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Izv. Akad. Nauk SSSR Ser. Mat., 1985, Volume 49, Issue 3, Pages 530–565 (Mi izv1365)  

This article is cited in 35 scientific papers (total in 35 papers)

“Isomonodromy” solutions of equations of zero curvature

A. R. Its

Abstract: A detailed discussion is given of the analytic properties (explicit description, asymptotic representations, etc.) of a new class of solutions of completely integrable evolution systems – the class of “isomonodromy solutions” – introduced in recent works of a group of Japanese mathematicians: M. Sato, M. Jimbo, T. Miwa, and others. The role of the concept of an isomonodromy solution is demonstrated in such important questions of the theory of equations of zero curvature as finding the time asymptotics of solutions of corresponding Cauchy problems in the class of rapidly decreasing initial data.
Bibliography: 30 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1986, 26:3, 497–529

Bibliographic databases:

UDC: 517.946
MSC: Primary 35Q20; Secondary 30E25, 30E15
Received: 06.01.1983

Citation: A. R. Its, ““Isomonodromy” solutions of equations of zero curvature”, Izv. Akad. Nauk SSSR Ser. Mat., 49:3 (1985), 530–565; Math. USSR-Izv., 26:3 (1986), 497–529

Citation in format AMSBIB
\by A.~R.~Its
\paper ``Isomonodromy'' solutions of equations of zero curvature
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1985
\vol 49
\issue 3
\pages 530--565
\jour Math. USSR-Izv.
\yr 1986
\vol 26
\issue 3
\pages 497--529

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    This publication is cited in the following articles:
    1. V. Yu. Novokshenov, “Movable poles of the solutions of Painleve's equation of the third kind and their relation with mathieu functions”, Funct. Anal. Appl., 20:2 (1986), 113–123  mathnet  crossref  mathscinet  zmath  isi
    2. A. V. Kitaev, “The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equation”, Math. USSR-Sb., 62:2 (1989), 421–444  mathnet  crossref  mathscinet  zmath
    3. A. R. Its, A. A. Kapaev, “The method of isomonodromy deformations and connection formulas for the second Painlevé transcendent”, Math. USSR-Izv., 31:1 (1988), 193–207  mathnet  crossref  mathscinet  zmath
    4. A. A. Kapaev, “Asymptotic expressions for the second Painlevй functions”, Theoret. and Math. Phys., 77:3 (1988), 1227–1234  mathnet  crossref  mathscinet  zmath  isi
    5. R. F. Bikbaev, A. R. Its, “Asymptotics at $t\to\infty$ of the solution of the Cauchy problem for the Landau–Lifshitz equation”, Theoret. and Math. Phys., 76:1 (1988), 665–675  mathnet  crossref  mathscinet  isi
    6. R. F. Bikbaev, “Korteweg–de Vries equation with finite-gap boundary conditions, and the Whitham deformations of Riemann surfaces”, Funct. Anal. Appl., 23:4 (1989), 257–266  mathnet  crossref  mathscinet  zmath  isi
    7. Jorge P. Zubelli, “On a zero curvature problem related to the ZS–AKNS operator”, J Math Phys (N Y ), 33:11 (1992), 3666  crossref  mathscinet  zmath
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    9. A V Kitaev, A V Rybin, J Timonen, J Phys A Math Gen, 26:14 (1993), 3583  crossref  mathscinet  zmath  adsnasa
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    11. A.Yu. Orlov, S. Rauch-Wojciechowski, “Dressing method, Darboux transformation and generalized restricted flows for the KdV hierarchy”, Physica D: Nonlinear Phenomena, 69:1-2 (1993), 77  crossref
    12. A R Its, A S Fokas, A A Kapaev, Nonlinearity, 7:5 (1994), 1291  crossref  mathscinet  zmath  isi
    13. A. O. Smirnov, “Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg–de Vries equation”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 461–470  mathnet  crossref  mathscinet  zmath  isi
    14. B. I. Suleimanov, “Influence of weak nonlinearity on the high-frequency asymptotics in caustic rearrangements”, Theoret. and Math. Phys., 98:2 (1994), 132–138  mathnet  crossref  mathscinet  zmath  isi
    15. 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
    16. D. Korotkin, H. Nicolai, “Isomonodromic quantization of dimensionally reduced gravity”, Nuclear Physics B, 475:1-2 (1996), 397  crossref
    17. H. M. Babujian, A. V. Kitaev, “Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping”, J Math Phys (N Y ), 39:5 (1998), 2499  crossref  mathscinet  zmath  adsnasa  elib
    18. V. R. Kudashev, B. I. Suleimanov, “Small-amplitude dispersion oscillations on the background of the nonlinear geometric optic approximation”, Theoret. and Math. Phys., 118:3 (1999), 325–332  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    19. V. L. Vereshchagin, “The asymptotic behavior of solutions of the sine-Gordon equation with singularities zero”, Math. Notes, 67:3 (2000), 274–285  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    20. Vartanian A.H., “Higher order asymptotics of the modified non-linear Schrodinger equation”, Communications in Partial Differential Equations, 25:5–6 (2000), 1043–1098  crossref  isi
    21. A Kapaev, “On the asymptotic expansion of the solutions of the separated nonlinear Schrödinger equation”, Physics Letters A, 285:3-4 (2001), 150  crossref  elib
    22. A V Kitaev, A H Vartanian, “Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: I”, Inverse Problems, 20:4 (2004), 1165  crossref  elib
    23. Y. Chen, A. R Its, “A Riemann–Hilbert approach to the Akhiezer polynomials”, Phil Trans R Soc A, 366:1867 (2008), 973  crossref
    24. Katrin Grunert, Gerald Teschl, “Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent”, Math Phys Anal Geom, 2009  crossref  isi
    25. A V Kitaev, A Vartanian, “Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: II”, Inverse Probl, 26:10 (2010), 105010  crossref
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    29. A. O. Smirnov, G. M. Golovachev, “Trekhfaznye resheniya nelineinogo uravneniya Shredingera v ellipticheskikh funktsiyakh”, Nelineinaya dinam., 9:3 (2013), 389–407  mathnet
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  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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