Discrete versions of the Painlevé equations

Ramani, A.; Grammaticos, B.; Hietarinta, Jarmo

doi:10.1103/physrevlett.67.1829

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Cited by 330 publications

(351 citation statements)

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“…Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).…”

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confidence: 98%

New exact solutions of the discrete fourth Painlevé equation

Bassom

1994

Physics Letters A

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confidence: 98%

New exact solutions of the discrete fourth Painlevé equation

Bassom

1994

Physics Letters A

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“…A. Ramani, B. Grammaticos and J. Hietarinta discovered discrete Painlevé equations such as d-P III , d-P IV , and d-P V (or q-PðA 3 Þ, d-PðA Ã 2 Þ and q-PðA 2 Þ respectively for their rational surfaces), using the method of singularity confinement (see [7]). Since then further discrete Painlevé equations have been found.…”

Section: Introductionmentioning

confidence: 99%

On Solutions of q-Painleve Equation of Type A7(1)

Nishioka

2009

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“…Since the singularity confinement method was introduced by Grammaticos et al [6], the discrete Painlevé equations have been studied extensively ( [13] for example). Emphasizing the fact that each discrete Painlevé equation preserves a family of rational surfaces, Sakai constructed the discrete Painlevé equations from families of rational surfaces (called generalized Halphen surfaces) and subsequently classified them.…”

Section: Introductionmentioning

confidence: 99%

Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in P 3 (C)

Takenawa

2004

Communications in Mathematical Physics

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A 3 dimensional analogue of Sakai's theory concerning the relation between rational surfaces and discrete Painlevé equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in P 3 , we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the Weyl group of type E (1) 7 . By normalizing the configuration space by means of elliptic curves, the action of the Weyl group and the dynamical system associated with a translation are explicitly described. As a result, it is found that the action of the Weyl group on P 3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on P 1 × P 1 .

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Discrete versions of the Painlevé equations

Cited by 330 publications

References 11 publications

New exact solutions of the discrete fourth Painlevé equation

New exact solutions of the discrete fourth Painlevé equation

On Solutions of q-Painleve Equation of Type A7(1)

Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in P 3 (C)

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