Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve Equation: II

where the prime denotes differentiation with respect to τ , and b ∈ R \ {0} and a ∈ C are parameters.

For other Painlevé equations, there exist families of solutions that have zeroes and poles accumulating at the point at infinity (see, for example, [2]). In Part I [1], such solutions were not distinguished; therefore, the main purpose of this work (Part II) is to determine whether or not such solutions exist, and if so, to find their asymptotics. As shown in [1], Equation (1.1) can be presented as a Hamiltonian system, with the Hamiltonian function, H(τ ), given by

In [1] we introduced the auxiliary function

which defines Bäcklund transformations for Equation (1.1):

where q(τ ) = u(τ ) and p(τ ) is a Bäcklund transformation of u(τ ). In the Hamiltonian setting, the functions q(τ ) and p(τ ) are, respectively, the generalized coordinate and impulse. (More detailed information about the functions H(τ ) and f (τ ), in particular, the corresponding ODEs they satisfy, can be found in Proposition 1.3 and Remark 1.3 of [1].) In this work, these functions play an important role in the study of the zeroes and poles of u(τ ). Section 1 of [1] contains a review of the literature on the theory and applications of Equation (1.1); so here we mention only those papers that are related to Equation (1.1) and which have appeared since the publication of [1]. According to the algebro-geometric classification scheme given in [3], the space of initial conditions of Equation (1.1) can be characterized by the extended Dynkin diagram of type D 7 . There is another case of the degenerate third Painlevé equation which can be obtained by a similarity reduction of the well-known Sine-Gordon equation: in the classification scheme of [3], this equation is of type D 8 . The latter equation is known to be equivalent, via a quadratic transformation, to a special case of the “complete” third Painlevé equation (type D 6 in the classification of [3]); therefore, we use the term “degenerate” to specify only Equation (1.1), or its equivalent forms. Asymptotics of solutions of the D 8 equation were studied via the Isomonodromy Deformation Method in [2]. Recently, asymptotics of the so-called instanton solutions of the D 8 equation were obtained in [4] via the exact WKB analysis. We also mention the recent work [5], where a class of semi-flat Calabi-Yau metrics is obtained in terms of real solutions of Equation (1.1) with a = 0.

In this work we apply the Isomonodromy Deformation Method: the reader can find some basic ideas and references concerning this method in Sections 1 and 2 of [1]. We also mention the new monograph [6], which reflects some recent developments of this method and of a closely related technique based on a steepest-descent-type analysis of the associated Riemann-Hilbert problem [7]. Although Equation (1.1) resembles one of the standard representatives of the list of the Painlevé equations, its asymptotic study via the Isomonodromy Deformation Method contains, in contrast to the other Painlevé equations, additional technical difficulties: the problem is that the corresponding Fuchs-Garnier (or Lax) pair is degenerate (see [1] for details); therefore, in contrast to the non-degenerate versions of the Painlevé equations, its associated WKB analysis requires a much more careful evaluation of the correction terms. In fact, this is one of the reasons why the present work took 6 years to complete since the appearance of [1].

In order to make the presentation as self-contained as possible, we now embark on succinctly reminding the reader about some of the basic facts introduced in Sections 1 and 2 of [1].

Our study of Equation (1.1) is based on the following Fuchs-Garnier (or Lax) pair (see Proposition 2.1 of [1]):

∂ µ Ψ(µ, τ ) = U(µ, τ )Ψ(µ, τ ), ∂ τ Ψ(µ, τ ) = V(µ, τ )Ψ(µ, τ ), (1.4) where

with σ 3 = 1 0 0 -1 , and

ia -A(τ )B(τ ) +τ (A(τ )D(τ )+B(τ )C(τ )) .

Proposition 1.1. The Frobenius compatibility condition of System (1.4) for arbitrary values of µ ∈ C and for differentiable, scalar-valued functions A(τ ), B(τ ), C(τ ), and D(τ ) is that these functions satisfy the following system of isomonodromy deformations:

( -A(τ )B(τ ) ) ′ = 2(A(τ )D(τ )-B(τ )C(τ )).

(1.5)

Remark 1.1. Hereafter, all explicit τ dependencies are suppressed, except where confusion may arise.

A relation between the Fuchs-Garnier pair (1.4) and the degenerate third Painlevé equation (1.1) is given by Proposition 1.2 ( [1]). Let u = u(τ ) and ϕ = ϕ(τ ) solve the system Then b is independent of τ , and u(τ ) and ϕ(τ ) solve System (1.6).

In this work asymptotics (as τ → ±∞ and τ → ±i∞) of solutions of Equation (1.1) are parametrized in terms of the monodromy data of System (1.4). This parametrization is equivalent to finding the corresponding connection formulae; indeed, given asymptotics of some solution as τ → +∞, say, one uses it to determine the corresponding

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