On the connection formulas of the third Painlevé transcendent

We consider the connection problem for the sine-Gordon PIII equation 
$u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general third 
Painleve transcendents. The connection 
formulas are derived by the method of "uniform asymptotics" 
proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. 
Anal., 1998).

1. Introduction. In this paper we consider the nonlinear differential equation u xx + 1 x u x + sin u = 0, (1.1) which is known as the sine-Gordon PIII equation. It is deduced from the third Painlevé equation (PIII) via the transformation when α = β = 0 and γ = −δ = − 1 4 . In view of its simplicity and its similarity to the sine-Gordon equation, (1.1) is the most commonly studied equation among all general PIII equations.
It has been known for some time that for any given real numbers r and s, there exists a solution of equation (1.1) which satisfies u(x) = r ln x + s + O(x 2 ) as x → 0; (1.4) see [4, p.460]. It is also known that with the given asymptotic behavior at the origin, the solution of equation (1.1) behaves like u(x) = 2πk + αx −1/2 cos x − α 2 16 ln x + β + o(x −1/2 ) as x → ∞, (1.5) where α, β are real constants and k is an integer; see [4, p.470]. Furthermore, the asymptotic formula (1.5) can be differentiated with respect to x. The relationship between the parameters α, β in (1.5) and the parameters r, s in (1.4) is provided by the connection formulas and n is an integer. Moreover, the integer k in (1.5) is given by k = integer part of 1 2π π − s + 3r ln 2 − 4 arg Γ 1 2 + ir 4 . (1.9) The above connection formulas can be established by using the method of isomonodromic deformation; see [4, p.446-448]. For convenience, we recall some of the relevant facts from the isomonodromy formalism for the sine-Gordon PIII reduction. The Lax pair of the sine-Gordon PIII reduction is the system of linear ordinary differential equations (1.11) where σ 1 , σ 2 and σ 3 are the Pauli matrices and u, u x are complex parameters. It is easily verified that the Lax pair satisfies the compatibility condition ∂ 2 Ψ ∂x∂λ = ∂ 2 Ψ ∂λ∂x if and only if u is a solution of the sine-Gordon PIII equation. In the neighborhood of infinity and of the origin , we define, respectively, the canonical sectors Ω (∞) = {λ : −π < arg λ < π, |λ| > r 1 } (1.13) and Ω (0) = {λ : −π < arg λ < π, |λ| < r 2 }, (1.14) where r 1 and r 2 are positive constants with r 1 < r 2 , and we assume that x ∈ R.
Since the Lax pair (1.10) − (1.11) has two irregular singular points of the first rank, one at λ = 0 and the other at λ = ∞, we may suppose that the system (1.10)−(1.11) has two fundamental solutions Ψ (∞) (x, λ) and Ψ (0) (x, λ) in the respective sectors Ω (∞) and Ω (0) with the prescribed asymptotic behaviors as |λ| → ∞ in Ω (∞) , and (1.17) Since Ψ (∞) (x, λ) and Ψ (0) (x, λ) are both fundamental solutions, there must be a connection matrix Q(x) independent of λ but, in general, dependent on x such that (1.18) The matrix Q is also known as the monodromy data. Differentiating both sides of equation (1.18) with respect to x, and making use of the fact that both Ψ (∞) and Ψ (0) satisfy (1.18), it is easily found that Q ′ (x) = 0; i.e., Q(x) is a constant matrix, which is equivalent to saying that the problem is isomonodromic in x.
The connection problem (1.6) − (1.9) has previously been solved by computing, for both large and small values of x, the asymptotic behaviors of the fundamental solutions as λ → ∞ and also as λ → 0 (in the canonical sectors Ω (∞) and Ω (0) , respectively). In an overlapping region, one then uses the WKB asymptotics to do the matching; see [4, pp.465-468]. This procedure is complicated, and is difficult to make rigorous, a comment made in [2, p.245].
In this paper we shall solve the connection problem for equation (1.1) by using the method of "uniform asymptotics" proposed in [2] and applied to the second Painlevé (PII) equation. Although the basic idea is taken from [2], there are some differences in the details. For instance, the number of irregular singular points in the Lax pair for the sine-Gordon PIII equation is two, while the PII equation has only one irregular singular point. If we only calculate the Stokes multipliers as was done in [2], we will miss some of the important information for the monodromy data we need. This inspired us to consider the connection matrix Q which, as we shall see, involves two parameters, instead of the Stokes multiplier which involves only one parameter. The difficulty in extending the techniques for PII to other transcendents is also acknowledged by the authors of [2, p.244, lines 21-22].
The material in this paper is arranged as follows: In Section 2, we express the solutions of the Lax pair (1.10) − (1.11) in terms of the parabolic cylinder functions D ν (z) and D −ν−1 (iz). Based on the asymptotic behavior of these latter functions, we work out in Section 3 the asymptotic formulas of the fundamental solutions Ψ (∞) (x, λ) and Ψ (0) (x, λ) as x → ∞, which hold uniformly with respect to λ. Once this is done, there is no further rigorous analysis required. All we need to do is to use the known asymptotic formula of u(x) given in (1.5) to obtain the monodromy data; this is also done in Section 3. The last section is devoted to deriving corresponding results as x → 0. Since the procedure of getting asymptotic formulas for the fundamental solutions and the monodromy data as x → 0, as given in [4, pp.462-463], is very simple and rigorous, we will just follow that method and correct several mistakes (typographic errors) in the calculations presented there.

R. WONG AND H. Y. ZHANG
Comparing the coefficients of e −iξη and e iξη on both sides of the above two asymptotic equations, we have which in turn gives q 11 = e νπi and q 21 = u x 2 By the definition of q in (2.2), we have where we have also made use of (2.39). In the same manner, since ψ from which it follows that π − β + 2nπ, (3.48) where n in an integer. Since |Γ(iy)| 2 = π/(y sinh πy), we conclude and 4. Monodromy data as x → 0. As explained in Section 1, we shall calculate the monodromy data as x → 0 by following the method given in [4] and [6]. First, we take 1 ≪ r 1 < r 2 ≪ x −2 in (1.13) and (1.14) so that the two sectors Ω (∞) and Ω (0) are overlapping. Again, we prescribe the asymptotic behavior of the two fundamental solutions Ψ (∞) and Ψ (0) in these sectors by (1.15) and (1.16), respectively. Since u(x) → ∞ as x → 0 + by (1.4), we can choose a sequence of values of u satisfying cos u = 1 and sin u = 0. Furthermore, since the connection matrix Q is independent of λ, we may without loss of generality restrict λ to be real and positive (i.e., arg λ = 0). Recall the Bessel equation which has the two linearly independent solutions H two linearly independent solutions of which are √ zH (1) ν (z) and √ zH (2) ν (z). We first investigate the asymptotic behavior of the solution Ψ (∞) . Let η 1 = x 2 λ/16, and consider the system: which is an approximate equation of (1.11). Write Φ = (φ 1 , φ 2 ) T , and define It is readily verifiable that and Differentiating each of the above two equations one more time with respect to η 1 gives and then we find that both φ + and φ − satisfy a Bessel equation (4.2) and we may take Let Φ (1) , Φ (2) be two linearly independent solutions of equation (4.3), and put Λ = (Φ (1) , Φ (2) ). Then Λ is a 2 × 2 matrix, and is of the form , (4.11) where δ 1 ,δ 1 , δ 2 andδ 2 are constants to be determined. Furthermore, we may always write Ψ (∞) = Λ(1 + ε 1 (x, λ)).