We introduce a new representation for the discrete Painlevé equations. It is based on the trihomographic form previously used in conjunction with discrete Painlevé equations associated with the affine Weyl group E. We show here that all discrete Painlevé equations do possess a trihomographic form. The latter is often more convenient than the usual representation in particular when it comes to the application of the singularity confinement integrability criterion. We show how trihomographic forms can be combined in order to construct generic discrete Painlevé equations or limits thereof. A trihomographic form for the generic elliptic Painlevé equation is also derived.
REFERENCES
1.
2.
J. A.
Shohat
, Duke Math. J.
5
, 401
(1939
).3.
A.
Ramani
, B.
Grammaticos
, and J.
Hietarinta
, Phys. Rev. Lett.
67
, 1829
(1991
).4.
H.
Sakai
, Commun. Math. Phys.
220
, 165
(2001
).5.
B.
Grammaticos
, F. W.
Nijhoff
, and A.
Ramani
, “Discrete Painlevé equations
,” in The Painlevé Property – One Century Later
, edited by R.
Conte
(Springer-Verlag
, New York
, 1999
), p. 413
.6.
G. R. W.
Quispel
, J. A. G.
Roberts
, and C. J.
Thompson
, Physica D
34
, 183
(1989
).7.
B.
Grammaticos
, A.
Ramani
, and V.
Papageorgiou
, Phys. Rev. Lett.
67
, 1825
(1991
).8.
A.
Ramani
, B.
Grammaticos
, R.
Willox
, T.
Mase
, and M.
Kanki
, J. Phys. A
48
, 11FT02
(2015
).9.
J.
Hietarinta
and C.-M.
Viallet
, Phys. Rev. Lett.
81
, 325
(1998
).10.
A.
Ramani
and B.
Grammaticos
, Phys. Lett. A
373
, 3028
(2009
).11.
A.
Ramani
, S.
Carstea
, B.
Grammaticos
, and Y.
Ohta
, Physica A
305
, 437
(2002
).12.
Y.
Ohta
, A.
Ramani
, and B.
Grammaticos
, J. Phys. A
35
, L653
(2002
).13.
A.
Ramani
and B.
Grammaticos
, “Discrete Painlevé equations associated with the affine Weyl group E8
,” J. Phys. A
(to be published).14.
B.
Grammaticos
and A.
Ramani
, J. Phys. A
48
, 16FT02
(2015
).15.
Y.
Ohta
, A.
Ramani
, and B.
Grammaticos
, J. Phys. A
34
, 10523
(2001
).16.
A.
Ramani
and B.
Grammaticos
, J. Phys. A
47
, 385201
(2014
).17.
B.
Grammaticos
and A.
Ramani
, Regular Chaotic Dyn.
5
, 53
(2000
).18.
B.
Grammaticos
, T.
Tamizhmani
, A.
Ramani
, A. S.
Carstea
, and K. M.
Tamizhmani
, J. Phys. Soc. Jpn.
71
, 443
(2002
).19.
A.
Ramani
and B.
Grammaticos
, Physica A
228
, 160
(1996
).20.
M.
Murata
, Funkcialaj Ekvacioj
47
, 291
(2004
).21.
M.
Murata
, H.
Sakai
, and J.
Yoneda
, J. Math. Phys.
44
, 1396
(2003
).22.
M.
Noumi
, S.
Tsujimoto
, and Y.
Yamada
, “Padé interpolation for elliptic Painlevé equation
,” in Symmetries, Integrable Systems and Representations
, Springer Proceedings in Mathematics and Statistics
, edited byK.
Iohara
, et al. (Springer-Verlag
, 2013
), Vol. 40
, p. 463
.23.
A.
Ramani
and B.
Grammaticos
, Chaos, Solitons Fractals
24
, 1331
(2005
).24.
A.
Ramani
and B.
Grammaticos
, J. Nonlinear Math. Phys.
20
, 153
(2013
).25.
B.
Grammaticos
, A.
Ramani
, J.
Satsuma
, and R.
Willox
, J. Math. Phys.
53
, 023506
(2012
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
