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Weak nonlinear asymptotic solutions for the fourth order analogue of the second Painlevé equation

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Abstract

The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the P 22 equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays ϕ = \(\frac{2}{5}\)π(2n + 1) on the complex plane have been found by the isomonodromy deformations technique.

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References

  1. Kudryashov, N.A, Two Hierarchies of Ordinary Differential Equations and Their Properties, Phys. Lett. A, 1999, vol. 252, nos. 3–4, pp. 173–179.

    Article  MathSciNet  MATH  Google Scholar 

  2. Kudryashov, N.A., Analytic Theory of Nonlinear Differential Equations, Izhevsk: R&C Dynamics, Institute of Computer Science, 2004 (Russian).

    Google Scholar 

  3. Kudryashov, N.A. and Demina, M.V, Special Polynomials Associated with the Fourth Order Analogue to the Painlevé Equations, Phys. Lett. A, 2007, vol. 363, nos. 5–6, pp. 346–355.

    Article  MATH  Google Scholar 

  4. Demina, M.V. and Kudryashov, N.A, The Yablonskii–Vorob’ev Polynomials for the Second Painlevé Hierarchy, Chaos Solitons Fractals, 2007, vol. 32, no. 2, pp. 526–537.

    Article  MathSciNet  MATH  Google Scholar 

  5. Painlevé, P, Sur les équations différentielles du second ordre et d’ordre supérieure dont l’intégrale générale est uniforme, Acta Math., 1902, vol. 25, pp. 1–85.

    Article  MathSciNet  MATH  Google Scholar 

  6. Boutroux P. Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup., 1913, vol. 30, pp. 255–375.

  7. Bruno, A.D, Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations, Int. J. Differ. Equ., 2015, Art. 340715, 13 pp.

    MATH  Google Scholar 

  8. Kitaev, A.V, Elliptic Asymptotics of the First and Second Painlevé Transcendents, Russian Math. Surveys, 1994, vol. 49, no. 1, pp. 81–150; see also: Uspekhi Mat. Nauk, 1994 vol. 49, no. 1(295), pp. 77–140.

    Article  MathSciNet  MATH  Google Scholar 

  9. Novokshenov, V.Yu., The Boutroux Ansatz for the Second Painlevé Equation in the Complex Domain, Math. USSR-Izv., 1991, vol. 37, no. 3, pp. 587–609; see also: Izv. Akad. Nauk SSSR Ser. Mat., 1990 vol. 54, no. 6, pp. 1229–1251.

    Article  MathSciNet  MATH  Google Scholar 

  10. Its, A.R., “Isomonodromy” Solutions of Equations of Zero Curvature, Math. USSR-Izv., 1986, vol. 26, no. 3, pp. 497–529; see also: Izv. Akad. Nauk SSSR Ser. Mat., 1985 vol. 49, no. 3, pp. 530–565.

  11. Kudryashov, N.A. and Demina, M.V, Power and Non-Power Expansions of the Solutions for the Fourth-Order Analogue to the Second Painlevé Equation, Chaos Solitons Fractals, 2007, vol. 32, no. 1, pp. 124–144.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kudryashov, N.A. and Pickering A, Rational solutions for Schwarzian integrable hierarchies, J. Phys. A. Math. Gen., 1998, vol. 31, no. 47, pp. 9505–9518.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Applied Mathematics, National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409, Russia

    Ilia Yu. Gaiur & Nikolay A. Kudryashov

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  1. Ilia Yu. Gaiur
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  2. Nikolay A. Kudryashov
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Correspondence to Ilia Yu. Gaiur.

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Gaiur, I.Y., Kudryashov, N.A. Weak nonlinear asymptotic solutions for the fourth order analogue of the second Painlevé equation. Regul. Chaot. Dyn. 22, 266–271 (2017). https://doi.org/10.1134/S1560354717030066

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  • DOI: https://doi.org/10.1134/S1560354717030066

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