The road to the discrete analogue of the Painlevé property: Nevanlinna meets singularity confinement. (English) Zbl 1057.39018

Ramani, A.; Grammaticos, B.; Tamizhmani, T.; Tamizhmani, K. M.

Comput. Math. Appl. 45, No. 6-9, 1001-1012 (2003).

Authors’ summary: The question of integrability of discrete systems is analyzed in the light of the recent findings of M. J. Ablowitz, A. Ramani and H. Sagur [Lett. Nuovo Cimento (2) 23, No. 9, 333–338 (1978)], 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.

Reviewer: D. M. Bors (Iaşi)

MSC:

39A12	Discrete version of topics in analysis
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests

Keywords:

Painlevé equations; discrete systems; discrete integrability; singularity confinement; meromorphic functions; order and Nevanlinna theory

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