scholarly journals Singularity confinement for matrix discrete Painlevé equations

Nonlinearity ◽  
2014 ◽  
Vol 27 (9) ◽  
pp. 2321-2335 ◽  
Author(s):  
Giovanni A Cassatella-Contra ◽  
Manuel Mañas ◽  
Piergiulio Tempesta
Keyword(s):  
Painlevé Equations ◽  
Painleve Equations ◽  
Singularity Confinement ◽  
Discrete Painlevé Equations

INVESTIGATING THE INTEGRABILITY OF DISCRETE SYSTEMS

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3551-3565 ◽  
Author(s):  
B. GRAMMATICOS ◽  
A. RAMANI
Keyword(s):  
Discrete Systems ◽  
Singularity Analysis ◽  
Discrete Analog ◽  
Painlevé Equations ◽  
Discrete Version ◽  
Painleve Equations ◽  
Singularity Confinement ◽  
Discrete Painlevé Equations ◽  
Systematic Derivation ◽  
Close Parallel

The question of the integrability of discrete systems is examined with emphasis on the singularity confinement method that is the discrete analog of the “Painlevé” singularity analysis for differential systems. One, all-important, result of this new method is the systematic derivation of discrete Painlevé equations. The existence of a discrete version for the continuous Painlevé equations allows us to establish a close parallel between the continuous and the discrete case.

scholarly journals What is the discrete analogue of the Painlevé property?

2002 ◽  
Vol 44 (1) ◽  
pp. 21-32 ◽  
Author(s):  
A. Ramani ◽  
B. Grammaticos
Keyword(s):  
Discrete Systems ◽  
Discrete Analogue ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
Exact Procedure ◽  
Painlevé Property ◽  
Singularity Confinement ◽  
Discrete Painlevé Equations ◽  
Growth Properties ◽  
Special Growth

AbstractWe analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Hierarchies of q-discrete Painlevé equations

2020 ◽  
Vol 27 (3) ◽  
pp. 453-477 ◽  
Author(s):  
Huda Alrashdi ◽  
Nalini Joshi ◽  
Dinh Thi Tran
Keyword(s):  
Painlevé Equations ◽  
Painleve Equations ◽  
Discrete Painlevé Equations
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