Correlation between pole location and asymptotic behavior for Painlevé I solutions

Ovidiu Costin

Department of Mathematics The University of Chicago

Abstract

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is applied to Painlevé’s first equation. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions, obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_{p}(C) on ℝ^{+} of a solution is monotonic in a parameter C describing its asymptotics on antistokes lines^{1}, and obtain rigorous bounds for x_{p}(C). We also derive the behavior of x_{p}(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles.

1 Introduction and notations

We consider the solutions of Painlevé’s first equation, P1

y′′ = 6y^{2} + z (1.1)

and study the connection between their asymptotic behavior and the location of their poles.

The analysis and proofs in the present paper are mostly self-contained, but the technique used, as well as the nature of results obtained, are much easier to interpret from the more general perspective of exponential asymptotics. We start by briefly presenting some concepts and results in this area which are relevant to the problem under discussion.

Jean Écalle has introduced and developed the theory of analyzable functions and tools of generalized resummation of transseries (generalized exponential-power series) as a natural and very comprehensive generalization of the theory of analytic functions [8, 11, 12]. The grand picture is the construction of a class of functions, closed under all imaginable algebraic and analytic operations, which are into an isomorphic correspondence with the space of transseries. In a sense, the space of transseries represents the formal closure of power series under all operations. A wide class of analytic problems admit formal solutions in the space of transseries (obtained in a more or less straightforward way). Transseries solutions contain ample analytic information and in many cases (for instance generic linear or nonlinear differential systems [16]) the actual solutions of the problem can be reconstructed from their transseries by modern resummation methods. Unlike the isomonodromic deformation approach (see [2, 10, 5, 9, 4]) the method introduced in this paper is suitable to be used in nonintegrable systems as well.

1.1 Notes on asymptotics beyond all orders for P1

Proofs of the various technical statements in this section can be found in [16], in the more general context of analytic rank-one differential systems.

The change of variables

±i\sqrt{6z^{−1}}y(z) = −1 + \frac{4}{25x^{2}} − h(x); 30x = (−24z)^{5/4} (1.2)

brings (1.1) to the (normal) form

(1.3)  h′′ + \frac{1}{x}h′ − h − \frac{1}{2}h^{2} − \frac{392}{625x^{4}} = 0

There is a one-parameter family of solutions of (1.3) that tend to zero as x → +∞, x ∈ ℝ, and they have a common asymptotic series (cf. also § 3):

h(x) ∼ h̃_{0}(x) = ∑_{k=4}^{∞} c_{k}x^{−k} (1.4)

The classical expansion (1.4) does not distinguish different solutions in this family. However there exists a one complex parameter family of transseries solutions, of which h̃_{0} is only a special instance:

h̃(x, C) = h̃_{0}(x) + ∑_{k=1}^{∞} C^{k}x^{−k/2}e^{−kx}h̃_{k}(x) (C ∈ ℂ) (1.5)

In (1.5), h̃_{k} are formal power series, h̃_{k} = ∑_{m=0}^{∞} c_{km}x^{−m}, that can be computed algorithmically by substituting (1.5) into (1.3) and identifying the coefficients. With the choice c_{10} = 1, which is possible for (1.3), the series h̃_{k} do not depend on C. Every h̃_{k} is factorially divergent. There is nevertheless a very close connection between (1.5) and the family of actual solutions of (1.3) which decrease on ℝ for x → +∞: any such solution can be reconstructed by generalized Borel resummation of h̃(x, C) for some C. Denote by LB the generalized Borel resummation operator (as defined in [16]). There exists x_{0} > 0 (independent of k) so that the functions h_{k} = LBh̃_{k} are analytic in the half plane H_{x_{0}} = {x : ℜ(x) > x_{0}}. In addition, h_{k} ∼ h̃_{k} as x → e^{iθ}∞ in H_{x_{0}}, θ ∈ (−π/2, π/2). For any C ∈ ℂ, the function series

h(x; C) = LBh̃_{0}(x) + ∑_{k=1}^{∞} C^{k}x^{−k/2}e^{−kx}LBh̃_{k}(x) (1.6)

converges for ℜ(x) > x_{1} = x_{1}(C) to a solution of (1.3); conversely, any solution h which vanishes for real x → +∞ is necessarily of the form (1.6) for a unique C. LB commutes with the algebraic operations, complex conjugation (thus a series with real coefficients sums to a real function), with differentiation, and has good continuity properties. Along directions in ℂ other than ℝ^{+} similar results hold.

In addition, the constant C in (1.6) is related by a simple formula to the large |x| behavior of h(x; C) on the antistokes lines ±iℝ^{+} (cf. (2.8)). We show that for any C > 0, the boundary of the domain where the series (1.6) converges contains an actual pole of h(x; C), much as the boundary of the convergence disk of the Taylor series of an analytic function must contain a singularity.

1.2 Heuristic discussion of transasymptotic matching

The value of C is related to the position of the outermost singularities of solutions of P1. The following heuristic argument illustrates the reason.

For simplicity, we assume that C is large. When x is very large, the exponentials in (1.5) are smaller than the leading term in h̃_{0}, and the behavior of h(x; C) is roughly c_{04}x^{−4}.

For smaller values of x, such that 1 ≫ |Ce^{−x}x^{−1/2}| ≫ |x|^{−1}, the relative magnitude of the terms of (1.5) changes: the largest term becomes Ce^{−x}x^{−1/2} (instead of c_{04}x^{−4}), the next order of magnitude is the term C^{2}e^{−2x}x^{−1}c_{20} (not Ce^{−x}x^{−3/2}c_{11}) and so on. In this intermediate regime, the terms of (1.5) need to be reshuffled accordingly, and h(x; C) is represented by

h̃(x; C) = ∑_{m=0}^{∞} \frac{1}{x^{m}} (∑_{k=0}^{∞} \frac{c_{km}}{(e^{x}x^{1/2}/C)^{k}} ) := ∑_{m=0}^{∞} \frac{G_{m}(e^{x}x^{1/2}/C)}{x^{m}} (1.7)

It turns out that, for x not too small, the inner sums converge (while, obviously, the expansion h̃ as a whole is still divergent), and G_{m}(t) are analytic. In fact, for (1.3), G_{m}(t) are rational functions and G_{0} has a particularly simple expression, G_{0}(t) = t(t − 1/12)^{−2} (see the proof of Lemma 7). Moreover, for any m ≥ 0 the function G_{m}(t) has only one pole in the right half plane, at t = 1/12.

The representation (1.7) is normally accurate as long as no Stokes phenomena occur and the following asymptoticity condition holds

|G_{m+1}(e^{x}x^{1/2}/C)x^{−m−1}| ≪ |G_{m}(e^{x}x^{1/2}/C)x^{−m}| (1.8)

A formal calculation shows that (1.8) holds for x in a region R that starts well to the left of the array of poles of g(x) = G_{0}(e^{x}x^{1/2}/C), located at x_{N}, where

x_{N} + \frac{1}{2} ln x_{N} = ln(C/12) + 2Nπi; N ∈ ℤ (1.9)

and extends all the way (small neighborhoods of the poles themselves excluded) to ℜ(x) → ∞. The region R is thus substantially wider than the region where (1.4) is valid.

Accepting for the moment that (1.7) is the asymptotic expansion of some solution h in R, (1.7) can be used to estimate ∮ (∫ h) on a circuit around a pole x_{N}. The calculation gives ∮ (∫ h) = −12πi + O(1/ ln C) for large |C|, showing that h is singular at some point x_{p}(C; N) = x_{N} + o(1). x_{p}(C; N) can be estimated to all orders in 1/ ln C from (1.7).

On the other hand, if x is large, and away from x_{N}, h(x; C) must be bounded (by (1.7)). The location of the poles of h in R should thus be x_{p}(C; N) = x_{N} + o(1).

This method gives information not only about the location, but also about the nature of the singularities of h(x; C) in R. A straightforward calculation based on the Cauchy integral argument explained above indicates, to any order in 1/ ln C, that x_{p}(C; N) are double poles.

The fact that P1 is an integrable equation plays no role, except that G_{m} can be computed in closed form.

We note that in the matching process all the terms of the transseries (1.5) are necessary, so that the matching is transasymptotic. The procedure makes the connection between the behavior of h in the pole-free region near infinity, and its singularities for large x in ℂ.

In the next sections these conclusions are made rigorous, and further results are proved on the connection between C and the position of x_{p}(C; 0), for C not necessarily large. The proofs yield global estimates for (1.6) and (1.7).

We can also interpret (1.7) from the perspective of multiscale analysis. Apparently an expansion of the form ∑ c_{m}(x)x^{−m} (where the coefficient of x^{−m} may depend on x) is subject to substantial arbitrariness. Not so, however, if c_{m}(x) are required to be sufficiently regular and the expansion to be valid over a wide enough range of x. This is a crucial point in multiscale analysis. We note that (1.5) determines the asymptotic scales x^{−1} and Ce^{−x}x^{1/2}. Once the scales are found, the terms in (1.7) are more conveniently determined by inserting (1.7) in (1.3) and identifying the coefficients of x^{−m}.

The proper choice of the asymptotic scales is essential. For instance looking for a representation of the more general form

h̃_{a} = ∑ x^{−m}G_{m}(Ce^{−x}x^{a})

only for a = 1/2 will the expansion be asymptotic (even formally) for large x. For any other value of a, G_{m}(Ce^{−x}x^{a}) = O(x^{m}) instead of O(1), and then h̃_{a} is rather meaningless.

2 Main results

Notations. By Lemma 2 in § 3 there is a one (real) parameter family of solutions of (1.3) which are real-valued for x > const. on ℝ^{+} and such that h_{0}(x) → 0 as x → +∞. Their behavior for large x is h_{0} ∼ c_{4}x^{−4} + c_{6}x^{−6}(1 + o(1)) where c_{4} = −392/625 and c_{6} < 0. We choose h_{0} to be one such solution. (The choice h_{0} = LBh̃_{0} has however important additional properties). We let A ≥ 9/4 be such that, for x > A,

h_{0}(x) ∈ (−x^{−4}, c_{4}x^{−4}) (2.1)

Proposition 1 (i) Any solution h of (1.3) such that |h(x)| < 1 for large positive x is given by

(2.2)  h(x; C) = ∑_{k=0}^{∞} C^{k}x^{−k/2}e^{−kx}h_{k}(x)

for some C ∈ ℂ, where the functions h_{k}(x) are determined recursively as solutions of linear differential equations (cf. (3.2)) and do not depend on C.

The series (2.2) converges for all x > x_{0}(C) = max{A, x_{1}} where x_{1} + \frac{1}{2} ln x_{1} = ln(|C|/12). (ii) The functions h_{k} satisfy the inequalities

0 < h_{k}(x) < lim_{x→∞} h_{k}(x) = \frac{k}{12^{k−1}} (k ∈ ℕ and x > A) (2.3)

Furthermore,

\overline{lim_{k→∞}}h_{k}(x)^{−1/k} = 12 + O(x^{−1}) (as x → +∞)

(iii) (Outermost pole of h(x; C) on ℝ^{+}.) If C > C_{0} = 12 min_{t≥A} \sqrt{t}e^{t}(1 − \frac{9}{4t})^{−1}, then h(x; C) has a double pole at the point x_{p} = x_{p}(C; 0), the unique solution of

x^{1/2}e^{x}\overline{lim_{k→∞}}h_{k}(x)^{−1/k} = C (2.4)

for x > A. The function h(x; C) has no other singularities on [x_{p}, ∞). (iv) (Monotonicity and bounds for x_{p}.) For C > C_{0}, x_{p}(C; 0) is increasing in C and satisfies the inequalities

(2.5) x_{p} + \frac{1}{2} ln x_{p} ≤ ln(C/12) ≤ x_{p} + \frac{1}{2} ln x_{p} − ln (1 − \frac{9}{4x_{p}} )

Thus, x_{p} has the asymptotic behavior,

(2.6)  x_{p} = ln(C/12) − \frac{1}{2}ln(ln(C/12)) + O(1/ ln C) (C → +∞)

(v) (Poles of h(x; C) in ℂ, for large C complex.) Let N ∈ ℕ and ϵ > 0. As |C| → ∞, h(x; C) has double poles at the points:

(2.7)

x_{p}(C; k) = ln(C/12) − \frac{1}{2} ln(ln(C/12)) + 2kπi + o(1) (k ∈ ℤ ∩ [−N, N])

and no poles in the region {x : |ℑ(x)| ≤ N, ℜ(x) > ℜ(x_{p}(C; 0)) + ϵ}. (vi) (Connection to behavior on antistokes lines.) For the special choice h_{0} = LBh̃_{0} we have h_{k} = LBh̃_{k}, and

h(x; C) ∼ (C ± \frac{1}{2}S)x^{−1/2}e^{−x} as x → ±i∞ (2.8)

where C ∈ ℂ is the same as in (2.2) and S is the Stokes constant of the equation (1.3).

Notes. (1) The property that the solutions which vanish in some direction can be expressed, as in (2.2), as a power series in the constant C of (2.8) is very general [16] (integrability plays no role). Thus, for P1, C is also a natural parameter for pole location. A formula of type (2.4) is expected to be quite general and is not a connection formula stricto sensu, because it involves limits. For P1 however, there are indications that these limits can be calculated in closed form.

(2) For the choice h_{0} = LBh̃_{0} it can be shown, using the methods in [17], that (2.1) holds for x > 4/3.

(3) The fact that P1 is integrable makes it possible to calculate explicitly the Stokes constant: S = i\sqrt{\frac{6}{5π}}. This was done in [9] using the isomonodromic deformation method; its computation using exponential is the object of a forthcoming paper. A detailed description of the asymptotic behavior of solutions of P1, derived through isomonodromy, can be found in [4, 5, 6, 9].

3 Proofs and further results

Before proving the main proposition we describe some asymptotic properties of the solutions of (1.3) which are bounded as x → +∞.

Lemma 2 (i) If h is a solution of (1.3) and |h(x)| < 1 for large positive x, then

h ∼ h̃_{0} = ∑_{k=4}^{∞} c_{k}x^{−k} (x → e^{iθ}∞ with θ ∈ (−\frac{π}{2} , \frac{π}{2} )) (3.1)

(the first few terms of h̃_{0} are −\frac{392}{625}x^{−4} − \frac{6272}{625}x^{−6} − \frac{141196832}{390625}x^{−8} − ...). (ii) There is a one (complex/real) parameter of such solutions (complex/real valued on ℝ^{+}); if h_{1} and h_{2} satisfy (3.1) then

h_{1} − h_{2} = Ce^{−x}x^{−1/2} + o(e^{−x}x^{−1/2})

for some C ∈ ℂ, as x → ∞. If in addition C = 0 then h_{1} = h_{2}.

Proof of Lemma 2: (i) is standard (see, e.g. [14]); it is also easy to reprove in our case, since (1.3) can be rewritten as

h′′ + \frac{1}{x}h′ − (1 + \frac{1}{4x^{2}} ) h = \frac{1}{2}h^{2} − \frac{1}{4x^{2}} h + \frac{392}{625x^{4}} := g(x)

which, in integral form

h(x) = \frac{1}{2}x^{−1/2} (e^{x} ∫_{∞}^{x} e^{−t}t^{1/2}g(t)dt − e^{−x} ∫_{a}^{x} e^{t}t^{1/2}g(t)dt)

is a contraction in the unit ball in the sup norm, for large x. If a ∈ ℝ then h is real-valued.

Part (ii) is proven without difficulty in essentially the same way, by writing an integral contractive equation for δ = h_{2}−h_{1} (see [16] for details under more general assumptions).

3.1 Properties of (2.2) on ℝ^{+}

Let h_{0} and A be as in § 2. Formal substitution of (2.2) in (1.3) yields

(3.2)

H′′_{k} − (1 + h_{0}(x) − \frac{1}{4x^{2}} ) H_{k} = \frac{\sqrt{x}}{2}e^{−kx} ∑_{0<j<k} h_{j}h_{k−j} := R_{k}(x) (k ≥ 1)

where H_{k}(x) = x^{−(k−1)/2}e^{−kx}h_{k}(x). Step 1. Study of the homogeneous part of (3.2) (same as (3.2) for k = 1):

(3.3)  y′′ − (1 + h_{0}(x) − \frac{1}{4x^{2}} ) y = 0

Lemma 3 (i) Eq. (3.3) has two solutions y_{+}, y_{−} such that: y_{±}(x) > 0 for x > A, y_{±}(x) = e^{±x}(1 + O(x^{−1})) as x → +∞ and their Wronskian (y_{−}y′_{+} − y′_{−}y_{+}) is W = 2.

If y(x) solves (3.3) and y(x) = O(e^{−x}) as x → ∞ then y = const.y_{−}. (ii) The function u(t, s) = y_{+}(t)y_{−}(s) − y_{−}(t)y_{+}(s) is positive for t > s > A.

Proof: (i) Except for the explicit region of positivity of y_{±}, this is standard asymptotic theory (see e.g. [14]; note also that W = const and W → 2 for x → ∞). We prove the statement only for y_{−}; the argument for y_{+} is similar. Associating to (3.3) the integral equation

(3.4)  f(x) = 1 − ∫_{x}^{∞} e^{x−s} sinh(s − x) (\frac{1}{4s^{2}} − h_{0}) f(s)ds

where f(x) = e^{x}y(x), we define

J(Ψ)(x) = ∫_{x}^{∞} e^{x−s} sinh(s − x) (\frac{1}{4s^{2}} − h_{0}) Ψ(s)ds (3.5)

on L^{∞}(x_{0}, ∞), for x_{0} > A.

Remark 1 ∥J∥_{L^{∞}↦L^{∞}} < \frac{1}{8x_{0}} (1 + \frac{1}{3x_{0}^{2}} ) and J is positivity preserving.

Thus (3.4) has a unique solution f_{1} in L^{∞}(A, ∞); moreover, 0 < f_{1} < 1 and f_{1}(x) ∼ 1 + ∑_{j=1}^{∞}(−1)^{j}|c_{j}|x^{−j} (x → +∞).

Proof: The first part is straightforward. Now,

∥f_{1}∥_{∞} ≤ (1 − ∥J∥)^{−1}∥1∥_{∞}

thus |f_{1}| ≤ \frac{6}{5} implying |f_{1} − 1| ≤ \frac{6}{5}J(1) ≤ \frac{1}{5}, in particular f_{1} > 0. Since J preserves positivity, we have f_{1} = 1−Jf_{1} < 1. The asymptotic expansion of f_{1} follows from the representation f_{1} = (∑_{j=0}^{∞}(−J)^{j}) 1.

Proof of Lemma 3: (ii) note that u(t, s) is a solution of u_{tt} = (1 + h_{0}(t) − \frac{1}{4}t^{−2})u with initial condition u(s, s) = 0; u_{t}(s, s) = 2. Since 1 + h_{0}(t) − \frac{1}{4}t^{−2} > 0 for t > A, we have u(t, s) > 0 for all t > s.

Lemma 4 There is a unique solution of (3.2) with H_{1} = y_{−} such that for k > 0, H_{k}(x) = O(e^{−kx}) as x → ∞. Namely, the H_{k}(x) are inductively given by

H_{k}(x) = \frac{1}{2} ∫_{x}^{∞} u(t, x)R_{k}(t)dt, k ≥ 2 (3.6)

In addition H_{k}(x) > 0 for x > A.

Proof: Immediate induction using Lemma 3. Uniqueness follows from the behavior for large x of the solutions of the homogeneous equation, y_{±} (Lemma 3).

Step 2. Upper bounds for H_{k}. We have, by (3.2) and Lemma 4

(3.7) H_{k}(x) = ∫_{x}^{∞} sinh(t − x)R_{k}(t)dt − ∫_{x}^{∞} sinh(t − x)(\frac{1}{4t^{2}} − h_{0}(t))H_{k}(t)dt

≤ ∫_{x}^{∞} sinh(t − x)e^{−kt}R_{k}(t)dt

From Lemma 3 and Remark 1 we get H_{1}(x)e^{x} ∈ (0, 1) on (A, ∞). Assuming by induction that H_{m}(x) ≤ \frac{m}{12^{m−1}} x^{−(k−1)/2}e^{−mx} if 1 ≤ m < k, we get

(3.8) H_{k}(x) ≤ \frac{1}{2} ∫_{x}^{∞} sinh(t − x)t^{−k/2+1/2}e^{−kt} ∑_{0<j<k} \frac{j(k − j)}{12^{k−1}}dt ≤

\frac{(k^{3} − k)}{12^{k}}x^{−(k−1)/2} ∫_{x}^{∞} sinh(t − x)e^{−kt}dt = \frac{k}{12^{k−1}} x^{−(k−1)/2}e^{−kx}

□ Step 3. Monotonicity

Lemma 5 The functions H_{k}(x) and F_{k}(x) = x^{−1/2}e^{−kx}h_{k}(x), k ≥ 1, are decreasing for x > A.

Proof: The functions F_{k} satisfy the equations

F′′_{k} = (1 + h_{0})F_{k} − \frac{1}{x}F′_{k} + \frac{1}{2} ∑_{0<j<k} F_{j}F_{k−j} (k ≥ 1) (3.9)

By (3.8) and Lemma 4, the F_{k} are positive and vanish as x → ∞. On the other hand, F_{k} can have no extremum for x > A (otherwise, by (3.9) all extrema would be minima), thus are decreasing in x. For H_{k} the proof is very similar.

Step 4. Lower bounds for H_{k}.

Lemma 6 For x > A we have

H_{m}(x) ≥ (1 − \frac{1}{8x}) (1 − \frac{9}{4x})^{(m−1)/2} \frac{m}{12^{m−1}} x^{−(k−1)/2}e^{−mx} (3.10)

From Remark 1 we get f = 1 − Jf ≥ 1 − J1 ≥ 1 − (1 − (8x)^{−1})^{−1} for x > 5/3. Thus H_{1}(x) ≥ (1 − (8x)^{−1}). Since the functions H_{k} are decreasing, we have

(3.11) H_{k}(x) ≥ ∫_{x}^{∞} dt sinh(t − x)R_{k}(t)

− H_{k}(x) ∫_{x}^{∞} dt sinh(t − x)(\frac{1}{4t^{2}} − h_{0}(t))

Thus

H_{k}(x) ≥ (1 − \frac{1}{8x}) ∫_{x}^{∞} sinh(t − x)R_{k}(t)dt (3.12)

From this point on, the proof of Lemma 6 follows the same track as Step 2, noting that (1 − t^{−1})^{(k−1)/2} is increasing in t and that ∫_{x}^{∞} sinh(t − x)t^{−(k−1)/2}e^{−kt} ≥ (1 − x^{−1})(k^{2} − 1)^{−1}x^{−(k−1)/2}e^{−kx}. □

Step 5 Completing the proof of Proposition 1 (i) and (ii): With C ∈ ℂ given, (2.2) converges for x > x_{0}(C), by (3.8). Furthermore, since the H_{k} are decreasing, if (2.2) converges at x = x_{1}, then it converges uniformly on [x_{1}, ∞) (together with its derivatives, in view of the equality in (3.7)) and thus is a solution of (1.3). By (3.8) and (3.10), we have h(x; C)−h_{0}(x) = Ce^{−x}x^{−1/2}(1+o(1)) for large x. Therefore, by Lemma 2, if g is solution of (1.3) which vanishes as x → ∞, then g = h(x; C) for some C ∈ ℂ.

Step 6. Divergence of the series (2.2) implies singularities of the function h(x; C). Let C > C_{0} and x_{div} := inf{x| ∑_{k=1}^{∞} C^{k}F_{k}(x) < ∞}. We have just shown that (2.2) solves (1.3) for x > x_{div}. By the definition of C_{0} and (3.7) we have x_{div} > A. We now want to show that the solution h(x; C) is nonanalytic at x_{div}. Let S_{N}(x) = ∑_{k=1}^{N} C^{k}F_{k}(x).

Lemma 7 Assume h(x; C) is a C^{2}(x_{1}, ∞) solution of (1.3), and x_{1} ∈ (A, x_{div}].

(i) The function δ_{N}(x) = h(x; C)−h_{0}−S_{N}(x) is positive for all x > x_{1} and N ∈ ℕ^{+}.

(ii) x_{1} = x_{div} = x_{p}(C; 0).

Proof: (i) We have, by Proposition 1 (ii) δ_{N}(x) = ∑_{j>N} C^{j}F_{j}(x) > 0 for x > x_{div}. Now, since the function δ_{N} satisfies the equation

δ′′_{N} = \frac{1}{2}δ_{N}^{2} + (1 + S_{N} + h_{0})δ_{N} − \frac{δ′_{N}}{x} + \frac{1}{2} ∑_{\begin{matrix}i,j∈(1,N−1) \\ i+j>N \end{matrix}} F_{i}F_{j} (3.13)

we see, as in Lemma 5, that δ_{N} is decreasing, therefore positive, for all x > x_{1}. (ii) By (i), h(x; C) is unbounded at x_{div}. But the only singularities of h(x; C) for x ≠ 0 are double poles (see the appendix).

Step 7.

Concluding the proof of Proposition 1, (iii) and (iv): The monotonicity of x_{p}(C; 0) in C is now immediate, while (2.5) follows from Lemma 7, (3.8) and (3.10).

Note. Formula (2.4) in itself is not a practical numerical way to calculate x_{p}(C), given a particular C. An efficient (and rigorously controllable) method is to use the procedure in this section, to obtain successively better upper and lower bounds for H_{k}.

3.2 Correspondence between (1.6), (2.2) and (2.8)

Lemma 8 If h_{0} = LBh̃_{0} then h_{k} = LBh̃_{k} for all k ∈ ℕ. Thus the representations (1.6) and (2.2) coincide. Formula (2.8) holds.

Proof of Lemma 8 and of Proposition 1, (vi): We let g_{1,2}(x, C) = h(x; C) as given by (1.6) and (2.2), respectively. In [16], formula (2.8) for g_{1}(x, C) and the estimate g_{1}(x, C) − h_{0}(x) = Ce^{−x}x^{−1/2}(1 + o(1)) are proved in a general setting. Lemma 2 then implies, as in Step 5 § 3.1, that g_{1}(x; C) = g_{2}(x; C).

Note. Using Lemma 8 it follows from [16] that (2.2) converges in fact for all large x with arg(x) ∈ (−π/2, π/2).

3.3 Uniform representation for large x

We are now interested in the region of validity of (1.7) and deriving the behavior of x_{p}(C) for large C ∈ ℂ. While the results in the previous sections used specific properties of (1.3) (positivity of h_{k}), the technique of transasymptotic matching introduced in §1.2 is expected to work for more general nonlinear systems. We will only prove (1.7) to leading order, sufficient to obtain (2.7); cf. also Remark 3. The expansion (1.7) is studied in a region of x containing points near x_{N} (see (1.9)) and extending to infinity. Let ϵ > 0 be small, M ∈ ℝ^{+} and s = s(x; C) = 12e^{x}x^{1/2}/C. Define R = {x : |ℑ(x)| ≤ M |s − 1| ≥ ϵ and ℜ(s) > 1 − ϵ}.

Lemma 9 Let C_{1} ∈ ℝ^{+} be large. For any C with |C| > C_{1} we have

h(x; C) = \frac{12s}{(s − 1)^{2}} + ψ(x; C) (3.14)

where |ψ(x; C)| ≤ K| ln C|^{−1} in Clos(R), for some K not depending on C.

Step 1: Proof of (3.14) for x ∈ ℝ^{+}. From (2.2) and Lemma 6 we obtain

(3.15) h(x; C) = h_{0} + ∑_{k=1}^{∞} 12ks^{−k} + ∑_{k=1}^{∞} 12ks^{−k}[(1 + r_{k}(x)x^{−1})^{k} − 1] =

\frac{12s}{(s − 1)^{2}} + ∑_{k=1}^{⌊x⌋} \frac{12k^{2}c_{k}(x)}{x}s^{−k} + ∑_{⌊x⌋+1}^{∞} 12k [(1 + r_{k}(x)x^{−1})^{k} − 1]+O(x^{−4}) =

\frac{12s}{(s − 1)^{2}} + \frac{E_{1}(x)}{ln(C)} with |E_{1}(x)| ≤ const. (x ∈ ℝ^{+} ∩ R)

The estimates in (3.15) are obtained in the following way: by Lemma 6 we have |r_{k}(x)| ≤ 9/4; thus, |c_{k}(x)| = |(1 + r_{k}/x)^{k} − 1| ≤ (e^{9/4} − 1)kx^{−1} for k ≤ x; the tail ∑_{⌊x⌋+1}^{∞}(·) is exponentially small, while for x ∈ R we have h_{0} = O(x^{−4}) and |x/ ln C| ≤ 2 as C → ∞. Similarly,

|\frac{d}{dx}(h(x, C) − 12s(s − 1)^{−2})| ≤ \frac{const}{ln(C)} for x ∈ ℝ^{+} ∩ R) (3.16)

Step 2. Extension to R. Let ψ(x) = h(x; C)−12s/(s−1)^{2}. It is convenient to change variables to v = x + \frac{1}{2} ln x − ln(C/12) and define R_{v} = \{v|x ∈ R}. Then ψ = (ψ, ψ′) satisfies the system

\frac{d}{dv} (\begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix}) = (\begin{matrix}0 & 1 \\ F (v) & 0 \end{matrix}) (\begin{matrix}ψ_{1} \\ ψ_{2} \end{matrix}) + (\begin{matrix}0 \\ (1 + \frac{1}{2x})^{−2} r \end{matrix}) := Aψ + R (3.17)

where F (v) = (10e^{v} + e^{2v} + 1)(e^{v} − 1)^{−2} and

r = \frac{ψ_{1}^{2}}{2x^{1/2}} +\frac{(x + \frac{1}{4})ψ_{1}}{(x + \frac{1}{2})^{2}} +\frac{2ψ_{2} − ψ_{1}}{4x^{2}} −3\frac{e^{3v} + 4e^{2v} + e^{v}}{(e^{v} − 1)^{4}x^{3/2}} +\frac{24e^{v}(2e^{v} + 1)}{x^{1/2}(e^{v} − 1)^{4}}−\frac{c_{4}}{x^{7/2}}

Let U(t; v) be the fundamental matrix of U′ = AU (the solution with U(t; t) = I, the identity matrix). ^{2}

Remark 2 U(t; v) and U^{−1}(t; v) are uniformly bounded on

R_{M;t} = {(t, v) ∈ R_{v}^{2}| |t − v| < M}

Proof: U is uniformly bounded since F is uniformly continuous on R_{v}. Since Tr(A) = 0 we have det(U) = const = 1, so U^{−1} is bounded as well.

Let v_{0} ∈ R_{v} ∩ ℝ. By (3.15) and (3.16) we have |ψ(v_{0})| ≤ const./ ln C. We write (3.17) in integral form

ψ(v) = U(v_{0}; v)ψ(v_{0}) + U(v_{0}; v) ∫_{v_{0}}^{v} U^{−1}(v_{0}; t)R(t)dt (3.18)

The integral equation (3.18) is contractive in L^{∞}(R_{M;v_{0}}) in a ball of radius K/ ln C for some K which is, by (3.15) (3.16) and Remark 2, independent of v_{0} (note that all terms on the r.h.s. of (3.18) are in fact O(1/ ln C) if ∥ψ∥ = 1). Lemma 9 is proven. □ Step 3: Pole location for large C ∈ ℂ.

Proof of Proposition 1, (v): Let

K = {x : |s − 1| = ϵ and |v − 2Nπi| < 2π}

If ϵ is small enough, then for large C the curve K is a simple closed curve in Clos(R). We have by Lemma 9 ∮_{K} (∫ h(·; C)) = −12πı + o(1) thus h is not analytic in the interior of K. The only singularities of h(x; C) for x ≠ 0 are double poles, cf. the appendix.

2Incidentally, the equation U′ = AU can be solved explicitly: for the associated second order equation the general solution is ψ = C_{1}t(t + 1)(t − 1)^{−3} + C_{2}(t^{−1}P_{5} + 60tv(t + 1))(t − 1)^{−3}, where t = e^{v} and P_{5} is a quintic polynomial in t.

Remark 3 Using the asymptotic behavior of the solution of U′ = AU and more careful estimates, it is possible to prove contractivity and thus (3.14) in a sector of opening less than π centered on ℝ^{+} (the opening cannot equal π because of (2.8)). The asymptotic expansion (3.14) can be determined then to higher orders in a straightforward way. (The equation for G_{k} is (3.17), with r.h.s. a bilinear expression in G_{m}, m < k (cf. also footnote 2)). The result is

g_{C} ∼ G_{0}(s) + \frac{1}{x}G_{1}(s) + ... + \frac{1}{x^{n}} G_{n}(s) + ... (3.19)

with

G_{0} = \frac{12s}{(s − 1)^{2}} , G_{1} = −\frac{15s^{3} + 175s^{2} + 30s − 2}{10s(s − 1)^{3}}, ⋯ (3.20)

It is also worth noting that for all G_{k}(t) to be rational functions, the value −392/625 in (1.3) is crucial. A different numerical value produces a valid expansion (3.19) but with G_{k}, k ≥ 5 not rational.

Remark 4 The arguments that were used to determine the asymptotic position of poles work even if C is not large, as long as |x| is large, yielding the location of poles in the neighborhood of the imaginary axis, for large x (while on the positive real line C small implies x_{p}(C) small, by Proposition 1).

Appendix

All solutions of P1 are meromorphic.

The fact that the only singularities of solutions of P1 are double poles was indicated by Painlevé, but it was not until the advent of the isomonodromic approach (see [1] and [7]) that a rigorous proof was given. For completeness, we present an independent proof of this crucial property of P1 (and thus of (1.3), for x ≠ 0), based on the idea that near poles, the equation P1 is meromorphically equivalent to an elliptic equation [18].

Notations: z_{0} denotes a point where initial conditions for y are given, z̃ will denote singular points of y. Let D̃ be a disk without a discrete set of points. Λ(z_{1}, z_{2}) will denote smooth curves in D̃, with the endpoints z_{1} and z_{2} ∈ Clos(D̃) removed, such that the length of any subsegment Λ(z_{3}, z_{4}) of Λ(z_{1}, z_{2}) is less than 2|z_{3} − z_{4}|.

Lemma 10 In a neighborhood of any point z̃ ∈ ℂ there exists a oneparameter family of solutions h of (1.1), having a double pole at z̃, of the form h(z) = (z − z̃)^{−2}A(z) where A(z) is analytic at z̃, and A(z̃) = 1.

Proof: Substituting y = ∑_{k=−2}^{∞} c_{k}(z − z̃)^{k} in (1.1) we get

(k + 1)(k + 2)c_{k+2} = 6 ∑_{m=−2}^{k+2} c_{m}c_{k−m} + z̃δ_{k=0} + δ_{k=1} (A.1)

so that c_{−2} = 0 or 1; choosing c_{−2} = 1 we have c_{−1} = 0 and

[(k + 1)(k + 2) − 12]c_{k+2} = 6 ∑_{m=0}^{k} c_{m}c_{k−m} + z̃δ_{k=0} + δ_{k=1} (A.2)

Carrying on the calculation we get

(c_{−2}, c_{−1}, c_{0}, c_{1}, c_{2}, c_{3}, c_{4}) = (1, 0, 0, 0, \frac{−1}{10}z̃, −\frac{1}{6}, c_{4}) (A.3)

i.e., c_{4} is a free constant (Note: The specific form of the equation (1.1) is crucial in obtaining an integer power series solution. Generic perturbations of (1.1) lead to an inconsistent equation for c_{4}.

Finally, for some ρ > 0 and all k ≥ 0 we have |c_{k}| ≤ \frac{1}{3}(k + 1)ρ^{k+2}. Indeed, let ρ be such that |c_{i}| ≤ \frac{1}{3}(i + 1)ρ^{i+2} for i ≤ 2. Then, assuming this property for c_{0}, c_{1}, ..., c_{k+1} we have

|c_{k+2}| ≤ \frac{1}{3} \frac{(k + 1)(k + 2)(k + 3)}{(k − 2)(k + 5)}ρ^{k+4} ≤ (k + 3)ρ^{k+4}

Lemma 11 If y is a solution of (1.1), analytic at z_{0}, then the radius of analyticity at z_{0} is at least R = 1/ max{|y(z_{0})|^{1/2}, |y′(z_{0})/2|^{1/3}, |y_{0}^{2} + z_{0}/6|^{1/4}}.

Proof: With y = ∑_{k=0}^{∞} c_{k}(z − z_{0})^{k} we have

(k + 1)(k + 2)c_{k+2} = 6 ∑_{m=0}^{k} c_{m}c_{k−m} + z_{0}δ_{k=0} + δ_{k=1} (A.4)

By the definition of R we have |c_{j}| ≤ (j + 2)R^{−j−2} for j ≤ 2. Inductively, if the inequality holds ∀j ≤ k + 1 we have

|c_{k+2}| ≤ \frac{(k + 1)(k + 2)(k + 3)R^{−k−4}}{(k + 1)(k + 2)} = (k + 3)R^{−k−4}

Lemma 12 Let y be a solution of (1.1), analytic at any z ∈ Λ(z_{0}, z̃) and that y(z) = ∑_{k=−2}^{4} a_{k}(z − z̃)^{k} + O(z − z̃)^{5} as z → z̃ along Λ(z_{0}, z̃), with a_{−2} ≠ 0.

Then y is meromorphic at z̃, and y is uniquely determined by z̃ and a_{4}.

Proof: Since ∑_{k=−2}^{4} a_{k}(z − z̃)^{k} is an O(z − z̃)^{5} formal solution of P1, for i ≤ 3 we must have a_{k} = c_{k} defined in (A.3) (note: since y solves (1.1), the asymptotic expansion is differentiable). Let y_{0}(z) be the solution given in Lemma 10 with c_{4} = a_{4}. Let v(z) = y_{0}(z) − (z − z̃)^{−2} and f(z) = y(z) − y_{0}(z). Then v = O((z − z̃)^{2}), f = O(z − z̃)^{5} and

f′′ − \frac{12}{(z − z̃)^{2}} f = 12fv + 6f^{2} (A.5)

or

f(z) = C_{1}(z − z̃)^{−3} + C_{2}(z − z̃)^{4} − \frac{7}{(z − z̃)^{3}} ∫ z ; z̃(s − z̃)^{4}g(s)ds

+\frac{(z − z̃)^{4}}{7} ∫ z ; z̃(s − z̃)^{−3}g(s)ds = N (f) (A.6)

where g = 12fv + 6f^{2} = O(z − z̃)^{7} near z̃. Thus C_{1} = C_{2} = 0. Let z_{1} ∈ Λ and S be the Banach space of continuous functions on Λ(z_{1}, z̃) with the norm sup_{Λ} |(z − z̃)^{−5}f(z)|. It is immediate to check that if z_{1} − z̃ is small enough N is contractive in the unit ball B ⊂ S. The solution of (A.6) is thus unique in B, f = 0.

Let y be a solution of (1.1), and D an open disk where y is meromorphic. By definition, the singularities of y in D are isolated. Let z̃ ∈ ∂D be a singular point of y. We will show that z̃ is a pole: this implies that y is meromorphic on ℂ.

Let D̃ be D without the poles of y.

Remark 5 (i) As z → z̃ along Λ(z, z̃) we have |y(z)|+|y′(z)| → ∞. (ii) y is unbounded as z → z̃.

Proof: (i) Since y is singular at z̃ then by Lemma 11 y(z) or y′(z) must be large for z near z̃. (ii) Multiplication of (1.1) by y′ and integration yields:

(y′)^{2} = 4y^{3} + 2zy − 2 ∫_{a}^{z} y(s)ds + C (A.7)

If y were bounded along Λ, then by (A.7) y′ would be bounded as well.

Proposition 13 Let z_{0} ∈ D be a regular point of y and Λ = Λ(z_{0}, z̃) ⊂ D. We have y^{−3}(s) ∫_{a}^{s} y(t)dt → 0, |y(s)| → ∞, and |y′(s)| → ∞ as s → z̃ on Λ.

Proof: For any continuous function f unbounded at z̃ we have

lim inf_{z→z̃} |f|^{−3} ∫_{a}^{z} |f|(t)d|t| ≤ lim inf_{z→z̃} |f|^{−3} max_{Λ(a,z)} |f| length(Λ) = 0

Thus, if we had y^{−3}(s) ∫_{a}^{s} y(t)dt ↛ 0, then for any γ > 0 small enough

| ∫_{a}^{s_{n}} y(t)dt| = γ|y^{3}(s_{n})| (A.8)

for some sequence s_{n} → z̃ on Λ. If γ < \frac{1}{2}, by (A.8), (A.7) and Remark 5 we have

|y(s_{n})| → ∞ and |y′(s_{n})| → ∞ and ∫_{z_{0}}^{s_{n}} |y(t)|d|t| → ∞ (A.9)

Consider now the solution h of the equation

(h′)^{2} = 4h^{3} + 2s_{n}h + C̃ (A.10)

satisfying h(0) = y(s_{n}), h′(0) = y′(s_{n}) (with C̃ chosen accordingly in (A.10)). It follows from (A.9) that C̃ = C − 2 ∫_{a}^{s_{n}} y(s)ds → ∞. Define the function H by

αH(βz) := h(z), where α = C̃^{1/3}, β = C̃^{1/6}

It follows that (H′)^{2} = 4H^{3} − 2θH + 1, where θ = C̃^{−2/3} (A.11)

H is thus an elliptic function with with periods Π_{1,2} = Π + o(θ) for some Π ∈ ℂ (independent of θ). By (A.8) |y(s_{n})| = (2γ^{−1/3})|C̃^{1/3}|(1 + o(1)), then using (A.7), we get |y′(s_{n})| = (4γ^{−1}e^{iϕ_{n}} + 2)^{−1/2}|C̃^{1/2}|(1 + o(1)) for some ϕ_{n} ∈ ℝ.

Therefore |H(0)| = (2γ)^{−1/3}+o(1), and |H′(0)| = (4γ^{−1}e^{iϕ_{n}}+2)^{−1/2}|+ o(1). With the substitution αY (βz) := y(s_{n} + z) we have

(Y′)^{2} = 4Y^{3} + 2θY + 1 + 2ϵ (zY − ∫_{0}^{z} Y (t)dt) , where ϵ = C̃^{−5/6},

or Y′′ = 6Y^{2} + θ + ϵs (A.12)

We substitute Y = H ∘G; G(z) = z+g(z) in (A.12) with g(0) = g′(0) = 0 and

2g′H′′ ∘ G + (g′)^{2}H′′ ∘ G + g′′H′ ∘ G = ϵz

or 2g′(H′ ∘ G)′ + g′′H′ ∘ G = ϵz + (g′)^{2}H′′ ∘ G (A.13)

which can be rewritten in the integral form (g(z) = ∫_{Λ(0,z)} w(t)dt, G = z + g):

w = (H′ ∘ G)^{−2} ∫_{Λ(0,z)} H′(G(u)) [ϵu + H′′(G(u))w^{2}(u)^{2}] du = N (w)

(A.14)

Let Δ_{ϵ} be the disk centered at 0, of diameter |ϵ|^{−1/4}, and Δ̃_{ϵ} ∋ 0 be Δ_{ϵ} without the disks of diameter d(γ) around the poles and zeros of H′(z). Let Λ ⊂ Δ_{ϵ}, let a ∈ (1/4, 1/2). Let S be the Banach space of continuous functions on Λ in the sup norm and B = {w : ∥w∥ ≤ |ϵ|^{a}} ⊂ S. If w ∈ B, then |g| ≤ | ∫_{Λ(0,z)} w(t)dt| ≤ |ϵ|^{a−1/4}. B is preserved by N provided that

|ϵ| Const(γ)|ϵ|^{−1/2} + Const(γ)|ϵ|^{2a−1/4} ≤ |ϵ|^{a} (A.15)

which holds for small ϵ. Similarly for u, v ∈ B, ∥N (u) − N (v)∥ = O(ϵ^{a−1/4})∥u − v∥. N has thus a unique fixed point in B.

Therefore |g(z)| ≤ |ϵ|^{a−1/4} for z ∈ Δ̃_{ϵ} and thus (when ϵ is small)

(A.16) sup_{Δ̃_{ϵ}} |α^{−1}y(s_{n} + β^{−1}·) − H(·)| = sup_{Δ̃_{ϵ}} |H(G(·)) − H(·)| = O(ϵ^{a−1/4})

Note now that as s_{n} → z̃ we have β^{−1} = C̃^{−1/6} → 0 while α → ∞ and |β^{−1}ϵ^{−1/4}| ∝ C̃^{1/24} → ∞. Thus L = s_{n} + α^{−1}Δ̃_{ϵ} is an α^{−1}−lattice for D (i.e., ∀z ∈ D , ∃z_{1} ∈ L : |z−z_{1}| ≤ α^{−1}), and (A.16) implies y is unbounded near z_{0}, contradiction.

Since y^{−3}(s) ∫_{a}^{s} y(t)dt → 0, the fact that min{|y|, |y′|} → ∞ now follows from Remark 5 and (A.7).

Proposition 14 z̃ is a second order pole.

Proof: The substitution y = 1/v^{2} transforms (A.7) into

(v′)^{2} = 1 + \frac{1}{2}v(z)^{4}z − \frac{1}{2}v(z)^{6} ∫_{z_{0}}^{z} v(s)^{−2}ds + Cv(z)^{6} (A.17)

It follows from Proposition 13 that along a curve Λ(z_{0}, z̃) avoiding the poles of y we have v^{6} ∫_{z_{0}}^{z} v^{−2}(s)ds → 0. With ζ = z − z̃, (A.17) implies (v′)^{2} = 1 + o(1) so that v^{2} = ζ^{2}(1 + o(1))^{2} near z̃. Using (A.17) again we now get v^{2} = (ζ + ∑_{k=2}^{6} v_{k}ζ^{k} + o(ζ^{6})). Lemma 11 implies that y is meromorphic at z̃.

Acknowledgment.

The author would like to thank Professors Percy Deift and Martin Kruskal for very interesting discussions on this subject.

Bibliography

[1] Fokas, A. S., Mugan, U., and Zhou, X. On the Solvability of PI, III, and V Inverse Problems 8: 757–785, 1991

[2] Jimbo, M. and Miwa, T. Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients I, I Physica D 2: 306–352, 1981

[3] Jimbo, M. and Miwa, T. Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients II, I Physica D 2: 407–448, 1981

[4] Kapaev, A. A. and Kitaev, A. V., Connection Formulae for the First Painlevé Transcendent in the Complex Domain Lectures in Mathematical Physics 27: 243– 252, 1993

[5] Kitaev, A. V. The method of Isomonodromy Deformations and the Asymptotics of Solutions of the “Complete” Thirf Painlevé Equation Math. USSR Sbornik 62: 421–444, 1989

POLE LOCATION VERSUS ASYMPTOTICS 21

[6] Kitaev, A. V. Elliptic Asymptotics of the First and Second Painlevé Transcendents Russian Math. Surveys 49:1: 81–150, 1994

[7] Its, A. and Novokshenov, V. Y. The isomonodromic deformation method in the theory of Painlevé equations. Springer-Verlag, 1986

[8] Écalle, J. Fonctions Resurgentes. Publications Mathematiques D’Orsay 81, 1981 [9] Kapaev, A. A. Asymptotics of Solutions of the Painlevé Equation of the First Kind.

Diff. Equ. 24: 1107–1115, 1989 (Translation from the Russian original (1988).)

[10] Flaschka, H. and Newell, A. C. Monodromy- and spectrum-preserving deformations. I Comm. Math. Phys. 76: 65–116, 1980

[11] Écalle, J. Finitude des cycles-limites et accéléro-sommation de l’application de retour. Lecture Notes in Math. 1455, Springer Berlin, 1990

[12] Écalle, J. in Bifurcations and periodic orbits of vector fields NATO ASI Series, Vol. 408 pp. 75–184, Kluwer Academic Publishers Dordrecht; Boston, 1993

[13] Écalle, J. and Menous, F. in The Stokes phenomenon and Hilbert’s 16th problem. pp. 71–101, World Sci. Publishing River Edge, NJ, 1995

[14] Wasow, W. Asymptotic expansions for ordinary differential equations. Interscience Publishers New York, 1968

[15] Costin, O. Exponential asymptotics, trans-series and generalized Borel summation for analytic nonlinear rank one systems of ODE’s. Internat. Math. Res. Notices 8: 377–417, 1995^{3}

[16] Costin, O. On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s. Duke Math. J., to appear ^{3}

[17] Costin, O. and Kruskal, M. D. On optimal truncation of divergent series solutions of nonlinear differential systems, submitted to Proc. Roy. Soc. Lond.^{3}

[18] O. Costin , R. D. Costin Singular normal forms of nonlinear equations, submitted to Nonlinearity^{3}.

Received Month 199X.

3Preprints available at http://www.math.uchicago.edu/∼costin