MR3468735
Tetsu MASUDA
Tetsu MASUDA
A $q$-Analogue of the Higher Order Painlevé Type Equations with the Affine Weyl Group Symmetry of Type $D$
Funkcialaj Ekvacioj. Serio Internacia
58
2015
405--430
http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-3/58_405.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3468735
We present a $q$-difference analogue of the higher order differential equations of Painlevé type whose symmetry can be described by the affine Weyl group of type $D$, so-called the Sasano system. It is derived from the birational realization of Weyl groups as a group of pseudo isomorphisms of certain algebraic varieties. The continuous limit is also investigated.
Cremona transformation, $q$-difference equation, Painlevé equation, Rational variety, Tropical representation, Weyl group, Continuous limit, Sasano system.
14E07, 14L30, 20F55, 34M55.
58-405
2015
A $q$-Analogue of the Higher Order Painlevé Type Equations with the Affine Weyl Group Symmetry of Type $D$
Tetsu MASUDA
Tetsu MASUDA
1
Fuji, K.; Inoue, K.; Shinomiya, K.; Suzuki, T.
Higher order Painlevé system of type $D_{2n+2}^{(1)}$ and monodromy preserving deformation
J. Nonlinear Math. Phys.
20
2013
suppl. 1, 57–69
MR3196472
2
Fuji, K.; Suzuki, T.
Higher order Painlevé system of type $D^{(1)}_{2n+2}$ arising from integrable hierarchy
Int. Math. Res. Not. IMRN
2008
2008
Art. ID rnm 129, 21 pp
MR2417788
3
Fuji, K.; Suzuki, T.
Drinfeld-Sokolov hierarchies of type $A$ and fourth order Painlevé systems
Funkcial. Ekvac.
53
2010
143-167
MR2668518
2668518
4
Garnier, R.
Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes
Ann. Sci. Ecole Norm. Sup. (3)
29
1912
1-126
MR1509146
5
Jimbo, M.; Sakai, H.
A $q$-analog of the sixth Painlevé equation
Lett. Math. Phys.
38
1996
145-154
MR1403067
6
Kajiwara, K.; Masuda, T.; Noumi, M.; Ohta, Y.; Yamada, Y.
Point configurations, Cremona transformations and the elliptic difference Painlevé equation
Sémin. Congr., 14, Soc. Math. France, Paris
2006
pp. 169-198
MR2353465
7
Kajiwara, K.; Noumi, M.; Yamada, Y.
Discrete dynamical systems with $W(A_{m-1}^{(1)}\times A_{n-1}^{(1)})$ symmetry
Lett. Math. Phys.
60
2002
211-219
MR1917133
8
Kajiwara, K.; Noumi, M.; Yamada, Y.
$q$-Painlevé systems arising from $q$-KP hierarchy
Lett. Math. Phys.
62
2002
259-268
MR1958118
9
Kawakami, H.; Nakamura, A.; Sakai, H., Degeneration scheme of 4-dimensional Painlevé-type equations, preprint
10
Noumi, M.; Yamada, Y.
Higher order Painlevé equations of type $A^{(1)}_l$
Funkcial. Ekvac.
41
1998
483-503
MR1676885
1676885
11
Sakai, H.
Rational surfaces associated with affine root systems and geometry of the Painlevé equations
Comm. Math. Phys.
220
2001
165-229
MR1882403
12
Sakai, H.
A $q$-analog of the Garnier system
Funkcial. Ekvac.
48
2005
273-297
MR2177121
2177121
13
Sakai, H., Isomonodromic deformation and 4-dimensional Painlevé type equations, preprint
14
Sasano, Y.
Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D^{(1)}_6$. II
RIMS Kōkyūroku Bessatsu, B5, Res. Inst. Math. Sci. (RIMS), Kyoto
2008
pp. 137-152
MR2405972
15
Sasano, Y.; Yamada, Y.
Symmetry and holomorphy of Painlevé type systems
RIMS Kōkyūroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto
2007
pp. 215-225
MR2310032
16
Suzuki, T.
A $q$-analogue of the Drinfel'd-Sokolov hierarchy of type $A$ and $q$-Painlevé system
Contemp. Math., 651, Amer. Math. Soc., Providence, RI
2015
pp. 25-38
17
Takenawa, T.
Weyl group symmetry of type $D^{(1)}_5$ in the $q$-Painlevé V equation
Funkcial. Ekvac.
46
2003
173-186
MR1996297
18
Tsuda, T.
On an integrable system of $q$-difference equations satisfied by the universal characters: its Lax formalism and an application to $q$-Painlevé equations
Comm. Math. Phys.
293
2010
347-359
MR2563787
19
Tsuda, T.; Masuda, T.
$q$-Painlevé VI equation arising from $q$-UC hierarchy
Comm. Math. Phys.
262
2006
595-609
MR2202304
20
Tsuda, T.; Takenawa, T.
Tropical representation of Weyl groups associated with certain rational varieties
Adv. Math.
221
2009
936-954
MR2511044