## Existence of finite-order meromorphic solutions as a detector of integrability in difference equations.(English)Zbl 1105.39019

The authors consider an analogue of the Painlevé property for discrete equations. They start from a difference equations of the type
$\bar{y}+\underline{y} = R(z,y)$
with $$R$$ being rational in both arguments, $$z\in C$$, $$\bar{y}=y(z+1)$$, $$\underline{y}=y(z-1)$$, $$y=y(z)$$. The discrete version of the Painlevé II equation for which
$R(z,y)={{(\lambda z+\mu)y+\nu}\over{1-y^2}}$
with constant parameters $$\lambda,\mu,\nu$$ belongs to this class of equations. For this equation the “integrability” test based on singularity confinement for meromorphic solutions is applied; this confinement is discussed using the Nevanlinna theory.

### MSC:

 39A12 Discrete version of topics in analysis 39A20 Multiplicative and other generalized difference equations 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
Full Text:

### References:

 [1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001 [2] Painlevé, P., Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. France, 28, 201-261 (1900) [3] Painlevé, P., Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math., 25, 1-85 (1902) [4] Fuchs, L., Sur quelques équations différentielles linéares du second ordre, C. R. Acad. Sci., Paris, 141, 555-558 (1905) [5] Gambier, B., Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math., 33, 1-55 (1910) [6] Ablowitz, M. J.; Segur, H., Exact linearization of a Painlevé transcendent, Phys. Rev. Lett., 38, 1103-1106 (1977) [7] Ablowitz, M. J.; Halburd, R. G.; Herbst, B., On the extension of the Painlevé property to difference equations, Nonlinearity, 13, 889-905 (2000) · Zbl 0956.39003 [8] Costin, O.; Kruskal, M., Movable singularities of solutions of difference equations in relation to solvability and a study of a superstable fixed point, Theoret. Math. Phys., 133, 1455-1462 (2002) [9] Grammaticos, B.; Ramani, A.; Papageorgiou, V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett., 67, 1825-1828 (1991) · Zbl 0990.37518 [10] Ramani, A.; Grammaticos, B.; Hietarinta, J., Discrete versions of the Painlevé equations, Phys. Rev. Lett., 67, 1829-1832 (1991) · Zbl 1050.39500 [11] Hietarinta, J.; Viallet, C.-M., Singularity confinement and chaos in discrete systems, Phys. Rev. Lett., 81, 325-328 (1998) [12] Veselov, A. P., Growth and integrability in the dynamics of mappings, Commun. Math. Phys., 145, 181-193 (1992) · Zbl 0751.58034 [13] Falqui, G.; Viallet, C.-M., Singularity, complexity, and quasi-integrability of rational mappings, Commun. Math. Phys., 154, 111-125 (1993) · Zbl 0791.58116 [14] Bellon, M. P.; Viallet, C.-M., Algebraic entropy, Commun. Math. Phys., 204, 425-437 (1999) · Zbl 0987.37007 [15] Roberts, J. A.G.; Vivaldi, F., Arithmetical method to detect integrability in maps, Phys. Rev. Lett., 154 (2003), 034102 · Zbl 1267.37058 [16] Lehto, O., On the birth of the Nevanlinna theory, Ann. Acad. Sci. Fenn. Ser. A I Math., 7, 5-23 (1982) · Zbl 0471.30017 [17] Frank, G.; Weissenborn, G., Rational deficient functions of meromorphic functions, Bull. London Math. Soc., 18, 29-33 (1986) · Zbl 0586.30025 [18] Steinmetz, N., Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math., 368, 134-141 (1986) · Zbl 0598.30045 [19] Yamanoi, K., The second main theorem for small functions and related problems, Acta Math., 192, 225-294 (2004) · Zbl 1203.30035 [20] Hayman, W. K., Meromorphic Functions (1964), Clarendon Press: Clarendon Press Oxford · Zbl 0115.06203 [21] H.L. Selberg, Über eine Eigenschaft der logaritmischen Ableitung einer meromorphen oder algebroiden Funktion endlicher Ordnung, Avhandlinger Oslo 14; H.L. Selberg, Über eine Eigenschaft der logaritmischen Ableitung einer meromorphen oder algebroiden Funktion endlicher Ordnung, Avhandlinger Oslo 14 [22] Selberg, H. L., Über die Wertverteilung der algebroiden Funktionen, Math. Z., 31, 709-728 (1930) [23] Selberg, H. L., Algebroide Funktionen und Umkehrfunktionen Abelscher Integrale, Avh. Norske Vid. Akad. Oslo, 8, 1-72 (1934) · Zbl 0010.12301 [24] Ullrich, E., Über den Einfluß der Verzweigtheit einer Algebroide auf ihre Wertverteilung, J. Reine Angew. Math., 167, 198-220 (1931) [25] Valiron, G., Sur la dérivée des fonctions algébroïdes, Bull. Soc. Math. France, 59, 17-39 (1931) [26] Yanagihara, N., Meromorphic solutions of some difference equations, Funkcial. Ekvac., 23, 309-326 (1980) · Zbl 0474.30024 [27] Laine, I., Nevanlinna Theory and Complex Differential Equations (1993), Walter de Gruyter: Walter de Gruyter Berlin [28] McMillan, E. M., A problem in the stability of periodic systems, (Brittin, E.; Odabasi, H., Topics in Modern Physics, A Tribute to E.V. Condon, Colorado Assoc. (1971), Univ. Press: Univ. Press Boulder, Colorado), 219-244 [29] Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), Academic Press, Inc.: Academic Press, Inc. London · Zbl 0538.60093 [30] Kimura, T., On the iteration of analytic functions, Funkcial. Ekvac., 14, 197-238 (1971) · Zbl 0237.30008 [31] Shimomura, S., Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28, 253-266 (1981) · Zbl 0469.30021 [32] Julia, G., Memoire sur l’iteration des fonctions rationnelles, J. Math. Pures Appl., 1, 47-245 (1918) [33] Laine, I.; Rieppo, J.; Silvennoinen, H., Remarks on complex difference equations, Comput. Methods Funct. Theory, 5, 77-88 (2005) · Zbl 1093.39018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.