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Mat. Sb., 1994, Volume 185, Number 8, Pages 103–114 (Mi msb920)  

This article is cited in 21 scientific papers (total in 21 papers)

Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg–de Vries equation

A. O. Smirnov

St. Petersburg State Academy of Aerospace Equipment Construction

Abstract: Two Ansätze concerning Krichever curves are considered for solutions elliptic in $x$ of the nonlinear Schrödinger equation and the Korteweg–de Vries equation. An example is given of a new two-gap elliptic solution of the nonlinear Schrödinger equation.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:2, 461–470

Bibliographic databases:

UDC: 517.95
MSC: Primary 58F07, 14K25; Secondary 35Q53
Received: 28.09.1993

Citation: A. O. Smirnov, “Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg–de Vries equation”, Mat. Sb., 185:8 (1994), 103–114; Russian Acad. Sci. Sb. Math., 82:2 (1995), 461–470

Citation in format AMSBIB
\by A.~O.~Smirnov
\paper Elliptic solutions of the~nonlinear Schr\"odinger equation and the~modified Korteweg--de Vries equation
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 8
\pages 103--114
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 2
\pages 461--470

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    This publication is cited in the following articles:
    1. A. O. Smirnov, “Elliptic in $t$ solutions of the nonlinear Schrödinger equation”, Theoret. and Math. Phys., 107:2 (1996), 568–578  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. 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
    3. A. O. Smirnov, “On a class of elliptic potentials of the Dirac operator”, Sb. Math., 188:1 (1997), 115–135  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. 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
    5. F. Gesztesy, R. Ratnaseelan, “An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy”, Rev. Math. Phys, 10:03 (1998), 345  crossref  mathscinet  zmath
    6. Gesztesy F., Weikard R., “Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies - an Analytic Approach”, Bull. Amer. Math. Soc., 35:4 (1998), 271–317  crossref  mathscinet  zmath  isi
    7. A. O. Smirnov, “Two-gap elliptic solutions of the Boussinesq equation”, Sb. Math., 190:5 (1999), 763–781  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Alisher Yakhshimuratov, “The Nonlinear Schrödinger Equation with a Self-consistent Source in the Class of Periodic Functions”, Math Phys Anal Geom, 2011  crossref  mathscinet  zmath
    9. A. O. Smirnov, “Elliptic breather for nonlinear Shrödinger equation”, J. Math. Sci. (N. Y.), 192:1 (2013), 117–125  mathnet  crossref  mathscinet
    10. A. O. Smirnov, “Solution of a nonlinear Schrödinger equation in the form of two-phase freak waves”, Theoret. and Math. Phys., 173:1 (2012), 1403–1416  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    11. A. O. Smirnov, G. M. Golovachev, “Trekhfaznye resheniya nelineinogo uravneniya Shredingera v ellipticheskikh funktsiyakh”, Nelineinaya dinam., 9:3 (2013), 389–407  mathnet
    12. 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
    13. V. B. Matveev, A. O. Smirnov, “Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: A unified approach”, Theoret. and Math. Phys., 186:2 (2016), 156–182  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. V. S. Oganesyan, “The AKNS hierarchy and finite-gap Schrödinger potentials”, Theoret. and Math. Phys., 196:1 (2018), 983–995  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. V. B. Matveev, A. O. Smirnov, “Two-phase periodic solutions to the AKNS hierarchy equations”, J. Math. Sci. (N. Y.), 242:5 (2019), 722–741  mathnet  crossref
    16. A. B. Hasanov, M. M. Hasanov, “Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions”, Theoret. and Math. Phys., 199:1 (2019), 525–532  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. A. B. Yakhshimuratov, “Integration of a higher-order nonlinear Schrödinger system with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 202:2 (2020), 137–149  mathnet  crossref  crossref  mathscinet  isi  elib
    18. A. B. Khasanov, M. M. Matjakubov, “Integration of the nonlinear Korteweg–de Vries equation with an additional term”, Theoret. and Math. Phys., 203:2 (2020), 596–607  mathnet  crossref  crossref  mathscinet  isi  elib
    19. V. B. Matveev, A. O. Smirnov, “Ellipticheskie solitony i «strannye volny»”, Algebra i analiz, 33:3 (2021), 129–168  mathnet
    20. Ufa Math. J., 13:2 (2021), 135–151  mathnet  crossref  isi
    21. A. B. Khasanov, T. G. Khasanov, “Zadacha Koshi dlya uravneniya Kortevega–de Friza v klasse periodicheskikh beskonechnozonnykh funktsii”, Matematicheskie voprosy teorii rasprostraneniya voln. 51, Zap. nauchn. sem. POMI, 506, POMI, SPb., 2021, 258–278  mathnet
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