Location of Poles for the HastingsMcLeod Solution to the Second Painlevé Equation
Abstract
We show that the wellknown HastingsMcLeod solution to the second Painlevé equation is polefree in the region
Keywords: HastingsMcLeod solution, Painlevé II equation, location of poles, quasisolutions
Mathematics Subject Classification (2000): 30D35, 30E10, 33E17, 41A10
1 Introduction
Painlevé equations are six second order nonlinear ordinary differential equations first studied by Painlevé and his colleagues around 1900. They are well known for the socalled Painlevé property, i.e., the only movable singularities of their solutions are (finite order) poles; see [30, §32.2]. Here ‘movable’ means the location of the singularities (which in general can be poles, essential singularities or branch points) of the solutions depend on the constants of integration associated with the initial or boundary conditions of the differential equations. The solutions of these equations, often called the Painlevé transcendents [30], in general cannot be represented in terms of elementary functions or known classical special functions. They play important roles in both pure and applied mathematics, and are widely thought of as the nonlinear counterparts of the classical special functions.
For the first two Painlevé equations
PI  
PII  (1.1) 
all solutions are meromorphic in the complex plane with being the only essential singularity. The locations of the movable poles for the Painlevé transcendents are crucial for understanding a number of problems arising from mathematical physics; cf. [4, 16, 26, 27]. In the pioneering works [6, 7], Boutroux established the “deformed” elliptic function approximations in appropriate sectors near infinity, which leads to the degeneration of lattices of poles along the critical rays
(1.2) 
where
This means that the poles tend to align themselves along certain smooth curves which tend to one of the rays near infinity. Furthermore, Boutroux also showed the existence of solutions which have no lines of poles near infinity near () of the critical rays , which are called truncated solutions.
An interesting feature of the  or truncated solutions is, as confirmed by numerical studies in [20, 19, 28], that the distributions of poles near infinity characterize the global behavior of the poles. More precisely, let be the sector bounded by two consecutive critical rays:
The following conjecture was made in [29] by Novokshenov:
Conjecture 1.1.
If the  or truncated solution of Painlevé equation has no pole at infinity in a sector , then it has no poles in the whole sector .
For the truncated solutions of PI, a special case of this conjecture is known as Dubrovin’s conjecture, which appeared in [16] with connections to the critical behavior of the nonlinear Schrödinger equation. It was proved recently in [13] with a technique developed in [12]; see also [25, 26, 27] for partial results.
In this paper, we will further improve the technique in [13] (see also a recent work [1] for other improvements of [13]) and give an analytic proof of Conjecture 1.1 in the context of a special truncated solution of PII, namely, the HastingsMcLeod solution [22]. This solution might be the most famous one among the Painlevé transcendents, due to its frequent appearances in applications, especially in mathematical physics. For instance, the cumulative distribution function of the celebrated TracyWidom distribution [31, 32] admits an integral representation involving the HastingsMcLeod solution. It is noted that the TracyWidom distribution is also applied to describe the length of the longest increasing subsequence in random permutations [2]. Another application is the appearance of functions associated with HastingsMcleod solution in building new universality class of limiting kernel for certain critical unitary random matrix ensembles [5, 8]; see also [21] for a nice review of this aspect and [15, 17] for its more recent applications related to nonintersecting Brownian motions. Our main result is stated in the next section.
2 Statement of results
The HastingsMcLeod solution is a special solution of (1.1) with , i.e., it satisfies the equation
(2.1) 
The solution is known to be polefree on the real axis ([22]), and has the following asymptotics:
where denotes the usual Airy function [30]. A plot of for real is shown in the left picture of Figure 1. The locations of poles for is illustrated in the right picture of Figure 1. The six dashed lines are the critical lines defined in (1.2), and it is clear from the picture that all the poles are located in the sectors , which is consistent with Conjecture 1.1.
Our main result is stated as follows.
Theorem 2.1.
The HastingsMcLeod solution of the second Painlevé equation (2.1) is polefree in the region
3 Strategy of proof
Although our method works for both of the two sectors
We first note that is also a solution to (2.1). This, together with the fact that is real on the real line and uniqueness of the solution, implies that . Therefore, for
Theorem 3.1.
The HastingsMcLeod solution is polefree in the region
As mentioned before, we will use the same idea as in [13] to prove the Theorem. To be precise, we will analyze in two regions
(3.1) 
and
(3.2) 
In each region we will construct an explicit quasisolution consisting of polynomials and exponential functions, and show that the difference between and the quasisolution is small in a suitable norm. This shows that is polefree in both and , and hence in .
The main challenge of the proof is to find an effective quasisolution approximation of the HastingsMcLeod solution which has sufficient accuracy for both small and large . This requires comprehensive knowledge of the asymptotics of the solution near infinity. To this end, we mention the following asymtotics of HastingsMcLeod solution relevant to our proof (see [18, Theorem 11.7] and [23]).
Proposition 3.2.
Let be the HastingsMcLeod solution of the second Painlevé equation (2.1), then
(3.3) 
as and ;
(3.4) 
as and
The constant is the socalled quasilinear Stokes’ multiplier, which reflects the quasilinear Stokes phenomenon for the second Painlevé transecedent; see [10, 23, 24] for more details. We emphasize two features of the asymptotics in Proposition 3.2:

The asymptotics (3.4) is valid along the critical line (i.e., the boundary of the relevant sector), where the asymptotics is oscillatory.
As we shall see later, the construction of quasisolutions in the regions away from the origin is based on these asymptotic behaviors.
The rest of this paper is organized as follows. The analysis of in is accomplished in Section 4. To get a concrete estimate of the initial values of at , we will also need to study along before we are able to construct a quasisolution in , which is carried out in Sections 5 and 6. Analysis in is accomplished in Section 7. We conclude this paper with the proofs of our main results in Section 8.
4 Analysis of in the region
We start with the analysis of the PII equation (2.1) in the region , . Our goal in this section is to prove the following result:
Proposition 4.1.
The HastingsMcLeod solution is polefree in the region , where is defined in (3.1).
As mentioned before, we will prove the above result by constructing a polefree quasisolution to PII, and showing that the difference between the quasisolution and the HastingsMcLeod solution is bounded. Our construction of this quasisolution is motivated by asymptotic expansions with exponential sums studied in [11], which suggests that we make the following change of variables:
(4.1) 
This brings (2.1) into the normalized form
(4.2) 
and the region of interest in Proposition 4.1 corresponds to
in the new variable .
Let denote the solution of (4.2) corresponding to the HastingsMcLeod solution. In view of (3.4), one naturally expects to have the decomposition
(4.3) 
where is a solution of (4.2) with pure power series behavior near (i.e., with zero quasilinear Stokes’ multiplier), and is exponentially small near . This is also consistent with the fact the Painlevé equations admit a oneparameter family of solution represented by the sum
which was found by Boutroux [6, 7], and particularly this includes as a special case of PII. Since we only need to prove is polefree in , we do not need to consider full expansions. Instead, we will only show the existence of a decomposition (4.3) with
where is a constant related to the quasilinear Stokes multiplier ; see (4.10) below.
4.1 Existence and uniqueness of the power series solution
Recall the asymptotics of in (3.4), by (4.1), this corresponds to a solution of (4.2) in . Formal asymptotic analysis of (4.2) indicates that there should exist a solution . We thus substitute
into (4.2), and get the equation
(4.4) 
Inverting the differential operator on the left side of (4.1), we get the integral equation
(4.5) 
where the first integral is along a horizontal line, while the second one is along a vertical ray starting from . We intend to prove existence of a solution by showing that is a contractive map in a suitable Banach space. The expressions of and indicate that it is necessary to estimate generalized exponential integrals in the complex plane. For this purpose, we introduce the following inequalities, which will also be used later:
Lemma 4.2.
Assume is analytic in the right half plane with where , , and . For , we have the estimates
(4.6)  
(4.7)  
(4.8) 
Proof.
We write where and . To prove the first inequality, we note that since is analytic with at least decay in the right half plane, we can rotate the integration path to a horizontal one, namely with ranging from to 0. Then by direct calculations we have
Alternatively, we can also rotate the integration path to a radial one, which gives
To prove the last inequality, we rotate the contour to a vertical one, namely with ranging from to 0. By direct calculations we have
This completes the proof of the lemma. ∎
Now we are ready to prove the main result below.
Proposition 4.3.
Proof.
We will prove the proposition using the contraction map theorem in the Banach space of analytic functions in the interior of , continuous up to the boundary, equipped with the weighted norm
Proof of statement (i):
Proof of statement (ii):
We only need to do similar estimates for the nonlinear terms in (4.1) using (4.7) and (4.8), as well as the simple facts that
Straightforward calculations give us
and
Adding up the above bounds we see that
4.2 Existence and uniqueness of the exponential correction
To analyze the exponential part of , we see from (3.4) that
(4.9) 
for , which means by (4.1) that,
(4.10) 
as .
Thus we write
and substitute this expression into (4.2), which gives the equation
(4.11) 
with
Based on the first few terms of the asymptotic expansion of , we construct a quasisolution
(4.12) 
and our goal is to show that there exists a solution to (4.11) of the form
(4.13) 
where is small in a suitable norm. The equation for can be found by substituting (4.13) into (4.11), which gives
(4.14) 
where
(4.15) 
We obtain the following integral equation by inverting the operator on the left side of (4.14):
(4.16) 
To estimate , we introduce a lemma similar to Lemma 4.2.
Lemma 4.4.
Assume is analytic in the right half plane with where and . For and , we have the estimate
(4.17) 
For and , we have the estimate
(4.18) 
Proof.
Since is analytic, we can deform the integration paths into horizontal ones as in Lemma 4.2. Denoting , where and , we have
We are then ready to prove
Proposition 4.5.
Proof.
The strategy is the same as in the proof of Proposition 4.3. We consider the Banach space of analytic functions in , continuous up to the boundary, equipped with the weighted norm
Proof of statement (i):
We first estimate in (4.15). Substituting the expression
with defined in (4.12) into (4.15), we get an expression of the form
where and are constants that can be written down explicitly, and in fact most of them are either zero or very small. For our purpose, it suffices to write out the terms with and estimate the rest crudely. Elementary calculations show that
where
and
To estimate , we take the absolute value of each term in , applying , and then adding them up. The last term in is special, and we use (4.6) to estimate the inner integral and a radial path for the outer integral, which gives
(4.19) 