The road to the discrete analogue of the Painlevé property: Nevanlinna meets singularity confinement

• Published in: Computers & mathematics with applications (1987) Vol. 45; no. 6; pp. 1001 - 1012
• Main Authors: Ramani, A., Grammaticos, B., Tamizhmani, T., Tamizhmani, K.M.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2003
• Subjects: Discrete integrability; Discrete systems; Meromorphic functions; Order and Nevanlinna theory; Painlevé equations; Singularity confinement; Singularity confinement; Painlevé equations; Discrete systems; Meromorphic functions; Order and Nevanlinna theory; Discrete integrability
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/S0898-1221(03)00076-2

The question of integrability of discrete systems is analyzed in the light of the recent findings of Ablowitz et al., who have conjectured that a fast growth of the solutions of a difference equation is an indication of nonintegrability. The study of the behaviour of the solutions of a mapping is based on the theory of Nevanlinna. In this paper, we show how this approach can be implemented in the case of second-order mappings which include the discrete Painlevé equations. Since the Nevanlinna approach does offer only a necessary condition which is not restrictive enough, we complement it by the singularity confinement requirement, first in an autonomous setting and then for deautonomisation. We believe that this three-tiered approach is the closest one can get to a discrete analogue of the Painlevé property.