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Israel Journal of Mathematics

, Volume 229, Issue 1, pp 1–38 | Cite as

Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

  • Tuen-Wai NgEmail author
  • Cheng-Fa Wu
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Abstract

In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth is at most two. Moreover, for the second order factorizable algebraic ODEs, all their meromorphic solutions (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

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References

  1. [1]
    M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher’s equation for a special wave speed, Bulletin of Mathematical Biology 41 (1979), 835–840.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. B. Bank, Some results on analytic and meromorphic solutions of algebraic differential equations, Advances in Mathematics 15 (1975), 41–61.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    K. F. Barth, D. A. Brannan and W. K. Hayman, Research problems in complex analysis, Bulletin of the London Mathematical Society 16 (1984), 490–517.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    N. Basu, S. Bose and T. Vijayaraghavan, A simple example for a theorem of Vijayaraghavan, Journal of the London Mathematical Society 12 (1937), 250–252.CrossRefzbMATHGoogle Scholar
  5. [5]
    E. Borel, Mémoire sur les séries divergentes, Annales Scientifiques de l’école Normale Supérieure 16 (1899), 9–131.CrossRefzbMATHGoogle Scholar
  6. [6]
    Y. M. Chiang and R. G. Halburd, On the meromorphic solutions of an equation of Hayman, Journal of Mathematical Analysis and Applications 281 (2003), 663–677.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. T. Chuang and C. C. Yang, Fix-points and Factorization Theory of Meromorphic Functions, World Scientific, Teaneck, NJ, 1990.zbMATHGoogle Scholar
  8. [8]
    R. Conte, The Painlevé approach to nonlinear ordinary differential equations, in The Painlevé Property, CRM Series in Mathematical Physics, Springer, New York, 1999, pp. 77–180.CrossRefGoogle Scholar
  9. [9]
    R. Conte, A. P. Fordy and A. Pickering, A perturbative Painlevé approach to nonlinear differential equations, Physica D. Nonlinear Phenomena 69 (1993), 33–58.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Conte and T. W. Ng, Meromorphic solutions of a third order nonlinear differential equation, Journal of Mathematical Physics 51 (2010), 033518.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Conte, T. W. Ng and C. F. Wu, Hayman’s classical conjecture on some nonlinear second-order algebraic ODEs, Complex Variables and Elliptic Equations 60 (2015), 1539–1552.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    O. Cornejo-Pérez and H. Rosu, Nonlinear second order ode’s: Factorizations and particular solutions, Progress of Theoretical Physics 114 (2005), 533–538.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Darboux, Sur les équations aux dérivées partielles, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 96 (1883), 766–769.zbMATHGoogle Scholar
  14. [14]
    A. E. Eremenko, Meromorphic traveling wave solutions of the Kuramoto–Sivashinsky equation, Journal of Mathematical Physics, Analysis, Geometry 2 (2006), 278–286.MathSciNetzbMATHGoogle Scholar
  15. [15]
    A. E. Eremenko, L. Liao and T. W. Ng, Meromorphic solutions of higher order Briot-Bouquet differential equations, Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009), 197–206.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    R. Fowler, Some results on the form near infinity of real continuous solutions of a certain type of second-order differential equation, Proceedings of the London Mathematical Society 13 (1914), 341–371.MathSciNetCrossRefGoogle Scholar
  17. [17]
    P. Gallagher, Some algebraic differential equations with few transcendental solutions, Journal of Mathematical Analysis and Applications 428 (2015), 717–734.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. A. Gol’dberg, On one-valued integrals of differential equations of the first order, Ukraininskiĭ Matematičeskiĭ Žurnal 8 (1956), 254–261.MathSciNetGoogle Scholar
  19. [19]
    R. G. Halburd and J. Wang, All admissible meromorphic solutions of Hayman’s equation, International Mathematics Research Notices 18 (2015), 8890–8902.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. H. Hardy, Some results concerning the behavior at infinity of a real and continuous solution of an algebraic differential equation of the first order, Proceedings of the London Mathematical Society 10 (1912), 451–468.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    W. K. Hayman, The growth of solutions of algebraic differential equations, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 7 (1996), 67–73.MathSciNetzbMATHGoogle Scholar
  22. [22]
    E. L. Ince, Ordinary Differential Equations, Dover, New York, 1944.zbMATHGoogle Scholar
  23. [23]
    L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bulletin of Mathematics 22 (1998), 385–405.MathSciNetzbMATHGoogle Scholar
  24. [24]
    A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Vestnik Moskovskogo Universiteta 1 (1937), 1–25.Google Scholar
  25. [25]
    N. A. Kudryashov, Meromorphic solutions of nonlinear ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation 15 (2010), 2778–2790.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    I. Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, Vol. 15, Walter de Gruyter, Berlin, 1993.CrossRefzbMATHGoogle Scholar
  27. [27]
    I. Laine, Complex differential equations, in Ordinary Differential Equations Handbook of Differential Equations, Vol. 4, Elsevier/North-Holland, Amsterdam, 2008, pp. 269–363.CrossRefGoogle Scholar
  28. [28]
    E. Lindelöf, Sur la croissance des intégrales des équations différentielles algébrique du premier order, Bulletin de la Société Mathématique de France 27 (1899), 205–215.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Loewy, Uber vollständig reduzible lineare homogene Differentialgleichungen, Mathematische Annalen 56 (1906), 89–117.CrossRefzbMATHGoogle Scholar
  30. [30]
    P. Painlevé, Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bulletin de la Société Mathématique de France 28 (1900), 201–261.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    L. A. Rubel, Some research problems about algebraic differential equations II, Illinois Journal of Mathematics 36 (1992), 659–680.MathSciNetzbMATHGoogle Scholar
  32. [32]
    F. Schwarz, Loewy Decomposition of Linear Differential Equations, Texts and Monographs in Symbolic Computation, Springer, Vienna, 2012.Google Scholar
  33. [33]
    F. Schwarz, Loewy decomposition of linear differential equations, Bulletin of Mathematical Sciences 3 (2013), 19–71.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S. P. Tsarev, Factorization of linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations, Theoretical and Mathematical Physics 122 (2000), 121–133.MathSciNetCrossRefGoogle Scholar
  35. [35]
    T. Vijayaraghavan, Sur la croissance des fonctions définies par les équations différentielles, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 194 (1932), 827–829.zbMATHGoogle Scholar

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Hong KongPokfulamHong Kong
  2. 2.Institute for Advanced StudyShenzhen UniversityShenzhenPR China

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