Finite order solutions of difference equations, and difference Painlevé equations IV

Wen, Zhi-Tao

doi:10.1090/proc/13210

Finite order solutions of difference equations, and difference Painlevé equations IV
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Proc. Amer. Math. Soc. 144 (2016), 4247-4260 Request permission

Abstract:

In this paper, from the non-linear difference equation \[ (\overline {w}+w)(w+\underline {w})=\frac {P(z,w)}{Q(z,w)} \] where $P(z,w)$ and $Q(r,w)$ are polynomials in $w(z)$ without common factors having small function coefficients related to $w(z)$, we present the form of difference Painlevé equation IV \[ (\overline {w}+w)(w+\underline {w})=\frac {(w^2-a^2)(w^2-b^2)}{(w+(\alpha z+\beta ))^2+\pi }, \] where $a$, $b$, $\alpha$, $\beta$ and $\pi$ are period small functions related to $w$. It shows that if the above difference equation admits at least one meromorphic solution $w(z)$ of finite order, then the difference equation can be transformed by Möbius tranformation in $w$ to difference Painlevé IV, unless $w$ is the solution of difference Riccati equations.

References

M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), no. 3, 889–905. MR 1759006, DOI 10.1088/0951-7715/13/3/321
Yik-Man Chiang and Shao-Ji Feng, On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105–129. MR 2407244, DOI 10.1007/s11139-007-9101-1
R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463–478. MR 2248826
R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477–487. MR 2185244, DOI 10.1016/j.jmaa.2005.04.010
R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 443–474. MR 2308234, DOI 10.1112/plms/pdl012
R. G. Halburd and R. J. Korhonen, Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations, J. Phys. A 40 (2007), no. 6, R1–R38. MR 2343636, DOI 10.1088/1751-8113/40/6/R01
Onni Ronkainen, Meromorphic solutions of difference Painlevé equations, Ann. Acad. Sci. Fenn. Math. Diss. 155 (2010), 59. Dissertation, University of Eastern Finland, Joensuu, 2010. MR 2809556
B. Grammaticos, T. Tamizhmani, A. Ramani, and K. M. Tamizhmani, Growth and integrability in discrete systems, J. Phys. A 34 (2001), no. 18, 3811–3821. MR 1840846, DOI 10.1088/0305-4470/34/18/309
B. Grammaticos, A. Ramani, and V. Papageorgiou, Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), no. 14, 1825–1828. MR 1125950, DOI 10.1103/PhysRevLett.67.1825
Valerii I. Gromak, Ilpo Laine, and Shun Shimomura, Painlevé differential equations in the complex plane, De Gruyter Studies in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2002. MR 1960811, DOI 10.1515/9783110198096
Rodney Halburd, Risto Korhonen, and Kazuya Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4267–4298. MR 3206459, DOI 10.1090/S0002-9947-2014-05949-7
Ilpo Laine and Chung-Chun Yang, Clunie theorems for difference and $q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556–566. MR 2377111, DOI 10.1112/jlms/jdm073
A. Z. Mohon′ko, The Nevanlinna characteristics of certain meromorphic functions, Teor. Funkciĭ Funkcional. Anal. i Priložen. 14 (1971), 83–87 (Russian). MR 0298006
A. A. Mohon′ko and V. D. Mohon′ko, Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations, Sibirsk. Mat. Ž. 15 (1974), 1305–1322, 1431 (Russian). MR 0404625
A. Ramani, B. Grammaticos, and J. Hietarinta, Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), no. 14, 1829–1832. MR 1125951, DOI 10.1103/PhysRevLett.67.1829
Georges Valiron, Sur la dérivée des fonctions algébroïdes, Bull. Soc. Math. France 59 (1931), 17–39 (French). MR 1504970