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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
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[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
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