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

✍ Scribed by A. Ramani; B. Grammaticos; T. Tamizhmani; K.M. Tamizhmani

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 998 KB
• Volume: 45
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(03)00076-2

✦ Synopsis

The question of integrability of discrete systems is analysed 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 csse of second-order mappings which include the discrete Painlevk 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 Painlevb property.