I. M. Gel'fand, L. A. Dikii, “Asimptotic benaviour of the resolvent of Sturm–LiouvilleI equations and the algebra of the Korteweg–de Vries equations”, Russian Math. Surveys, 30:5 (1975), 77–113  mathnet  crossref  mathscinet  zmath
    2. B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Russian Math. Surveys, 31:1 (1976), 59–146  mathnet  crossref  mathscinet  zmath
    3. B. A. Dubrovin, “Konechnozonnye lineinye differentsialnye operatory i abelevy mnogoobraziya”, UMN, 31:4(190) (1976), 259–260  mathnet  mathscinet  zmath
    4. O. I. Bogoyavlenskii, “Integrals of higher-order stationary KdV equations and eigenvalues of the Hill operator”, Funct. Anal. Appl., 10:2 (1976), 92–95  mathnet  crossref  mathscinet  zmath
    5. 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
    6. I. M. Gel'fand, L. A. Dikii, “The calculus of jets and nonlinear Hamiltonian systems”, Funct. Anal. Appl., 12:2 (1978), 81–94  mathnet  crossref  mathscinet  zmath
    7. I. M. Krichever, “Commutative rings of ordinary linear differential operators”, Funct. Anal. Appl., 12:3 (1978), 175–185  mathnet  crossref  mathscinet  zmath
    8. 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
    9. I. M. Gel'fand, L. A. Dikii, “Integrable nonlinear equations and the Liouville theorem”, Funct. Anal. Appl., 13:1 (1979), 6–15  mathnet  crossref  mathscinet  zmath
    10. 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
    11. E. D. Belokolos, “Peierls-Fröhlich problem and potentials with finite number of gaps. I”, Theoret. and Math. Phys., 45:2 (1980), 1022–1026  mathnet  crossref  mathscinet  isi
    12. A. P. Veselov, “Finite-zone potentials and integrable systems on a sphere with quadratic potential”, Funct. Anal. Appl., 14:1 (1980), 37–39  mathnet  crossref  mathscinet  zmath
    13. B. M. Levitan, “Pochti periodichnost beskonechno-zonnykh potentsialov”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 291–320  mathnet  mathscinet  zmath; B. M. Levitan, “Almost periodicity of infinite-zone potentials”, Math. USSR-Izv., 18:2 (1982), 249–273  crossref
    14. B. A. Dubrovin, “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. E. D. Belokolos, “Peierls-Fröhlich problem and potentials with finite number of gaps. II”, Theoret. and Math. Phys., 48:1 (1981), 604–610  mathnet  crossref  mathscinet  isi
    16. N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, V. G. Samoilenko, “Discrete periodic problem for the modified nonlinear Korteweg–de Vries equation”, Theoret. and Math. Phys., 50:1 (1982), 75–81  mathnet  crossref  mathscinet  zmath  isi
    17. E. D. Belokolos, D. Ya. Petrina, “Connection between the approximating Hamiltonian method and theta-function integration”, Theoret. and Math. Phys., 58:1 (1984), 40–46  mathnet  crossref  mathscinet  isi
    18. B. M. Levitan, “O zamykanii mnozhestva konechno-zonnykh potentsialov”, Matem. sb., 123(165):1 (1984), 69–91  mathnet  mathscinet  zmath; B. M. Levitan, “On the closure of the set of finite-zone potentials”, Math. USSR-Sb., 51:1 (1985), 67–89  crossref
    19. N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, A. M. Kurbatov, V. G. Samoilenko, “Nonlinear model of Schrödinger type: Conservation laws, Hamiltonian structure, and complete integrability”, Theoret. and Math. Phys., 65:2 (1985), 1154–1164  mathnet  crossref  mathscinet  zmath  isi
    20. 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
    21. 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
    22. 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
    23. B. M. Levitan, A. B. Khasanov, “Estimation of the Cauchy function for finite-zone nonperiodic potentials”, Funct. Anal. Appl., 26:2 (1992), 91–98  mathnet  crossref  mathscinet  zmath  isi
    24. Krichever, I, “Periodic and almost-periodic potentials in inverse problems”, Inverse Problems, 15:6 (1999), R117  isi
    25. A. A. Akhmetshin, Yu. S. Vol'vovskii, “The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation”, Funct. Anal. Appl., 35:4 (2001), 247–256  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. 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
    27. Yu. V. Brezhnev, “Finite-Band Potentials with Trigonal Curves”, Theoret. and Math. Phys., 133:3 (2002), 1657–1662  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. S. P. Suetin, “Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions”, Sb. Math., 193:12 (2002), 1837–1866  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    29. N. V. Ustinov, Yu. V. Brezhnev, “On the $\Psi$-function for finite-gap potentials”, Russian Math. Surveys, 57:1 (2002), 165–167  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    30. Yu. V. Brezhnev, “Dubrovin equations for finite-gap operators”, Russian Math. Surveys, 57:2 (2002), 415–417  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    31. S. P. Suetin, “On the dynamics of “wandering” zeros of polynomials that are orthogonal on certain intervals”, Russian Math. Surveys, 57:2 (2002), 425–427  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    32. S. P. Suetin, “Sravnitelnaya asimptotika reshenii i formuly sledov dlya nekotorogo klassa raznostnykh uravnenii”, Sovr. probl. matem., 6, MIAN, M., 2006, 3–74  mathnet  crossref  mathscinet  zmath; S. P. Suetin, “Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S96–S137  crossref  isi
    33. S. P. Suetin, “Trace formulae for a class of Jacobi operators”, Sb. Math., 198:6 (2007), 857–885  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    34. Gesztesy F., “Inverse spectral theory as influenced by Barry Simon”, Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon'S 60th Birthday - Ergodic Schrodinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory, Proceedings of Symposia in Pure Mathematics, 76, no. 2, 2007, 741–820  isi
    35. A. B. Khasanov, A. B. Yakhshimuratov, “The Korteweg–de Vries equation with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 164:2 (2010), 1008–1015  mathnet  crossref  crossref  adsnasa  isi
    36. Y. C. Hon, Engui Fan, “Super quasiperiodic wave solutions and asymptotic analysis for $\mathcal N=1$ supersymmetric $KdV$-type equations”, Theoret. and Math. Phys., 166:3 (2011), 317–336  mathnet  crossref  crossref  adsnasa  isi
    37. A. B. Yakhshimuratov, “Integrirovanie uravneniya Kortevega-de Friza so spetsialnym svobodnym chlenom v klasse periodicheskikh funktsii”, Ufimsk. matem. zhurn., 3:4 (2011), 144–150  mathnet  zmath
    38. Shou-fu Tian, Hong-qing Zhang, “Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV–Burgers equation”, Theoret. and Math. Phys., 170:3 (2012), 287–314  mathnet  crossref  crossref  adsnasa  isi  elib
    39. Dafeng Zuo, “Commuting differential operators of rank 3 associated to a curve of genus 2”, SIGMA, 8 (2012), 044, 11 pp.  mathnet  crossref  mathscinet
    40. Brezhnev Yu.V., “Spectral/Quadrature Duality: Picard-Vessiot Theory and Finite-Gap Potentials”, Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemporary Mathematics, 563, ed. AcostaHumanez P. Finkel F. Kamran N. Olver P., Amer Mathematical Soc, 2012, 1–31, ISBN: 978-0-8218-7584-1  crossref  isi
    41. M. M. Matyoqubov, A. B. Yakhshimuratov, “Integration of higher Korteweg-de Vries equation with a self-consistent source in class of periodic functions”, Ufa Math. Journal, 5:1 (2013), 102–111  mathnet  crossref  mathscinet  elib
    42. Badanin A., Korotyaev E., “Trace Formula for Fourth Order Operators on the Circle”, Dyn. Partial Differ. Equ., 10:4 (2013), 343–352  isi
    43. 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  adsnasa  isi  elib
    44. Zhai Yu. Geng X. He G., “Explicit Quasi-Periodic Solutions of the Vakhnenko Equation”, J. Math. Phys., 55:5 (2014), 053512  crossref  isi
    45. Aleksandr O. Smirnov, Sergei G. Matveenko, Sergei K. Semenov, Elena G. Semenova, “Three-Phase Freak Waves”, SIGMA, 11 (2015), 032, 14 pp.  mathnet  crossref  mathscinet
    46. A. B. Yakhshimuratov, M. M. Matyokubov, “Integration of loaded Korteweg–de Vries equation in a class of periodic functions”, Russian Math. (Iz. VUZ), 60:2 (2016), 72–76  mathnet  crossref
    47. A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russian Math. Surveys, 71:4 (2016), 751–779  mathnet  crossref  crossref  mathscinet  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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