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.


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 Painlevé equations [13]: (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 P VI degenerates step by step to the twodimensional degenerate Garnier system G(9/2) which is the extension of the first Painlevé equation P I .
The purpose of this paper is to obtain the special solutions to the system G(1112), for which the linear monodromy M 0 , M 1 = S (1) 1 S (1) 2 e 2πiT 1 , M t 2 , M ∞ can be calculated explicitly. The two-dimensional degenerate Garnier system G(1112) {K 1 , K 2 , λ 1 , λ 2 , µ 1 , µ 2 , t 1 , t 2 } is derived 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]: where K 1 and K 2 are Hamiltonians, λ 1 , λ 2 , µ 1 and µ 2 are the Garnier functions, t 1 and t 2 are deformation parameters and α j (j = 0, 1, 2, ∞), ν ∈ C and η ∈ C × are complex parameters. The Riemann scheme of (1.1) is This is also derived by the confluence of two regular singularities x = t 1 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 H 2 (1112){H 1 , H 2 , q 1 , q 2 , p 1 , p 2 , s 1 , s 2 } by the canonical transformations: H 2 has the Painlevé property and the polynomial Hamiltonian structure [7,13,16]. We obtain the special solutions in the Hamiltonian system H 2 and then inversely transform them to the solutions in the Hamiltonian system G(1112), which are substituted into the linear equation (1.1). We obtain eight meromorphic solutions with generic values of complex parameters around the origin (t 1 , t 2 ) = (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 (t 1 , t 2 ) → (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 M 0 and M t 2 are calculated as the confluent linear monodromy M t 2 M 0 . In the second step, we separate this confluent linear monodromy M t 2 M 0 . After transforming the linear equation (1.1) by putting x = t 2 ξ and substituting the solution into the linear equation (1.1), we take the limit (t 1 , t 2 ) → (0, 0). We call this step "the second limit". In the third step, we transform the linear equation (1.1) by putting x − 1 = ηt 1 /z which keeps the irregularity at x = 1 so that we can calculate the Stokes matrices S . We call this step "the third limit". Each of the obtained eight meromorphic solutions with generic values of complex parameters around the origin (t 1 , t 2 ) = (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 H 2 (1112) In this section, we write down the polynomial Hamiltonians H 1 , H 2 and the Hamiltonian system H 2 (1112).
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 {α 0 , α 1 , α 2 , α ∞ , ν, η}.
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,

The linear monodromy
In this section, we calculate the linear monodromy for the solutions (1) and (5).

The f irst limit
After substituting the solution (1) into the linear equation (1.1), we take the limit (t 1 , t 2 ) → (0, 0). Hereafter we call this as the first limit. Then the linear equation (1.1) becomes α∞ . This is a Heun's type equation with the Riemann scheme The general solution of (4.1) is By taking the first limit, two regular singular points become confluent as a regular singular point [2,8]. The linear monodromy around x = 0 · t 2 is obtained as a confluent one M t 2 M 0 . The linear monodromy We should separate the confluent linear monodromy M t 2 M 0 .
The Stokes regions S j around x = 1 are given by where ε and r are sufficiently small. There exist holomorphic functions Ψ j (x) of (1.1) on S j such that is a solution of (1.1) on S j . The Stokes matrix S j is defined by We notice that Ψ 3 = Ψ 1 (xe −2πi )e 2πiT 1 . First by taking a limit t 1 → 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 Ψ (1) k are finite in the limit t 1 → 0, the formal solution Ψ (1) exists even in the limit t 1 → 0. Therefore, has a regular singularity at x = 1. Thus the Stokes matrix S j become I 2 for j = 1, 2. Then taking a limit t 2 → 0 makes two regular singular points x = 1 and x = ∞ confluent [2,8].

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 then ψ 3 (z) = ψ( ηt 1 z + 1, t 1 , t 2 ) satisfies the following degenerate Kummer's equation after taking the limit (t 1 , t 2 ) → (0, 0), We have the general solution This means the formal solution around the irregular singular point x = 1 (z = ∞) becomes convergent and Stokes matrices around x = 1 become I 2 . Therefore, we have the linear monodromy for the solution (1) explicitly. (1), the linear monodromy of (1.1) is

For the solution (5)
In this section, we calculate the linear monodromy for the solution (5) by the similar way to Subsection 4.1.

The f irst limit
After substituting the solution (5) into the linear equation (1.1), we take the limit (t 1 , t 2 ) → (0, 0). At first we take a limit t 1 → 0 keeping t 2 as a non-zero constant, then we take t 2 → 0. Then the linear equation (1.1) becomes α∞ . This is a Heun's type equation with the Riemann scheme The general solution of (4.3) is The linear monodromy We should separate the confluent linear monodromy M t 2 M 0 .

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 then ψ 3 (z, t 1 , t 2 ) = (η −1 z) −ν ψ( ηt 1 z + 1, t 1 , t 2 ) satisfies the following Kummer's confluent hypergeometric equation after taking the limit (t 1 , t 2 ) −→ (0, 0), A system of the fundamental solutions is We have the Stokes matrices S 1 and S 2 around the irregular singular point x = 1 (z = ∞) (see Appendix A, Lemma 2).
Summarizing the calculations above, we have the following theorem: Theorem 3. For the solution (5), the linear monodromy of (1.1) is Remark 4. If we take a limit (t 1 , t 2 ) → (0, 0) along a curve s 2 ∼ At 2 1 (A ∈ C × ), the limit of the last term in (1.1) is not zero. In our calculation, we take a special path from (t 1 , t 2 ) 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 (t 1 , t 2 ) to (0, 0), which is the main part of the linear monodromy for the solution (5). (6), (7) and (8) We can determine the linear monodromy for the other solutions (6), (7) and (8), which are summarized as follows.

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 The Riemann scheme of (A.1) is We list fundamental systems of solutions for (A.1) around x = 0, 1, ∞.
Lemma 1. The Gauss hypergeometric function which is the solution of (A.1) has the following connection matrices between fundamental solutions around two singularities: where ψ (ν) (ν ∈ {0, 1, ∞}) is the fundamental solution around the singularity ν and C ij (α, β, γ) are the connection matrices which are shown as follows .