Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)

We have classified special solutions around the origin for the two-dimensional degenerate Garnier system G(1112) with generic values of complex parameters, whose linear monodromy can be calculated explicitly.

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Дата:	2014
Автор:	Kaneko, K.
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Опубліковано:	Інститут математики НАН України 2014
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) / K. Kaneko // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ.

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spelling	irk-123456789-1466072019-02-11T01:24:12Z Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) Kaneko, K. We have classified special solutions around the origin for the two-dimensional degenerate Garnier system G(1112) with generic values of complex parameters, whose linear monodromy can be calculated explicitly. 2014 Article Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) / K. Kaneko // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 33C15 DOI:10.3842/SIGMA.2014.069 http://dspace.nbuv.gov.ua/handle/123456789/146607 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We have classified special solutions around the origin for the two-dimensional degenerate Garnier system G(1112) with generic values of complex parameters, whose linear monodromy can be calculated explicitly.
format	Article
author	Kaneko, K.
spellingShingle	Kaneko, K. Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kaneko, K.
author_sort	Kaneko, K.
title	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)
title_short	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)
title_full	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)
title_fullStr	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)
title_full_unstemmed	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)
title_sort	special solutions and linear monodromy for the two-dimensional degenerate garnier system g(1112)
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146607
citation_txt	Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) / K. Kaneko // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 16 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kanekok specialsolutionsandlinearmonodromyforthetwodimensionaldegenerategarniersystemg1112
first_indexed	2025-07-11T00:19:07Z
last_indexed	2025-07-11T00:19:07Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 069, 15 pages Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112) Kazuo KANEKO Seki Kowa Institute of Mathematics, Yokkaichi University, Kayaucho, Yokkaichi, Mie, 512-8512, Japan E-mail: dr kaneko k@yahoo.co.jp Received October 24, 2013, in final form June 14, 2014; Published online July 05, 2014 http://dx.doi.org/10.3842/SIGMA.2014.069 Abstract. We have classified special solutions around the origin for the two-dimensional degenerate Garnier system G(1112) with generic values of complex parameters, whose linear monodromy can be calculated explicitly. Key words: two-dimensional degenerate Garnier system; monodromy data 2010 Mathematics Subject Classification: 34M55; 33C15 1 Introduction We have studied special solutions with generic values of complex parameters for the fourth, fifth, sixth and third Painlevé equations, for which the monodromy data of the associated linear equation (we call linear monodromy) can be calculated explicitly [9, 11, 10, 12]. These papers are based on A.V. Kitaev’s idea who calculated first the linear monodromy with generic value of complex parameter explicitly by taking examples of the first and second Painlevé equations [14]. We remark that P. Appell [1] also studied the symmetric solutions to the first and second Painlevé equations, but he did not study linear monodromy problems. The Garnier system was derived by R. Garnier (1912) as the extension of the sixth Painlevé equation [4]. The original Garnier system has n variables and is expressed in the nonlinear partial differential equations system, whose dimension of the solution space is 2n. There are few research for the special solutions to the Garnier system compared with Painlevé equations. We will study the Garnier transcendents by applying first the same method to the two-dimensional Garnier system, which we have used for the Painlevé equations above. Some new discovery is expected by viewing Painlevé equations from the Garnier system. Two-dimensional Garnier system has the following degeneration diagram similar to the Pain- levé equations [13]: G(11111) → G(1112) → G(122) → G(23) ↓ ↓ ↓ G(113) → G(14) → G(5) → G(9/2) (The degeneration from G(113) to G(23) also exists.) Numbers in brackets represents a partition of 5. The number 1 represents the regular singular point and the number r + 1 represents an irregular singular point of Poincaré rank r. The two-dimensional Garnier system G(11111) which is the extension of the sixth Painlevé equation PVI degenerates step by step to the two- dimensional degenerate Garnier system G(9/2) which is the extension of the first Painlevé equation PI. mailto:dr_kaneko_k@yahoo.co.jp http://dx.doi.org/10.3842/SIGMA.2014.069 2 K. Kaneko The purpose of this paper is to obtain the special solutions to the system G(1112), for which the linear monodromy { M0,M1 = S (1) 1 S (1) 2 e2πiT1 ,Mt2 ,M∞ } can be calculated explicitly. The two-dimensional degenerate Garnier system G(1112) {K1,K2, λ1, λ2, µ1, µ2, t1, t2} is de- rived as the extension of the fifth Painlevé equation by the isomonodromic deformation of the second kind, non-Fuchsian ordinary differential equation, which has three regular singularities and one irregular singularity of Poincaré rank 1 on the Riemann sphere [13, 16]: d2ψ dx2 + [ 1− α0 x + ηt1 (x− 1)2 + 2− α1 x− 1 + 1− α2 x− t2 − 1 x− λ1 − 1 x− λ2 ] dψ dx + [ ν(ν + α∞) x(x− 1) − t1K1 x(x− 1)2 − t2(t2 − 1)K2 x(x− 1)(x− t2) + λ1(λ1 − 1)µ1 x(x− 1)(x− λ1) + λ2(λ2 − 1)µ2 x(x− 1)(x− λ2) ] ψ = 0, (1.1) where K1 and K2 are Hamiltonians, λ1, λ2, µ1 and µ2 are the Garnier functions, t1 and t2 are deformation parameters and αj (j = 0, 1, 2,∞), ν ∈ C and η ∈ C× are complex parameters. The Riemann scheme of (1.1) is P x = 0 x = 1 x = t2 x = λ1 x = λ2 x =∞ 0 α0 ︷︸︸︷ 0 0 ηt1 α1 0 α2 0 2 0 2 ν ν + α∞ ;x  , α0 + α1 + α2 + α∞ = 1− 2ν. This is also derived by the confluence of two regular singularities x = t1 and x = 1 in the two-dimensional Garnier system G(11111). G(1112) has movable algebraic branch points and Hamiltonian structure expressed in rational function. We have the two-dimensional degenerate Garnier system H2(1112){H1, H2, q1, q2, p1, p2, s1, s2} by the canonical transformations: s1 = 1 t1 , s2 = t2 t2 − 1 , −t1(t2 − 1)q1 = (λ1 − 1)(λ2 − 1), (t2 − 1)2q2 = (λ1 − t2)(λ2 − t2), µi = q1p1 λi − 1 + q2p2 λi − t2 , i = 1, 2, H1 = −t21 K1 + 2∑ j=1 pj ∂qj ∂t1  , H2 = −(t2 − 1)2 K2 + 2∑ j=1 pj ∂qj ∂t2  , 2∑ j=1 (dpj ∧ dqj − dHj ∧ dsj) = 2∑ j=1 (dµj ∧ dλj − dKj ∧ dtj). H2 has the Painlevé property and the polynomial Hamiltonian structure [7, 13, 16]. We obtain the special solutions in the Hamiltonian system H2 and then inversely transform them to the solutions in the Hamiltonian system G(1112), which are substituted into the linear equa- tion (1.1). We obtain eight meromorphic solutions with generic values of complex parameters around the origin (t1, t2) = (0, 0), which we name the solutions (1), (2), . . . , (8). The calculation of the linear monodromy consists of three steps. The first step is taking the limit (t1, t2) → (0, 0) after substituting the solution into the linear equation (1.1). We call this step “the first limit”, in which the linear monodromy matrices M0 and Mt2 are calculated as the confluent linear monodromy Mt2M0. In the second step, we separate this confluent linear monodromy Mt2M0. After transforming the linear equation (1.1) by putting x = t2ξ and substituting the solution into the linear equation (1.1), we take the limit (t1, t2) → (0, 0). We Special Solutions and Linear Monodromy for G(1112) 3 call this step “the second limit”. In the third step, we transform the linear equation (1.1) by putting x− 1 = ηt1/z which keeps the irregularity at x = 1 so that we can calculate the Stokes matrices { S (1) 1 , S (1) 2 } . We call this step “the third limit”. Each of the obtained eight meromorphic solutions with generic values of complex parameters around the origin (t1, t2) = (0, 0) has the remarkable characteristics, respectively. The four solutions make the two monodromy matrices commutable and the Stokes matrices around x = 1 unity and the other four solutions make the three monodromy matrices commutable, which are summarized in Theorem 4. In Appendix A, we show the fundamental solutions and the associated monodromy matrices of Gauss hypergeometric equation and Kummer’s equation. In Appendix B, we show the Briot– Bouquet’s theorem for a system of partial differential equations in two variables and short comment on it, how it proves convergence of the eight solutions. 2 The two-dimensional degenerate Garnier system H2(1112) In this section, we write down the polynomial Hamiltonians H1, H2 and the Hamiltonian system H2(1112). • Hamiltonians H1 and H2: s2 1H1 = q2 1(q1 − s1)p2 1 + 2q2 1q2p1p2 + q1q2(q2 − s2)p2 2 − [ (α0 + α2 − 1)q2 1 + α1q1(q1 − s1) + η(q1 − s1) + ηs1q2 ] p1 − [ (α0 + α1 − 1)q1q2 + α2q1(q2 − s2)− η(s2 − 1)q2 ] p2 + ν(ν + α∞)q1, s2(s2 − 1)H2 = q2 1q2p 2 1 + 2q1q2(q2 − s2)p1p2 + [ q2(q2 − 1)(q2 − s2) + s2(s2 − 1) s1 q1q2 ] p2 2 − [ (α0 + α1 − 1)q1q2 + α2q1(q2 − s2)− η(s2 − 1)q2 ] p1 − [ (α0 − 1)q2(q2 − 1) + α1q2(q2 − s2) + α2(q2 − 1)(q2 − s2) + s2(s2 − 1) s1 (α2q1 + ηq2) ] p2 + ν(ν + α∞)q2. • Hamiltonian system H2(1112): −t1 ∂q1 ∂t1 = s1 ∂H1 ∂p1 , −t1 ∂q2 ∂t1 = s1 ∂H1 ∂p2 , t1 ∂p1 ∂t1 = s1 ∂H1 ∂q1 , t1 ∂p2 ∂t1 = s1 ∂H1 ∂p1 , ∂q1 ∂s2 = ∂H2 ∂p1 , ∂q2 ∂s2 = ∂H2 ∂p2 , −∂p1 ∂s2 = ∂H2 ∂q1 , −∂p2 ∂s2 = ∂H2 ∂q2 . Remark 1. We use t1 (= 1/s1) instead of s1 to apply the Briot–Bouquet’s theorem [3] at the origin. 3 Meromorphic solutions around the origin (t1 = 1/s1, s2) = (0, 0) In this section, we give the calculated meromorphic solutions around (t1 = 1/s1, s2) = (0, 0), which are satisfied with the Hamiltonian system H2(1112). 4 K. Kaneko When qi and pi (i = 1, 2) are meromorphic, they have at most a simple pole around (t1 = 1/s1, s2) = (0, 0). Let R = C{{t1, s2}} be the ring of convergent power series of t1 (= 1/s1) and s2 around the origin and for u1, u2, . . . , un ∈ R, let 〈u1, u2, . . . , un〉 be the ideal of R generated by u1, u2, . . . , un. Theorem 1. For generic values of the complex parameters {α0, α1, α2, α∞, ν, η}, the Hamilto- nian system H2(1112) has the following eight meromorphic solutions around (t1 = 1/s1, s2) = (0, 0): (1) q1 = η α∞ + 〈t1, s2〉, q2 = α∞ + α1 α∞ + 〈t1, s2〉, p1 = (α∞ + α1)(ν + α2) α∞(1− α∞ − α1) t1s2 + 〈t31, t21s2, s 2 2〉, p2 = −να∞ α∞ + α1 + 〈t1, s2〉, (2) q1 = −η α∞ + 〈t1, s2〉, q2 = α∞ − α1 α∞ + 〈t1, s2〉, p1 = (α∞ − α1)(ν + α2 + α∞) α∞(1 + α∞ − α1) t1s2 + 〈t31, t21s2, s 2 2〉, p2 = −(α∞ + ν) α∞ − α1 α∞ + 〈t1, s2〉, (3) q1 = −η α1 + 〈t1, s2〉, q2 = s2 ( α2 α0 + α2 + 〈t1, s2〉 ) , p1 = ν(ν + α∞) 1− α1 t1 + 〈s2, t 2 1〉, p2 = 1 s2 ( ν(ν + α∞) 1− α0 − α2 s2 + 〈t21, t1s2, s 2 2〉 ) , (4) q1 = −η α1 + 〈t1, s2〉, q2 = s2 ( α2 α2 − α0 + 〈t1, s2〉 ) , p1 = (ν + α2)(ν + α2 + α∞) 1− α1 t1 + 〈s2, t 2 1〉, p2 = 1 s2 (α0 − α2 + 〈t1, s2〉) , (5) q1 = 1 t1 ( α∞ + α0 + α2 α∞ + 〈t1, s2〉 ) , q2 = −α2 α∞ s2 + 〈t21, t1s2, s 2 2〉, p1 = t1 ( −να∞ α∞ + α0 + α2 + 〈t1, s2〉 ) , p2 = ηνα∞t1 (α1 + 2ν)(α∞ + α0 + α2) + 〈s2, t 2 1〉, (6) q1 = 1 t1 ( α∞ − α0 − α2 α∞ + 〈t1, s2〉 ) , q2 = α2 α∞ s2 + 〈t21, t1s2, s 2 2〉, p1 = t1 ( −(ν + α∞)α∞ α∞ − α0 − α2 + 〈t1, s2〉 ) , p2 = ηα∞(ν + α∞)t1 (α∞ − α0 − α2)(1− α∞ + α0 + α2) + 〈s2, t 2 1〉, (7) q1 = 1 t1 ( α∞ + α0 − α2 α∞ + 〈t1, s2〉 ) , q2 = s2 ( α2 α∞ + 〈t1, s2〉 ) , p1 = t1 ( −(ν + α2)α∞ α∞ + α0 − α2 + 〈t1, s2〉 ) , p2 = 1 s2 (α∞ + 〈t1, s2〉) , (8) q1 = 1 t1 ( α∞ − α0 + α2 α∞ + 〈t1, s2〉 ) , q2 = s2 ( −α2 α∞ + 〈t1, s2〉 ) , p1 = t1 ( −(ν + α2 + α∞)α∞ α∞ − α0 + α2 + 〈t1, s2〉 ) , p2 = 1 s2 (−α∞ + 〈t1, s2〉) . Remark 2. 1. Higher order expansions of these solutions are uniquely determined recursively by the Hamiltonian system and do not contain any other parameter than the complex parameters {α0, α1, α2, α∞, ν, η}. Special Solutions and Linear Monodromy for G(1112) 5 2. These solutions are convergent by Briot–Bouquet’s theorem (see Appendix B). 3. The values of complex parameters are generic and should be excluded the values with which the denominator of the coefficients become zero in the solutions above, that is, {α1, α∞, α1 ± α∞, α0 ± α2, α0 ± α2 ± α∞} /∈ Z. 4 The linear monodromy In this section, we calculate the linear monodromy for the solutions (1) and (5). 4.1 For the solution (1) 4.1.1 The first limit After substituting the solution (1) into the linear equation (1.1), we take the limit (t1, t2) → (0, 0). Hereafter we call this as the first limit. Then the linear equation (1.1) becomes d2ψ1 dx2 + ( 2− α0 − α2 x + 1− α1 x− 1 − 1 x− b0 ) dψ1 dx + [ ν(ν + α∞) x(x− 1) − k2 ( 1 x(x− 1) − 1 x2 ) + m2 x(x− 1)(x− b0) ] ψ1 = 0, (4.1) where b0 = α∞+α1 α∞ , k2 = ν(1 − α0 − α2 − ν) and m2 = −να1 α∞ . This is a Heun’s type equation with the Riemann scheme P  x = 0 · t2 x = 1 x = b0 x =∞ −ν −1 + α0 + α2 + ν 0 α1 0 2 ν ν + α∞ ;x  . The general solution of (4.1) is ψ1 = c1x −ν + c2x −1+α0+α2+ν(x− 1)α1 , c1, c2 ∈ C. By taking the first limit, two regular singular points become confluent as a regular singular point [2, 8]. The linear monodromy around x = 0 · t2 is obtained as a confluent one M̃t2M̃0. The linear monodromy {M̃t2M̃0, M̃1, M̃∞} of (4.1) is M̃t2M̃0 = ( e−2πiν 0 0 e2πi(α0+α2+ν) ) , M̃1 = ( 1 0 0 e2πiα1 ) , M̃∞ = ( e2πiν 0 0 e2πi(ν+α∞) ) , M̃∞M̃1M̃t2M̃0 = I2. We should separate the confluent linear monodromy M̃t2M̃0. 4.1.2 The second limit In this section, we separate the confluent linear monodromy M̃t2M̃0. After transforming the linear equation (1.1) with x = t2ξ and substituting the solution (1) into (1.1), we take the limit (t1, t2) → (0, 0). Hereafter we call this as the second limit. By taking the second limit, x = 0 and x = t2 are separated and x = 1 and x =∞ become confluent (see Remark 4). Then ψ2(ξ) = ψ(t2ξ, t1, t2) satisfies the following Gauss hypergeometric equation after taking the limit (t1, t2) −→ (0, 0), d2ψ2 dξ2 + ( 1− α0 ξ + 1− α2 ξ − 1 ) dψ2 dξ + k2 ξ(ξ − 1) ψ2 = 0 (4.2) 6 K. Kaneko with the Riemann scheme P  (x = 0) (x = t2) (x = 1 · ∞) ξ = 0 0 α0 ξ = 1 0 α2 ξ =∞ ν 1− α0 − α2 − ν ; ξ  . The linear monodromy {M0,Mt2 ,M∞M1} of (4.2) is M0 = ( 1 0 0 e2πiα0 ) , Mt2 = C−1 01 ( 1 0 0 e2πiα2 ) C01, M∞M1 = C−1 0∞ ( e2πiν 0 0 e2πi(α1+ν+α∞) ) C0∞, M∞M1Mt2M0 = I2, where C01 and C0∞ are the connection matrices of the Gauss hypergeometric function (see Appendix A, Lemma 1). The linear monodromy {M0,Mt2 ,M∞M1} of (4.2) is equivalent to {M̃t2M̃0, M̃1, M̃∞} of (4.1). We have M1 = P−1M̃1P, M∞ = P−1M̃∞P, M∞M1 = P−1M̃∞M̃1P for a matrix P ∈ GL(2,C). Therefore, PC−1 0∞ ∈ GL(2,C) is a diagonal matrix. We have M1 = C−1 0∞ ( 1 0 0 e2πiα1 ) C0∞, M∞ = C−1 0∞ ( e2πiν 0 0 e2πi(ν+α∞) ) C0∞. Remark 3. The formal solution of the linear equation (1.1) around x = 1 has the form Ψ(1) = ( I2 + ∞∑ k=1 Ψ̂ (1) k (x− 1)k ) (x− 1)T1e T x−1 , where T = diag(0, ηt1), T1 = diag(0, α1), Ψ̂ (1) k = Ψ̂ (1) k (λ1, λ2, µ1, µ2, t1, t2, αj , η). The series Ψ̂(1) = I2 + ∞∑ k=1 Ψ̂ (1) k (x− 1)k may be a divergent series since x = 1 is an irregular singularity of the Poincaré rank one. The Stokes regions S̃j around x = 1 are given by S̃j = {x ∈ C \| − ε+ (j − 1)π < arg(x− 1) < jπ + ε, \|x− 1\| < r} , where ε and r are sufficiently small. There exist holomorphic functions Ψ̃j(x) of (1.1) on S̃j such that Ψ̃j(x) ∼ Ψ̂(1) for x→ 1 and Ψj(x) = Ψ̃j(x)(x− 1)T1e T x−1 is a solution of (1.1) on S̃j . The Stokes matrix Sj is defined by Ψj+1 = ΨjSj . We notice that Ψ3 = Ψ1(xe−2πi)e2πiT1 . First by taking a limit t1 → 0 after substituting the solution (1) into the linear equation (1.1), x = 1 of (1.1) becomes a regular singular point. Since the coefficients Ψ̂ (1) k are finite in the limit t1 → 0, the formal solution Ψ(1) exists even in the limit t1 → 0. Therefore, Ψ(1)\|t1=0 = ( I2 + ∞∑ k=1 [ Ψ̂ (1) k ] t1=0 (x− 1)k ) (x− 1)T1 has a regular singularity at x = 1. Thus the Stokes matrix Sj become I2 for j = 1, 2. Then taking a limit t2 → 0 makes two regular singular points x = 1 and x =∞ confluent [2, 8]. Special Solutions and Linear Monodromy for G(1112) 7 4.1.3 The third limit In this section, we have Stokes matrices around the irregular singular point x = 1 by the transformation of the linear equation (1.1), which keeps the irregularity at x = 1. Put x− 1 = ηt1 z , then ψ3(z) = ψ(ηt1z + 1, t1, t2) satisfies the following degenerate Kummer’s equation after taking the limit (t1, t2)→ (0, 0), d2ψ3 dz2 + ( 1 + α1 z − 1 z − α1 − 1 ) dψ3 dz = 0. We have the general solution ψ3 = c3e zz−α1 + c4, c3, c4 ∈ C. This means the formal solution around the irregular singular point x = 1 (z = ∞) becomes convergent and Stokes matrices around x = 1 become I2. Therefore, we have the linear monodromy for the solution (1) explicitly. Theorem 2. For the solution (1), the linear monodromy of (1.1) is M0 = ( 1 0 0 e2πiα0 ) , Mt2 = C−1 01 ( 1 0 0 e2πiα2 ) C01, M1 = S (1) 1 S (1) 2 C−1 0∞ ( 1 0 0 e2πiα1 ) C0∞, S (1) 1 = S (1) 2 = I2, M∞ = C−1 0∞ ( e2πiν 0 0 e2πi(ν+α∞) ) C0∞, C01 =  Γ(1−α0)Γ(α2) Γ(1−α0−ν)Γ(α2+ν) Γ(1+α0)Γ(α2) Γ(1−ν)Γ(α0+α2+ν) Γ(1−α0)Γ(−α2) Γ(ν)Γ(1−α0−α2−ν) Γ(1+α0)Γ(−α2) Γ(α0+ν)Γ(1−α2−ν)  , C0∞ =  eπiνΓ(1−α0)Γ(1−α0−α2−2ν) Γ(1−α0−α2−ν)Γ(1−α0−ν) eπi(α0+ν)Γ(1+α0)Γ(1−α0−α2−2ν) Γ(1−ν)Γ(1−α2−ν) eπi(1−α0−α2−ν)Γ(1−α0)Γ(α0+α2+2ν−1) Γ(ν)Γ(α2+ν) eπi(1−α2−ν)Γ(1+α0)Γ(α0+α2+2ν−1) Γ(α0+α2+ν)Γ(α0+ν)  , M∞M1Mt2M0 = I2, for which [M1,M∞] = 0 holds. 4.2 For the solutions (2), (3) and (4) Also for the solutions (2), (3) and (4), we have the similar results, which are summarized as follows. • For the solution (2): M0 = ( 1 0 0 e2πiα0 ) , Mt2 = C (2)−1 01 ( 1 0 0 e2πiα2 ) C (2) 01 , M1 = S (1) 1 S (1) 2 C (2)−1 0∞ ( e2πiα1 0 0 1 ) C (2) 0∞, S (1) 1 = S (1) 2 = I2, M∞ = C (2)−1 0∞ ( e2πiν 0 0 e2πi(ν+α∞) ) C (2) 0∞, C (2) 01 = C01(ν + α1, ν + α∞, 1− α0), C (2) 0∞ = C0∞(ν + α1, ν + α∞, 1− α0), M∞M1Mt2M0 = I2, for which [M1,M∞] = 0 holds. 8 K. Kaneko • For the solution (3): M0 = ( 1 0 0 e2πiα0 ) , Mt2 = ( 1 0 0 e2πiα2 ) , M1 = S (1) 1 S (1) 2 C (3)−1 01 ( 1 0 0 e2πiα1 ) C (3) 01 , S (1) 1 = S (1) 2 = I2, M∞ = C (3)−1 0∞ ( e2πiν 0 0 e2πi(ν+α∞) ) C (3) 0∞, C (3) 01 = C01(ν, ν + α∞, 1− α0 − α2), C (3) 0∞ = C0∞(ν, ν + α∞, 1− α0 − α2), M∞M1Mt2M0 = I2, for which [M0,Mt2 ] = 0 holds. • For the solution (4): M0 = ( 1 0 0 e2πiα0 ) , Mt2 = ( 1 0 0 e2πiα2 ) , M1 = S (1) 1 S (1) 2 C (4)−1 01 ( 1 0 0 e2πiα1 ) C (4) 01 , S (1) 1 = S (1) 2 = I2, M∞ = C (4)−1 0∞ ( e2πiν 0 0 e2πi(ν+α∞) ) C (4) 0∞, C (4) 01 = C01(ν, ν + α∞, 1− α2), C (4) 0∞ = C0∞(ν, ν + α∞, 1− α2), M∞M1Mt2M0 = I2, for which [M0,Mt2 ] = 0 holds. 4.3 For the solution (5) In this section, we calculate the linear monodromy for the solution (5) by the similar way to Subsection 4.1. 4.3.1 The first limit After substituting the solution (5) into the linear equation (1.1), we take the limit (t1, t2) → (0, 0). At first we take a limit t1 → 0 keeping t2 as a non-zero constant, then we take t2 → 0. Then the linear equation (1.1) becomes d2ψ1 dx2 + ( 1− α0 − α2 x + 2− α1 x− 1 − 1 x− b0 ) dψ1 dx + [ ν(ν + α∞) x(x− 1) − k1 x(x− 1)2 + m1 x(x− 1)(x− b0) ] ψ1 = 0, (4.3) where b0 = α0+α2 −α∞ , k1 = ν(ν + α1 − 1) and m1 = ν(α0+α2) α∞ . This is a Heun’s type equation with the Riemann scheme P x = 0 · t2 x = 1 x = b0 x =∞ 0 α0 + α2 −ν ν + α1 − 1 0 2 ν ν + α∞ ;x  . The general solution of (4.3) is ψ1 = c5(x− 1)−ν + c6x α0+α2(x− 1)ν+α1−1, c5, c6 ∈ C. Special Solutions and Linear Monodromy for G(1112) 9 The linear monodromy {M̃t2M̃0, M̃1, M̃∞} of (4.3) is M̃t2M̃0 = ( 1 0 0 e2πi(α0+α2) ) , M̃1 = ( e−2πiν 0 0 e2πi(ν+α1) ) , M̃∞ = ( e2πiν 0 0 e2πi(ν+α∞) ) , M̃∞M̃1M̃t2M̃0 = I2. We should separate the confluent linear monodromy M̃t2M̃0. 4.3.2 The second limit In this section, we separate the confluent linear monodromy M̃t2M̃0. After transforming the linear equation (1.1) with x = t2ξ and substituting the solution (5) into (1.1), we take the limit (t1, t2)→ (0, 0) as the same as the first limit. By taking the second limit, x = 0 and x = t2 are separated and x = 1 and x = ∞ become confluent. Then ψ2(ξ) = ψ(t2ξ, t1, t2) satisfies the following degenerate Heun’s equation after taking the limit (t1, t2)→ (0, 0), d2ψ2 dξ2 + ( 1− α0 ξ + 1− α2 ξ − 1 − 1 ξ − ξλ2 ) dψ2 dξ = 0, (4.4) where ξλ2 = α0 α0+α2 . We have the general solution ψ2 = c7 + c8ξ α0(ξ − 1)α2 , c7, c8 ∈ C. The linear monodromy {M0,Mt2 ,M∞M1} of (4.4) is M0 = ( 1 0 0 e2πiα0 ) , Mt2 = ( 1 0 0 e2πiα2 ) , M∞M1 = ( 1 0 0 e−2πi(α0+α2) ) , M∞M1Mt2M0 = I2. 4.3.3 The third limit In this section, we calculate the Stokes matrices around the irregular singular point x = 1 by the transformation of the linear equation (1.1), which keeps the irregularity at x = 1. Put x− 1 = ηt1 z , ψ = ( η−1z )ν ψ3(z, t1, t2), then ψ3(z, t1, t2) = (η−1z)−νψ(ηt1z + 1, t1, t2) satisfies the following Kummer’s confluent hyper- geometric equation after taking the limit (t1, t2) −→ (0, 0), d2ψ3 dz2 + ( 2ν + α1 z − 1 ) dψ3 dz − ν z ψ3 = 0. A system of the fundamental solutions is( 1F1(ν, 2ν + α1; z), z1−2ν−α1 1F1(1− ν − α1, 2− 2ν − α1; z) ) . We have the Stokes matrices S (1) 1 and S (1) 2 around the irregular singular point x = 1 (z = ∞) (see Appendix A, Lemma 2). Summarizing the calculations above, we have the following theorem: 10 K. Kaneko Theorem 3. For the solution (5), the linear monodromy of (1.1) is M0 = C(5) ( 1 0 0 e2πiα0 ) C(5)−1, Mt2 = C(5) ( 1 0 0 e2πiα2 ) C(5)−1, M1 = S (1) 1 S (1) 2 e2πiT1 , e2πiT1 = ( 1 0 0 e2πiα1 ) , S (1) 1 = ( 1 0 −2πieπiα1 Γ(ν)Γ(1−ν−α1) 1 ) , S (1) 2 = ( 1 −2πie−2πiα1 Γ(1−ν)Γ(ν+α1) 0 1 ) , M∞ = C(5) ( e2πiν 0 0 e2πi(ν+α∞) ) C(5)−1, C(5) = C(ν, 2ν + α1) = Γ(2ν+α1) Γ(ν+α1) e πiν Γ(2−2ν−α1) Γ(1−ν) eπi(1−ν−α1) Γ(2ν+α1) Γ(ν) Γ(2−2ν−α1) Γ(1−ν−α1)  , M∞Mt2M0S (1) 1 S (1) 2 e2πiT1 = I2, for which [M0,Mt2 ] = 0, [M0,M∞] = 0 and [Mt2 ,M∞] = 0 hold and C(5) is the connection matrix of Kummer’s confluent hypergeometric function (see Appendix A, Lemma 2). Remark 4. If we take a limit (t1, t2)→ (0, 0) along a curve s2 ∼ At21(A ∈ C×), the limit of the last term in (1.1) λ2(λ2 − 1)µ2 x(x− 1)(x− λ2) is not zero. In our calculation, we take a special path from (t1, t2) to (0, 0), such that the numerator of the above term tends to zero. Therefore we obtain different limit equations when we choose different paths for the first and the second limit equations. It may be a contradiction. But the whole of linear monodromy is the same even though some limit equations are different, since we have the Riemann–Hilbert correspondence. In our case, the third limit is the same for any path from (t1, t2) to (0, 0), which is the main part of the linear monodromy for the solution (5). 4.4 For the solutions (6), (7) and (8) We can determine the linear monodromy for the other solutions (6), (7) and (8), which are summarized as follows. • For the solution (6): M0 = C(6) ( 1 0 0 e2πiα0 ) C(6)−1, Mt2 = C(6) ( 1 0 0 e2πiα2 ) C(6)−1, M∞ = C(6) ( e2πi(ν+α∞) 0 0 e2πiν ) C(6)−1, M1 = S (1) 1 S (1) 2 e2πiT1 , e2πiT1 = ( 1 0 0 e2πiα1 ) , C(6) = C(ν + α∞, 2ν + 2α∞ + α1), S (1) 1 = S (∞) 1 (ν + α∞, 2ν + 2α∞ + α1), S (1) 2 = S (∞) 2 (ν + α∞, 2ν + 2α∞ + α1), M∞Mt2M0S (1) 1 S (1) 2 e2πiT1 = I2, for which [M0,Mt2 ] = 0, [M0,M∞] = 0 and [Mt2 ,M∞] = 0 hold. • For the solution (7): M0 = C(7) ( 1 0 0 e2πiα0 ) C(7)−1, Mt2 = C(7) ( e2πiα2 0 0 1 ) C(7)−1, Special Solutions and Linear Monodromy for G(1112) 11 M∞ = C(7) ( e2πiν 0 0 e2πi(ν+α∞) ) C(7)−1, M1 = S (1) 1 S (1) 2 e2πiT1 , e2πiT1 = ( 1 0 0 e2πiα1 ) , C(7) = C(ν + α2, 2ν + 2α2 + α1), S (1) 1 = S∞1 (ν + α2, 2ν + 2α2 + α1), S (1) 2 = S∞2 (ν + α2, 2ν + 2α2 + α1), M∞Mt2M0S (1) 1 S (1) 2 e2πiT1 = I2, for which [M0,Mt2 ] = 0, [M0,M∞] = 0 and [Mt2 ,M∞] = 0 hold. • For the solution (8): M0 = C(8) ( e2πiα0 0 0 1 ) = C(8)−1, Mt2 = C(8) ( 1 0 0 e2πiα2 ) C(8)−1, M∞ = C(8) ( e2πiν 0 0 e2πi(ν+α∞) ) C(8)−1, M1 = S (1) 1 S (1) 2 e2πiT1 , e2πiT1 = ( 1 0 0 e2πiα1 ) , C(8) = C(ν + α2 + α∞, 2ν + 2α2 + 2α∞ + α1), S (1) 1 = S∞1 (ν + α2 + α∞, 2ν + 2α2 + 2α∞ + α1), S (1) 2 = S∞2 (ν + α2 + α∞, 2ν + 2α2 + 2α∞ + α1), M∞Mt2M0S (1) 1 S (1) 2 e2πiT1 = I2, for which [M0,M∞] = 0, [M0,Mt2 ] = 0 and [Mt2 ,M∞] = 0 hold. Summarizing the all calculations above, we have the following theorem: Theorem 4. The eight meromorphic solutions around the origin of the two-dimensional degene- rate Garnier system G2(1112) have the following characteristics: • For the solution (1) and (2), [M1,M∞] = 0 and S (1) 1 = S (1) 2 = I2 hold. • For the solution (3) and (4), [M0,Mt2 ] = 0 and S (1) 1 = S (1) 2 = I2 hold. • For the solution (5), (6), (7) and (8), [M0,M∞] = 0, [M0,Mt2 ] = 0 and [Mt2 ,M∞] = 0 hold. Remark 5. For the linear monodromy data { M0,M∞, S (1) 1 , S (1) 2 , e2πiT1 } of the fifth Painlevé equation, there are three meromorphic solutions around the origin: two solutions such that [M0,M∞] = 0 and one solution such that S (1) 1 = S (1) 2 = I2 [11]. A Gauss hypergeometric equation and Kummer’s equation In this appendix, we show the fundamental solutions and the associated monodromy matrices of Gauss hypergeometric equation [15] and Kummer’s equation [6]. A.1 Gauss hypergeometric equation The Gauss hypergeometric equation is x(1− x) d2ψ dx2 + (γ − (α+ β + 1)x) dψ dx − αβψ = 0. (A.1) 12 K. Kaneko The Riemann scheme of (A.1) is P x = 0 x = 1 x =∞ 0 1− γ 0 γ − α− β α β ;x  , We list fundamental systems of solutions for (A.1) around x = 0, 1,∞. • Around x = 0: ψ(0) = ( 2F1(α, β, γ;x) x1−γ 2F1(α+ 1− γ, β + 1− γ, 2− γ;x) ) . • Around x = 1: ψ(1) = ( ψ (1) 1 ψ (1) 2 ) , ψ (1) 1 = 2F1(α, β, α+ β − γ + 1; 1− x), ψ (1) 2 = (1− x)γ−α−β2F1(γ − α, γ − β, γ + 1− α− β; 1− x). • Around x =∞: ψ(∞) = ( ψ (∞) 1 ψ (∞) 2 ) , ψ (∞) 1 = x−α2F1 ( α, α− γ + 1, α+ 1− β;x−1 ) , ψ (∞) 2 = x−β2F1 ( β, β − γ + 1, β + 1− α;x−1 ) . The associated monodromy matrices Mj (j = 0, 1,∞) are as follows M0 = ( 1 0 0 e−2πiγ ) , M1 = C01(α, β, γ)−1 ( 1 0 0 e2πi(γ−α−β) ) C01(α, β, γ), M∞ = C0∞(α, β, γ)−1 ( e2πiα 0 0 e2πiβ ) C0∞(α, β, γ), M∞M1M0 = I2, where C01(α, β, γ) and C0∞(α, β, γ) are connection matrices which are shown in the following lemma. Lemma 1. The Gauss hypergeometric function which is the solution of (A.1) has the following connection matrices between fundamental solutions around two singularities: ψ(i) = ψ(j)Cij(α, β, γ), i, j ∈ {0, 1,∞}, where ψ(ν) (ν ∈ {0, 1,∞}) is the fundamental solution around the singularity ν and Cij(α, β, γ) are the connection matrices which are shown as follows C01(α, β, γ) = Γ(γ)Γ(γ−α−β) Γ(γ−α)Γ(γ−β) Γ(2−γ)Γ(γ−α−β) Γ(1−α)Γ(1−β) Γ(γ)Γ(α+β−γ) Γ(α)Γ(β) Γ(2−γ)Γ(α+β−γ) Γ(1+α−γ)Γ(1+β−γ)  , C0∞(α, β, γ) =  eαπiΓ(γ)Γ(β−α) Γ(β)Γ(γ−α) e(α−γ+1)πiΓ(2−γ)Γ(β−α) Γ(1−α)Γ(1−γ+β) eβπiΓ(γ)Γ(α−β) Γ(α)Γ(γ−β) e(β−γ+1)πiΓ(2−γ)Γ(α−β) Γ(1−β)Γ(1−γ+α)  , C∞1(α, β, γ) = ( Γ(1+α−β)Γ(γ−α−β) Γ(γ−β)Γ(1−β) Γ(1+β−α)Γ(γ−α−β) Γ(γ−α)Γ(1−α) e(γ−α−β)πiΓ(1+α−β)Γ(α+β−γ) Γ(1+α−γ)Γ(α) e(γ−α−β)πiΓ(1+β−α)Γ(α+β−γ) Γ(1+β−γ)Γ(β) ) . Special Solutions and Linear Monodromy for G(1112) 13 A.2 Kummer’s equation The Kummer’s equation is d2φ dx2 + (γ x − 1 ) dφ dx − α x φ = 0. (A.2) The Riemann scheme of (A.2) is P x = 0 x =∞ 0 1− γ ︷︸︸︷ 0 1 α γ − α ;x  . Fundamental systems of solutions for (A.2) is given by φ(0) = ( 1F1(α, γ;x) x1−γ 1F1(α+ 1− γ, 2− γ;x) ) . Asymtotic solutions around x =∞ is given by φ (∞) 1 ( e−πix ) ∼ x−α ∞∑ k=0 (−1)k(α)k(α+ 1− γ)k k!(e−πix)k , φ (∞) 2 (x) ∼ exxα−γ ∞∑ k=0 (γ − α)k(1− α)k k!xk . The associated monodromy matrices are as follows M0 = C(α, γ) ( 1 0 0 e−2πiγ ) C(α, γ)−1, M∞ = S (∞) 1 (α, γ)S (∞) 2 (α, γ)e2πiT∞ , e2πiT∞ = ( e2πiα 0 0 e2πi(γ−α) ) , M0M∞ = I2, where C(α, γ), S (∞) 1 (α, γ) and S (∞) 2 (α, γ) are connection matrix and Stokes matrices respectively which are shown in the following lemma. Lemma 2. The Kummer’s confluent hypergeometric function which is the solution of (A.2) has the following connection matrix between fundamental solutions around x = 0 and x = ∞ and Stokes matrices around x =∞: • Connection matrix:( 1F1(α, γ;x) x1−γ 1F1(α+ 1− γ, 2− γ;x) ) = ( φ (∞) 1 (e−πix) φ (∞) 2 (x) ) C(α, γ), where φ (∞) i (i ∈ {1, 2}) is the fundamental solutions around the singularity x = ∞ and C(α, γ) is the connection matrix which is shown as follows C(α, γ) = ( Γ(γ)eαπi Γ(γ−α) Γ(2−γ)eπi(1+α−γ) Γ(1−α) Γ(γ) Γ(α) Γ(2−γ) Γ(1+α−γ) ) . • Stokes matrices: S (∞) 1 (α, γ) = ( 1 0 −2πieπi(γ−2α) Γ(α)Γ(1+α−γ) 1 ) , S (∞) 2 (α, γ) = ( 1 −2πieπi(4α−2γ) Γ(1−α)Γ(γ−α) 0 1 ) . 14 K. Kaneko B Briot–Bouquet’s theorem for a system of partial differential equations in two variables Briot and Bouquet [3] showed that existence of a holomorphic solution for a special type of nonlinear ordinary differential equations. In this section we explain the Briot–Bouquet’s type theorem for a system of partial differential equations in two variables following [5]. B.1 Briot–Bouquet’s theorem Briot and Bouquet studied a nonlinear ordinary differential equation x dz dx = h(z, x), z = (z1, . . . , zn) (B.1) for h(0, 0) = 0. They have shown that if the eigenvalues of the Jacobi matrix (∂h∂z (0, 0)) are not positive integers, then (B.1) has a convergent holomorphic solution. R. Gerard and Y. Sibuya [5] studied the Briot–Bouquet’s type theorem for a system of partial differential equations in two variables. They have shown that a formal solution will be convergent: Lemma 3. Assuming that h1 and h2 are holomorphic functions of z, x1 and x2 and z(0, 0) = 0, h1(0, 0, 0) = h2(0, 0, 0) = 0. If the simultaneous equations x1 ∂z ∂x1 = h1(z, x1, x2), x2 ∂z ∂x2 = h2(z, x1, x2) have the formal solutions around (x1, x2) = (0, 0) expressed in power series of x1 and x2, they are convergent. B.2 Convergence of the solutions Solutions (1) and (2) are convergent by Lemma 3. For the solutions with a pole, for example, solutions (7) and (8), we put q1 = Q1 t1 , p1 = t1P1, q2 = s2Q2, p2 = P2 s2 , where Q1, P1, Q2 and P2 are holomorphic functions of t1 and s2 near (t1, s2) = (0, 0). Substi- tuting these into the Hamiltonian system H2, it becomes all Briot–Bouquet’s type differential equations with respect to Q1, P1, Q2 and P2. Acknowledgements The author wishes to thank Professor Y. Ohyama for his constant guidance and useful sugges- tions to complete this work. The author also gives thanks to the anonymous referees for their relevant contributions to improve this paper. This work was supported by JSPS KAKENHI Grant Number 22540237 and the Mitsubishi Foundation. References [1] Appell P., Sur les polynômes se rattachant à l’équation différentielle y′′ = 6y2 + x, Bull. Soc. Math. France 45 (1917), 150–153. [2] Bolibruch A.A., On isomonodromic confluences of Fuchsian singularities, Proc. Steklov Inst. Math. 221 (1998), 117–132. Special Solutions and Linear Monodromy for G(1112) 15 [3] Briot C., Bouquet J.C., Recherches sur les propriétés des fonctions définies par des équations différentielles, J. de l’Ecole Polytechnique 21 (1856), 133–198. [4] Garnier R., Étude de l’intégrale générale de l’équation VI de M. Painlevé dans le voisinage de ses singularités transcendantes, Ann. Sci. École Norm. Sup. (3) 34 (1917), 239–353. [5] Gérard R., Sibuya Y., Étude de certains systèmes de Pfaff au voisinage d’une singularité, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), A57–A60. [6] Heading J., The Stokes phenomenon and the Whittaker function, J. London Math. Soc. 37 (1962), 195–208. [7] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991. [8] Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137–1161. [9] Kaneko K., A new solution of the fourth Painlevé equation with a solvable monodromy, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 75–79. [10] Kaneko K., Local expansion of Painlevé VI transcendents around a fixed singularity, J. Math. Phys. 50 (2009), 013531, 24 pages. [11] Kaneko K., Ohyama Y., Fifth Painlevé transcendents which are analytic at the origin, Funkcial. Ekvac. 50 (2007), 187–212. [12] Kaneko K., Ohyama Y., Meromorphic Painlevé transcendents at a fixed singularity, Math. Nachr. 286 (2013), 861–875. [13] Kimura H., The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian struc- ture, Ann. Mat. Pura Appl. (4) 155 (1989), 25–74. [14] Kitaev A.V., Symmetric solutions for the first and the second Painlevé equation, J. Math. Sci. 73 (1995), 494–499. [15] Kohno M., Global analysis in linear differential equations, Mathematics and its Applications, Vol. 471, Kluwer Academic Publishers, Dordrecht, 1999. [16] Okamoto K., Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), 575–618. http://dx.doi.org/10.1112/jlms/s1-37.1.195 http://dx.doi.org/10.1007/978-3-322-90163-7 http://dx.doi.org/10.2977/prims/1195183300 http://dx.doi.org/10.2977/prims/1195183300 http://dx.doi.org/10.3792/pjaa.81.75 http://dx.doi.org/10.3792/pjaa.81.75 http://dx.doi.org/10.1063/1.3052083 http://dx.doi.org/10.1619/fesi.50.187 http://dx.doi.org/10.1002/mana.200810241 http://dx.doi.org/10.1007/BF01765933 http://dx.doi.org/10.1007/BF02364571 http://dx.doi.org/10.1007/978-94-011-4605-0 1 Introduction 2 The two-dimensional degenerate Garnier system H2(1112) 3 Meromorphic solutions around the origin (t1=1/s1,s2)=(0,0) 4 The linear monodromy 4.1 For the solution (1) 4.1.1 The first limit 4.1.2 The second limit 4.1.3 The third limit 4.2 For the solutions (2), (3) and (4) 4.3 For the solution (5) 4.3.1 The first limit 4.3.2 The second limit 4.3.3 The third limit 4.4 For the solutions (6), (7) and (8) A Gauss hypergeometric equation and Kummer's equation A.1 Gauss hypergeometric equation A.2 Kummer's equation B Briot–Bouquet's theorem for a system of partial differential equations in two variables B.1 Briot–Bouquet's theorem B.2 Convergence of the solutions References

Special Solutions and Linear Monodromy for the Two-Dimensional Degenerate Garnier System G(1112)

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