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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 6, Pages 1291–1315 (Mi izv1157)  

This article is cited in 27 scientific papers (total in 27 papers)

Commuting differential operators of rank 3, and nonlinear differential equations

O. I. Mokhov


Abstract: Complete solutions of the commutation equations of ordinary differential operators are obtained, to which there corresponds a three-dimensional vector bundle of common eigenfunctions over an elliptic curve. The deformation of the commuting pair by the Kadomtsev–Petviashvili equation is studied. The finite-zone solutions of the Kadomtsev–Petviashvili equation of rank 3 and genus 1 are explicitly expressed in terms of functional parameters satisfying a Boussinesq-type system of two evolution equations.
Bibliography: 40 titles.

Full text: PDF file (2226 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1990, 35:3, 629–655

Bibliographic databases:

UDC: 517.9+512.7
MSC: Primary 47E05, 14K07, 14K25; Secondary 25D25, 34B25, 58F37, 14G10, 12F10
Received: 27.04.1988

Citation: O. I. Mokhov, “Commuting differential operators of rank 3, and nonlinear differential equations”, Izv. Akad. Nauk SSSR Ser. Mat., 53:6 (1989), 1291–1315; Math. USSR-Izv., 35:3 (1990), 629–655

Citation in format AMSBIB
\Bibitem{Mok89}
\by O.~I.~Mokhov
\paper Commuting differential operators of rank~3, and nonlinear differential equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 6
\pages 1291--1315
\mathnet{http://mi.mathnet.ru/izv1157}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1039965}
\zmath{https://zbmath.org/?q=an:0707.34009}
\transl
\jour Math. USSR-Izv.
\yr 1990
\vol 35
\issue 3
\pages 629--655
\crossref{https://doi.org/10.1070/IM1990v035n03ABEH000720}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. Fritz Gesztesy, Rudi Weikard, “A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy”, Acta Math, 181:1 (1998), 63  crossref  mathscinet  zmath  isi
    3. 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
    4. 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
    5. A. E. Mironov, “Discrete analogues of Dixmier operators”, Sb. Math., 198:10 (2007), 1433–1442  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. A. E. Mironov, “O kommutiruyuschikh differentsialnykh operatorakh ranga $2$”, Sib. elektron. matem. izv., 6 (2009), 533–536  mathnet  mathscinet  elib
    7. A. B. Zheglov, A. E. Mironov, “Moduli Beikera – Akhiezera, puchki Krichevera i kommutativnye koltsa differentsialnykh operatorov v chastnykh proizvodnykh”, Dalnevost. matem. zhurn., 12:1 (2012), 20–34  mathnet
    8. Dafeng Zuo, “Commuting differential operators of rank 3 associated to a curve of genus 2”, SIGMA, 8 (2012), 044, 11 pp.  mathnet  crossref  mathscinet
    9. V. N. Davletshina, “O samosopryazhennykh kommutiruyuschikh differentsialnykh operatorakh ranga dva”, Sib. elektron. matem. izv., 10 (2013), 109–112  mathnet
    10. O. I. Mokhov, “On Commutative Subalgebras of the Weyl Algebra Related to Commuting Operators of Arbitrary Rank and Genus”, Math. Notes, 94:2 (2013), 298–300  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. A.E.. Mironov, “Self-adjoint commuting ordinary differential operators”, Invent. math, 2013  crossref
    12. 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
    13. N Delice, F.W. Nijhoff, S Yoo-Kong, “On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants”, J. Phys. A: Math. Theor, 48:3 (2015), 035206  crossref
    14. 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
    15. V. N. Davletshina, “Self-Adjoint Commuting Differential Operators of Rank 2 and Their Deformations Given by Soliton Equations”, Math. Notes, 97:3 (2015), 333–340  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. A. E. Mironov, B. T. Saparbayeva, “On the eigenfunctions of the one-dimensional Schrödinger operator with a polynomial potential”, Dokl. Math, 91:2 (2015), 171  crossref
    17. 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
    18. A. B. Zheglov, A. E. Mironov, “On commuting differential operators with polynomial coefficients corresponding to spectral curves of genus one”, Dokl. Math, 91:3 (2015), 281  crossref
    19. A.E.. Mironov, A.B.. Zheglov, “Commuting Ordinary Differential Operators with Polynomial Coefficients and Automorphisms of the First Weyl Algebra”, Int Math Res Notices, 2015, rnv218  crossref
    20. 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
    21. 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
    22. 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
    23. A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russian Math. Surveys, 71:4 (2016), 751–779  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. A. B. Zheglov, A. E. Mironov, B. T. Saparbayeva, “Commuting Krichever–Novikov differential operators with polynomial coefficients”, Siberian Math. J., 57:5 (2016), 819–823  mathnet  crossref  crossref  isi  elib  elib
    25. 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
    26. V. S. Oganesyan, “Alternative proof of Mironov's results on commuting self-adjoint operators of rank 2”, Siberian Math. J., 59:1 (2018), 102–106  mathnet  crossref  crossref  isi  elib
    27. V. S. Oganesyan, “The AKNS hierarchy and finite-gap Schrödinger potentials”, Theoret. and Math. Phys., 196:1 (2018), 983–995  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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