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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 3(357), Pages 31–80 (Mi umn736)  

This article is cited in 93 scientific papers (total in 93 papers)

Asymptotic behaviour and expansions of solutions of an ordinary differential equation

A. D. Bruno

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: An ordinary differential equation of quite general form is considered. It is shown how to find the following near a finite or infinite value of the independent variable by using algorithms of power geometry: (i) all power-law asymptotic expressions for solutions of the equation; (ii) all power-logarithmic expansions of solutions with power-law asymptotics; (iii) all non-power-law (exponential or logarithmic) asymptotic expressions for solutions of the equation; (iv) certain exponentially small additional terms for a power-logarithmic expansion of a solution that correspond to exponentially close solutions. Along with the theory and algorithms, examples are presented of calculations of the above objects for one and the same equation. The main attention is paid to explanations of algorithms for these calculations.

DOI: https://doi.org/10.4213/rm736

Full text: PDF file (594 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2004, 59:3, 429–480

Bibliographic databases:

UDC: 517.925
MSC: Primary 34E05, 34A26; Secondary 34A25, 34A34, 34C20
Received: 07.11.2003

Citation: A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Uspekhi Mat. Nauk, 59:3(357) (2004), 31–80; Russian Math. Surveys, 59:3 (2004), 429–480

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth Painlevé equation in the cases $a=0$ and $b=0$”, Dokl. Math., 74:2 (2006), 660–665  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation”, Dokl. Math., 73:1 (2006), 117–120  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  elib
    3. A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Dokl. Math., 76:2 (2007), 729–733  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Dokl. Math., 76:3 (2007), 851–855  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    5. Efimova O.Yu., Kudryashov N.A., “Power expansions for the self-similar solutions of the modified Sawada-Kotera equation”, Regul. Chaotic Dyn., 12:2 (2007), 198–218  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Kudryashov N.A., Demina M.V., “Polygons of differential equations for finding exact solutions”, Chaos Solitons Fractals, 33:5 (2007), 1480–1496  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Kudryashov N.A., Demina M.V., “Special polynomials associated with the fourth order analogue to the Painlevé equations”, Phys. Lett. A, 363:5-6 (2007), 346–355  crossref  zmath  adsnasa  isi  elib
    8. Demina M.V., Kudryashov N.A., “The Yablonskii-Vorob'ev polynomials for the second Painlevé hierarchy”, Chaos Solitons Fractals, 32:2 (2007), 526–537  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Demina M.V., Kudryashov N.A., “Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation”, Chaos Solitons Fractals, 32:1 (2007), 124–144  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Guzzetti D., “The logarithmic asymptotics of the sixth Painlevé equation”, J. Phys. A, 41:20 (2008), 205201, 46 pp.  crossref  mathscinet  zmath  isi  elib
    11. A. D. Bruno, “Power-logarithmic expansions of solutions to a system of ordinary differential equations”, Dokl. Math., 77:2 (2008), 215–218  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    12. A. D. Bruno, “Nonpower asymptotic forms of solutions to a system of ordinary differential equations”, Dokl. Math., 77:3 (2008), 325–328  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. A. D. Bruno, I. V. Goryuchkina, “Boutroux asymptotic forms of solutions to Painlevé equations and power geometry”, Dokl. Math., 78:2 (2008), 681–685  mathnet  crossref  mathscinet  isi  elib  elib
    14. A. D. Bruno, I. V. Goryuchkina, “Asymptotic forms of solutions to the third Painlevé equation”, Dokl. Math., 78:2 (2008), 765  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    15. A. D. Bruno, I. V. Goryuchkina, “Asymptotic forms of solutions to the fourth Painlevé equation”, Dokl. Math., 78:3 (2008), 868–873  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    16. M. V. Demina, N. A. Kudryashov, D. I. Sinel'shchikov, “The polygonal method for constructing exact solutions to certain nonlinear differential equations describing water waves”, Comput. Math. Math. Phys., 48:12 (2008), 2182–2193  mathnet  crossref  mathscinet  isi  elib
    17. Zhang Shan Qing, “Geometrical explanation and application of some algebraic direct methods”, Acta Phys. Sinica, 57:3 (2008), 1335–1338  crossref  mathscinet  zmath  isi
    18. M. S. Nelepko, “Stepennye i eksponentsialnye razlozheniya reshenii uravneniya ierarkhii $(K_{II})$”, Tr. In-ta matem., 16:2 (2008), 97–104  mathnet  zmath
    19. A. D. Bryuno, I. V. Goryuchkina, “Vse razlozheniya reshenii shestogo uravneniya Penleve vblizi ego neosoboi tochki”, Preprinty IPM im. M. V. Keldysha, 2008, 075, 30 pp.  mathnet
    20. I. V. Goryuchkina, “Basic asymptotic expansions of solutions to the sixth Painlevé equation”, J. Math. Sci. (New York), 160:1 (2009), 1–9  crossref  zmath  elib
    21. A. D. Bruno, I. V. Goryuchkina, “All expansions of solutions to the sixth Painlevé equation near its nonsingular point”, Dokl. Math., 79:3 (2009), 397–402  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    22. A. D. Bruno, I. V. Goryuchkina, “Basic expansions of solutions to the sixth Painlevé equation in the generic case”, Differ. Equ., 45:1 (2009), 18–32  crossref  mathscinet  mathscinet  zmath  adsnasa  isi  elib  elib
    23. A. D. Bryuno, I. V. Goryuchkina, “Ellipticheskie asimptotiki reshenii uravnenii Penleve”, Preprinty IPM im. M. V. Keldysha, 2009, 006, 26 pp.  mathnet
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    26. Goryuchkina I., “Methods of Power Geometry in Asymptotic Analysis of Solutions to Algebraic or Differential Equations”, Lie Theory and its Applications in Physics, AIP Conference Proceedings, 1243, 2010, 289–298  crossref  zmath  adsnasa  isi
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    38. A. D. Bryuno, A. V. Parusnikova, “Razlozheniya reshenii pyatogo uravneniya Penleve vblizi ego neosoboi tochki”, Preprinty IPM im. M. V. Keldysha, 2011, 018, 16 pp.  mathnet
    39. I. V. Goryuchkina, “Teorema o skhodimosti formalnogo resheniya ODU”, Preprinty IPM im. M. V. Keldysha, 2011, 025, 16 pp.  mathnet
    40. A. D. Bryuno, “Eksponentsialnye razlozheniya reshenii ODU”, Preprinty IPM im. M. V. Keldysha, 2011, 036, 16 pp.  mathnet
    41. A. D. Bryuno, “Stepenno-eksponentsialnye razlozheniya reshenii ODU”, Preprinty IPM im. M. V. Keldysha, 2011, 054, 11 pp.  mathnet
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    44. I. V. Goryuchkina, “Reshenie Pikara shestogo uravneniya Penleve i asimptotiki, poluchennye s pomoschyu stepennoi geometrii”, Preprinty IPM im. M. V. Keldysha, 2011, 066, 8 pp.  mathnet
    45. A. D. Bruno, A. V. Parusnikova, “Expansions of solutions to the fifth Painlevé equation near its nonsingular point”, Dokl. Math, 85:1 (2012), 87  crossref  mathscinet  zmath  isi  elib
    46. Guzzetti D., “Solving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae”, Int Math Res Not, 2012, no. 6, 1352–1413  crossref  mathscinet  zmath  isi  elib
    47. A. D. Bruno, “Power-exponential expansions of solutions to an ordinary differential equation”, Dokl. Math, 85:3 (2012), 336  crossref  mathscinet  zmath  isi  elib
    48. Davide Guzzetti, “Tabulation of Painlevé 6 transcendents”, Nonlinearity, 25:12 (2012), 3235  crossref  mathscinet  zmath  isi
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    53. Goryuchkina I.V., “O kratnykh i nekratnykh resheniyakh algebraicheskikh odu”, Preprinty IPM im. M.V. Keldysha, 2012, no. 83, 1–12  elib
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