2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0

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1 1473 : de nouvelles perspectives pp VI q 1 Tetsuya Kikuchi Sabro Kakei Drinfel d-sokolov Painlevé [KK1] [KK2] [KK3] [KIK] [ ] [ ] [KK3] three-wave equation Painlevé VI q q Drinfel d-sokolov ĝl 3 homogeneous q-painlevé VI 3 -Wilson W z; t z; t i α i β i i = ii Laplace q-painlevé VI 2 2 Schlesinger ii Laplace Harnad [H] Mazzocco [M1] 3 3 Painlevé VI Dubrovin-Mazzocco [DM] 1 Boalch [Bo1] [Bo2] [KK2] Painlevé VI i q q Painlevé VI [JS] 2 3 q 1

2 2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := z + 2 z 2 + W i = W i t i 1 i = i t i W 1 0 W 1 = w ij 0 = w ij W -Wilson : W = B a W W Λ a 21 = B a Λa a = Λ a = ze aa B a = B a z; t := W Λ a W 0 23 E aa aa 1 B a 0 z z w12 w 13 0 w12 0 B 1 = w B 2 = w 21 z w 23 w w w13 B 3 = 0 0 w 23 w 31 w 32 z -Wilson [KK3] -Wilson [Ta1] -Wilson 22 z 0 W = B a0 0 a = B a0 B a z = 0 w bc = w ba w ac a b w ac = b a w ab w bc 25 2

3 α = α 1 α 2 α 3 C 3 Baker-Akhiezer Ψ z; t α = W z; t expzt a E aa z Dα Ψ 0 z; t α = z; t expzt a E aa -Wilson : Ψ = zψ z Dα Ψ = B a Ψ a = [ ] B a B b = 0 t b 1 a b KP reduction Lax 6 reduction w ij = w ia w aj i j a 28 w ij = 0 = t 1 + t 2 + t 3 29 three-wave three-wave [KvdL] 0 Ψ 0 Painlevé VI B a t B a := 0 B a 0 a 0 24 B a B a = 0 B a = 0 Λ a 0 0 ij ω ij ω ij = ji 0 det ji 0 0 ji -Wilson det 0 t a 0 Ψ 0 [ ] t B a a t B b = 0 1 a b b 3

4 w ij [ B a B b ] = 0 B b = B a 212 t b Three-wave α i β j C i j = Dα Dβ Dα := diagα 1 α 2 α 3 Dβ := diagβ 1 β 2 β 3 λ C W λ z; t := λ Dα W λ z; λtλ Dα 213 λ z; t := λ Dα λ z; λtλ Dβ 214 λt := λt 1 λt 2 λt 3 W 1 0 w ij t λ α i α j +1 w ij λt w ij t λ α i βj w ij λt 1 W z; t z; t -Wilson Wλ z; t λ z; t Drinfeld-Sokolov Painlevé [ ] Painlevé VI reduction 1 Painlevé VI [AvdL] [Du] : W λ z; t = W z; t λ z; t = z; t Baker-Akhiezer Ψ z; t α Ψ 0 z; t β z Ψ z = Dα + t a B a Ψ Ψ = B a Ψ a =

5 dressing method [KK2] [KK3] 215 λ λ = W 1 0 t a w ij = α j α i 1w ij t a w ij = β j α i w ij Painlevé VI [FY] [Kit] Painlevé VI α i w ij w ij Painlevé VI z z Ψ z = zt + V Ψ T := diagt 1 t 2 t 3 V V := Dα + t a B a0 = α 1 t 2 t 1 w 12 t 3 t 1 w 13 t 1 t 2 w 21 α 2 t 3 t 2 w t 1 t 3 w 31 t 2 t 3 w 32 α z = 0 z = rank 1 t α β z = 0 Baker-Akhiezer Ψ 0 z = 0 Ψ z = z = 0 1 Ψ = 0 Ψ 0 Ψ 0 z Ψ z = z 0 T 0 + Dβ Ψ Ψ = B a Ψ a = Painlevé VI Schlesinger 218 V λ λ = 1 z z = Dα Dβ + t a 0 0 V 0 = Dβ 220 5

6 23 Laplace 219 Schlesinger Laplace Ψz Φζ = L[Ψz]ζ := e zζ Ψzdz 221 Ψz Φζ z k ζ k z k ζ k : 3 Φζ; t = L[ 0 Ψ 0 ] L[ 0 Ψ ] γ Φζ; t ζ Φζ; t = A a t ζ t a Φζ; t 222 = A at ζ t a Φζ; t a = A a t := 0 E aa 0 Dβ + I = 0 E aa V + I Laplace xi + 0 T 0 Φζ ζ = Dβ + IΦζ Φζ ζ = 0 ζi + T 0 Dβ + IΦx 3 4 t 1 t 2 t Schlesinger 24 Painlevé VI reduction Painlevé VI Schlesinger β W z; t = λ Dα β 3+1I W λ z; λtλ Dα β 3+1I z; t = λ Dα β 3+1I λ z; λtλ Dβ+β 3+1I 6

7 V ij V ij = ω 1j w i1 β 1 β 3 1 w i2 β 2 β 3 1 w i3 ω 2j ω 3j = w i1 β 1 β 3 w i2 β 2 β 3 ω 1j w ω ia ω aj 225 2j ω ij 0 ij Laplace A a = 0 E aa 0 Dβ β 3 I ω 1a = ω 2a w a1 β 1 β 3 w a2 β 2 β 3 0 ω 3a ω1a à a = wa1β 1 β 3 w a2β 2 β ω 2a à ξ := t 1 ζ t := t 1 t 3 t 1 t 2 t 1 t 2 Y Ã1 ξ = ξ + Ã2 ξ 1 + Ã3 Y ξ t Y t = Ã3 ξ t Y [JM] [Ok1] y = Painlevé VI d 2 y dt 2 =1 2 t 3 t 1 ω 1a w a2 β 2 β 3 t 1 t 2 ω 1a w a2 β 2 β 3 + t 1 t 3 ω 1a w a2 β 2 β 3 1 y + 1 y yy 1y t 2t 2 t dy 1 y t dt t + 1 t dy y t dt { } θ θ1 2 t y + t 1 2 θ2 3 y tt 1 2 θ2 2 y t 2 α i β j θ 1 = α 1 β 3 θ 2 = α 3 β 3 θ 3 = α 2 β 3 θ 4 = β 1 β A 1 2 affine Weyl -Wilson [KK2] [KK3] Boalch [Bo2] 3 3 Lax F 1 4 Weyl [Ok2] 7

8 3 q n-wave equation Lax q 31 q -Wilson q x = x 1 x 2 x 3 q-shft T 1 T 3 T 3 T j x k = q δ jk x k j k = 1 n x f T j f j x := T j fx f jk x := T j T k fx q -Wilson [Ta2] : W = W z; x := I + W 1 xz + W 2 xz 2 + = z; x := 0 x + 1 xz + 2 xz 2 + W j = W j x j = j x 3 3 x W a I ɛx a Λ a = I ɛx a B a W 31 a I ɛx a Λ a = I ɛx a B a 32 ɛ = 1 q Λ a = ze aa B a z; x : B a := W a Λ a W 0 = Λ a + W a 1 E aa E aa W 1 33 q I T a ɛx a W = B a W W a Λ a = W a Λ a W <0 W I T a ɛx a = Ba a Λ a -Wilson q Baker-Akhiezer q ẽ q z := ɛzq ; q = 1 ɛzq i 34 i=0 8

9 q > 1 q Painlevé VI q -shift q ẽ q qz = 1 ɛzẽ q z 1 T q ẽ q z = ẽ q z 35 ɛz q Baker-Akhiezer Ψ z; x α = W z; t ẽ q zx a E aa z Dα 36 Ψ 0 z; x α = z; t ẽ q zx a E aa z Dα 37 q -Wilson q 35 q Baker-Akhiezer : I T a ɛx a Ψ = B a Ψ T a Ψ = I ɛx a B a Ψ 38 q q-shift 38 z I ɛx a B a = I ɛx a Λ a + W a 1 E aa E aa W 1 =: zɛx a + V a X a := x a E aa V a := I ɛw a 1 X a X a W 1 39 W 1 w ij 0 w ij V 3 = I ɛx 3 w 1 11 w 11 w 12 w 13 V 1 = I ɛx 1 w w w V 2 = I ɛx 2 w 21 w 2 22 w 22 w 23 0 w w w 3 23 w 31 w 32 w 3 33 w 33 -Wilson 32 0 w bc w a bc ɛx a = w a ba w ac b a w ac w a ac ɛx a = w a aa w aa w ab w bc 310 b=1 9

10 25 q q Ψ 0 top term 0 Ψ 0 a T a Ψ = I ɛx a 0 Λ a 0 Ψ a = q Painlevé VI 311 I ɛz V a z V a a := x a 0 a Λ a 0 = z 0 X a 0 32 q three-wave q 38 T a T b Ψ = T b T a Ψ a b = ɛzx b + V a b ɛzx a + V a = ɛzx a + V a b ɛzx b + V b q-kp reduction q three-wave equation 312 z z 2 0 z 1 z 0 X b V a + V a b X a = X a V b + V a b X b 313 V a b V a = V a b V b 314 [KNY] q KP -Wilson V a [ ] 39 W trivial 314 w aa a w aa + w aa b w aa ab + ɛx b w b ab w ba w ab ab w a ba = x a w b ab w ab + x b w ab ab w b ab + ɛx a x b w b ab w bb w aa ab w b ab + i=1 w b ai wb ia = w b ac w ac + ɛx b w b ab w bc = q three-wave 317 w ac w ac b = w b ab ɛx w bc b three-wave 28 q 10

11 311 I ɛz a V b I ɛz V b a = I ɛz V a I ɛz V b 318 z z 1 z 2 x a Va + x b V a b V a b b = x b Vb + x a V a 319 V a = V b trivial 319 x a ω a ia w aj + x b ω ab ib w a bj V b a = x b ω b ib w bj + x a ω ab ia w b aj 321 ω ij 0 ij T b ω a ia ɛx b w aj = 1 T a ω b ib ɛx w bj a 212 q 33 q scaling symmetry W λ z; x := λ Dα W λ z; λxλ Dα λ z; x := λ Dα λ z; λxλ Dβ λx = λx 1 λx 2 λx 3 4 W z; x z; x q -Wilson W λ z; x λ z; x W λ z; x = W z; x λ z; x = z; x 322 λ = q Baker-Akhiezer Ψqz; x = q Dα Ψz; qx = q Dα ɛzx 1 + V 23 1 ɛzx2 + V 3 ɛzx3 + V 3 Ψz; x z 3 z 3 z 2 0 V a W 1 z Ψqz; x = q Dα ɛzx 1 + X 2 + X 3 + V 23 1 V 3 2 V 3 Ψz; x Painlevé VI 3 3 Lax q X := X 1 + X 2 + X 3 = diagx 1 x 2 x

12 5 322 q Baker-Akhiezer Ψqz; x = q Dα ɛzx + V 23 1 V 3 2 V 3 Ψz; x 323 T a Ψz; x = zɛx a + V a Ψz; x a = z λ = q grade 0 0 x = q Dα 0 qxq Dβ T a 0 = V a 0 0 qx = T 1 T 2 T 3 0 x = V 23 1 V 3 2 V 3 0 x q Dα V 23 1 V 3 2 V 3 = 0 xq Dβ 0 x Ψ 0 Ψ 0 2 Ψ = 0 Ψ 0 Ψ 0 : Ψqz; x = 0 q Dα ɛzx + V 23 1 V 3 2 V 3 0 Ψz; x = q Dβ ɛz 0 q Dα X 0 Ψz; x 328 T a Ψz; x = I ɛz V a a Ψz; x Va := 0 X a q Laplace Laplace q > 1 q Jackson 0 ftd q t = 1 q n= fq n q n 330 q Laplace q ẽ q z 34 Jackson Ψz = 0 Φζẽ q zζd q ζ = 1 q 0 ft fqt gtd q t = ɛt 12 0 n= Φq n ẽ q zq n q n ft gt gq t d q t 331 ɛt

13 q D qz := 1 T qz ɛz z Ψz Φζ z D q ζ D qz ζ q Laplace q 2 Ψqz q Φq ζ q Baker-Akhiezer Ψ = 0 Ψ 0 Ψ 0 q Laplace : Φq ζ; x = ζ 0 ζi q Dα+I X q Dα+I Xζ + V 23 1 V 3 2 V 3 0 Φζ; x 333 ζa a x = I + Φζ; x 334 ζ q α a+1 x a a 0 E aa 0 I q Dβ+I T a Φζ; x = I x a Φζ; x 335 ζ q α a+1 x a A a x : = a = = 0 E aa 0 I q Dβ+I E aa I q Dα+I V 23 1 V 3 2 V V a q Laplace 0 I q Dα+I Xζ 0 Φq ζ; x = 0 q Dα+I Xζ + V 23 1 V 3 2 V 3 0 Φζ; x q Laplace ζi 0 q Dα+I X 0 Φq ζ; x = ζq Dβ+I 0 q Dα+I X 0 Φζ; x 337 = ζq Dβ+I I + 0 ζi q Dα+I X 0 Φζ; x 13

14 ζi q Dα+I = E aa ζ q αa q-laplace q Painlevé VI q Painlevé VI t f g [JS]: T g = f ta 1f ta 2 b 3 b 4 T f = g tb 1g tb 2 a 3 a 4 gf a 3 f a 4 fg b 3 g b T t q a i b i i = q Y qζ t = Aζ ty x t Aζ t = A 0 t + A 1 tζ + A 2 tζ κ1 0 A 2 t = A 0 κ 0 t tθ 1 tθ 2 2 θ 1 θ 2 κ 1 κ 2 {q ±1 q ±2 } ζ = 0 t Aζ t det Aζ t = κ 1 κ 2 x ta 1 x ta 2 x a 3 x a 4 t Y ζ qt = Bζ ty ζ t Bζ t = ζζi + B 0 t ζ qta 1 ζ qta f g f = A12 0 A 12 1 b j g = A ta 1 A 12 1 A ta 2 A 12 1 q A 11 0 A A 11 1 A 12 0 A κ 1 A b 1 = a 1a 2 θ 1 b 2 = a 1a 2 θ 2 b 3 = 1 qκ 1 b 4 = 1 κ 2 i A i t jk q-three wave Lax Laplace q q Painlevé VI A jk 14

15 1 x 3 = 0 x 1 = γt γ x 2 q q Painlevé VI κ 1 = q β 1+1 κ 2 = q β 2+1 θ 1 = γx 2 q α 1+α 2 +2 θ 2 = γx 2 q α 1+α 2 +α a 1 = γ a 2 = γq α 1+1 a 3 = x 2 a 4 = x 2 q α 2+1 x 3 = 0 39 V 3 = I 6 ζ Φq A1 x A 2 x ζ; x = I + ζ + + A 3x Φζ; x ζ q 1+α 1 x1 ζ q 1+α 2 x2 ζ = 0 diag ζ q α1+1 x 1 ζ q α2+1 x 2 ζ q Dα+I ζ X 1 ζ + V 23 1 X2 ζ + V Φζ; x x E 11 0 I q Dβ+I T 1 Φζ; x = I x 1 Φζ; x 342 ζ q α 1+1 x 1 ζ q -shift x 1 q-shift 342 Φζ x = Y ζ x i=0 1 qα 1 i x 1 ζ i=0 1 qα 2 i x 2 ζ Y ζ x ζ Ãζ; x = Ã2ζ 2 + Ã1ζ + Ã0 Ãζ; x =ζ 2 I + A 1 + A 2 + A 3 ζ q 1+α 1 x 1 I + A 2 + A 3 + q 1+α 2 x 2 I + A 1 + A 3 = + q α 1+α 2 +2 x 1 x 2 I + A diag ζζ q α2+1 x 2 ζζ q α1+1 x 1 ζ q α1+1 x 1 ζ q α2+1 x 2 q Dα+I X 1 ζ + V 23 1 X2 ζ + V [JS] q ζ Ã 2 = I + A 1 + A 2 + A 3 = q I+Dβ Ã 1 = q 1+α 1 x 1 A 1 q Dβ+I + q 1+α 2 x 2 A 2 q Dβ+I Ã 0 = q α 1+α 2 +2 x 1 x 2 I + A 3 15

16 2 β 3 = 336 A i Aζ x = A 2 ζ 2 + A 1 ζ + A 0 q β A 2 = 0 q β 2+1 A 0 q α 1+α 2 +2 x 1 x 2 q α 1+α 2 +α 3 +3 x 1 x 2 det Aζ; x ζ = x 1 x 2 q α1+1 x 1 q α1+1 x x 1 x 2 q Painlevé VI 338 f g q three-wave 341 f = A12 0 A 12 1 g = A x 1 A 12 1 A x 1 q α1+1 A 12 1 q A 11 0 A A 11 1 A 12 0 A q β 2+1 A A 12 0 = q α 1+α 2 +2 x 1 x 2 ω 13 w 32 A 12 1 = q α 1+1 x 1 ω 11 w 12 + q α 2+1 x 2 ω 12 w 22 A 11 0 = q α 1+α 2 +2 x 1 x ω 13 w 31 A 11 1 = q α 1+1 x ω 12 w 21 + ω 13 w 31 q α 2+1 x ω 11 w 11 + ω 13 w Wilson n Garnier n n Lax [M2] n q q-garnier [Sa] Lax q three-wave q Painlevé VI affine Weyl τ [TM] 16

17 References [AvdL] Aratyn H and van de Leur J: Integrable structure behind WDVV equations Theoret Math Phys [Bo1] Boalch P P: From Klein to Painlevé via Fourier Laplace and Jimbo Proc London Math Soc [Bo2] Boalch P P: Six results on Painlevé VI preprint arxiv:mathag/ [Du] [DM] [FY] [H] [JM] [JS] Dubrovin B: Geometry of 2d topological field theories integrable systems and quantum groups Springer Lecture Notes in Math 1620 Springer Berlin Dubrovin B and Mazzocco M: Monodromy of certain Painlevé VI transcendents and reflection groups Invent Math Fokas A S and Yortsos Y C: The transformation properties of the sixth Painlevé equation and one-parameter families of solutions Lett Nuovo Cimento Harnad J: Dual isomonodromic deformations and moment maps to loop algebras Commun Math Phys Jimbo M and Miwa T: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II Physica D Jimbo M and Sakai H: A q-analog of the sixth Painlevé equation Lett in Math Phys [KvdL] Kac V G and van de Leur J: The n-component KP hierarchy and representation theory in Important developments in soliton theory Springer Berlin [KNY] Kajiwara K Noumi M and Yamada Y: q-painlevé systems arising from q-kp hierarchy Lett in Math Phys [KK1] Kakei S and Kikuchi T: Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction Int Math Res Not [KK2] Kakei S and Kikuchi T: Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction Glasgow Math J 47 Issue A [KK3] Kakei S and Kikuchi T: The sixth Painlevé equation as similarity reduction of ĝl 3 hierarchy Preprint arxiv:nlinsi/ [ ]

18 [ ] [KIK] Kikuchi T Ikeda T and Kakei S: Similarity reduction of the modified Yajima- Oikawa equation J Phys A: Math Gen [Kit] [M1] [M2] Kitaev A V: On similarity reductions of the three-wave resonant system to the Painlevé equations J Phys A: Math Gen M Mazzocco Painlevé sixth equation as isomonodromic deformations equation of an irregular system In The Kowalevski property: CRM Proceedings and Lecture Notes M Mazzocco Irregular isomonodromic deformations for Garnier systems and Okamoto s canonical transformations J London Math Soc [Ok1] Okamoto K: Isomonodromic deformation and Painlevé equations and the Garnier system J Fac Sci Univ Tokyo Sect IA Math [Ok2] Okamoto K: Studies on the Painlevé equations I Sixth Painlevé equation P VI Ann Mat Pura Appl [Sa] Sakai H: A q-analog of the Garnier system Preprint UTMS [TM] [Ta1] [Ta2] Tsuda T and Masuda T: q-painlevé VI equation arising from q-uc hierarchy Preprint UTMS Takasaki K: q-analogue of modified KP hierarchy and its quasi-classical limit Lett Math Phys

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