SINGULAR NORMAL FORM FOR THE PAINLEVÉ EQUATION P1

OVIDIU COSTIN AND RODICA D. COSTIN

Abstract. We show that there exists a rational change of coordinates of Painlevé’s P1 equation y′′ = 6y^{2}+x and of the elliptic equation y′′ = 6y^{2} after which these two equations become analytically equivalent in a region in the complex phase space where y and y′ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlevé property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlevé property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures.

1. Introduction

The problem of determining which classes of nonlinear differential equations can define new transcendents (special functions having good properties), received a special attention in the last century, especially due to the emphasis on finding “explicit” solutions to differential equations. Fuchs had the intuition that the appropriate condition these equations must satisfy is that their solutions have no movable branch points. This feature of an equation is now known as the Painlevé property and proved to be a very relevant characteristic, in a wide range of problems. Fuchs’ study was pursued by Briot and Bouquet, and then by Painlevé [10] and Gambier who showed that there are no new transcendents coming from first order equations, but there are six second order equations which define new special functions. These equations (now denoted usually as P1 to P6) were discovered as a result of a purely theoretical quest, but they later arose naturally in many distinct physical applications (see, e.g., [4] and [7]). Linearization of second order Painlevé equations through the isomonodromic transformation method [2], [4], [8], [9] is one of the most important recent developments. To this date, higher order equations have not yet been classified from the point of view of Painlevé integrability.

Perhaps surprisingly, proving the Painlevé property of an equation turns out to be quite difficult (although if one assumes that singularities are described locally by convergent power-logarithmic series, then it is usually easy to check for the absence of movable branch points) and some of the classical proofs for Painlevé equations have been subsequently challenged. See also [5] and [6].

The Painlevé property, being shared by all solutions, must reveal a particular structure of the equation itself. We show in fact that P1 is equivalent with an equation integrable by quadratures, in a region in the phase space where the solutions y(x) are singular. The equivalence also has the implication that the solutions of P1 are meromorphic, and is a natural and rigorous way to prove the Painlevé property.

Key words and phrases. Integrability; Painlevé transcendents; Painlevé property; Normal forms.

Research at MSRI was supported in part by NSF grants DMS-9701755 and DMS-9704968.

We expect our technique to work for other equations having the Painlevé property, as well.

The existence of a simple integrable singular normal form of P1 is tied to the special integrability properties of this equation, and is not merely a consequence of the fact that (2.1) “approximately equals” y′′ = 6y^{2} when y, y′, y′′ are large. Approximations near singularities are usually very unstable. For instance, the modification Y′′ = 6Y^{2} + x^{2} of P1 also seems to be approximated by the equation y′′ = 6y^{2} for large Y, Y′′, especially if x is small, but these two equations are not equivalent in mentioned regime. The obstruction in the equivalence is the presence of “bad” logarithmic terms in the Frobenius series of Y (x) near its movable singularities, and the fact that their absence in y(x) is stable under combinations of analytic and rational transformations.

2. Main Results

We consider the Painlevé P1 equation

\frac{d^{2}y}{dx^{2}} = 6y^{2} + x (2.1)

and show that there exists a transformation of the independent variable only (i.e., of the form x = F (t, u, u′), y = u) which is an equivalence of (2.1) to the elliptic equation

\frac{d^{2}u}{dt^{2}} = 6u^{2} (2.2)

in regions of the phase space where the dependent variables are large.

The regularity of the transformation giving the equivalence appears more clearly after making a rational transformation of the dependent variables of (2.1) and (2.2)

v_{1} = −y^{−2} \frac{dy}{dx}, w_{1} = y^{3} (\frac{dy}{dx})^{−2} (2.3)

and, respectively

v = −u^{−2} \frac{du}{dt}, w = u^{3} (\frac{du}{dt})^{−2} (2.4)

Then (2.1) and (2.2), written in the new variables,

\{ \begin{matrix}\frac{dv_{1}}{dx} = \frac{2}{w_{1}} − 6 − xw_{1}^{2}v_{1}^{4} \\ \frac{dw_{1}}{dx} = \frac{12}{v_{1}} (w_{1} − \frac{1}{4}) + 2xw_{1}^{3}v_{1}^{3} \end{matrix} (2.5)

and, respectively

\{ \begin{matrix}\frac{dv}{dt} = \frac{2}{w} − 6 \\ \frac{dw}{dt} = \frac{12}{v} (w − \frac{1}{4}) \end{matrix} (2.6)

are analytically equivalent on polydisks centered at (x_{0}, 0, 1/4) (x_{0} ∈ ℂ arbitrary). We note that the centers of the polydisks correspond to the point at infinity for y and u. In remark 5 below we explain how this equivalence translates into properties of the solutions of P1.

Proposition 1. There exists a map (t, v, w) ↦ Γ(t, v, w) = (x, v_{1}, w_{1}), in the extended phase space, of the form

\{ \begin{matrix}x = γ(t, v, w) \\ v_{1} = η(t, v, w) \\ w_{1} = θ(t, v, w) = v^{2}wη(t, v, w)^{−2} \end{matrix} (2.7)

which transforms (2.5) into (2.6) and has the following properties:

(i) Γ is holomorphic in a neighborhood of each point (x_{0}, 0, 1/4), x_{0} ∈ ℂ. Namely, for every x_{0} ∈ ℂ there exists a polydisk Δ(x_{0})

Δ(x_{0}) = \{(t, v, w) ∈ ℂ^{3} : |t − x_{0}| < 1 , |v| < R^{−1} , |w − \frac{1}{4}| < ϵ \} (2.8)

(R ≥ 1 and ϵ < 1/4) where Γ is holomorphic and where its power series in v has the form

γ(t, v, w) = t + ∑_{k≥5} γ_{k}(t, w)v^{k} (2.9)

η(t, v, w) = v + ∑_{k≥4} η_{k}(t, w)v^{k} (2.10)

θ(t, v, w) = w + ∑_{k≥4} θ_{k}(t, w)v^{k} (2.11)

(ii) Γ is a biholomorphism of Δ(x_{0}) onto its image. Its inverse is holomorphic near (x_{0}, 0, 1/4) and Γ^{−1} = Id + O(v_{1}^{4}).

(iii) Let t → (v(t), w(t)) be a solution of (2.6) such that (t, v(t), w(t)) ∈ Δ(x_{0}) for t in some disk |t − t_{1}| < ϵ′.

Then the function t → γ (t, v(t), w(t)) is a biholomorphism of the disk |t − t_{1}| < ϵ′ onto its image.

The proof of Proposition 1 is given in § 3.1.

The local equivalence of (2.5) and (2.6) stated in Proposition 1 is straightforwardly translated into a local equivalence of (2.1) and (2.2):

Corollary 2. There exists a map (t, u, u′) ↦ Γ̃(t, u, u′) = (x, y, y′), in the extended phase space, of the form

\{ \begin{matrix}x = γ (t, −\frac{u′}{u^{2}} , \frac{u^{3}}{u′^{2}} ) \\ y = u \\ y′ = −u^{2}η (t, −\frac{u′}{u^{2}} , \frac{u^{3}}{u′^{2}} ) = u′ [1 + (\frac{u′}{u^{2}} )^{4} η_{1} (t, −\frac{u′}{u^{2}} , \frac{u^{3}}{u′^{2}} )] \end{matrix} (2.12)

where γ, η, η_{1} are holomorphic on the polydisks Δ(x_{0}) (x_{0} ∈ ℂ) which takes equation (2.1) into (2.2).

The transformation Γ̃ is one-to-one on each domain

Δ̃(x_{0}) = \{(t, u, u′) ∈ ℂ^{3} : |t − x_{0}| < 1, |\frac{u′}{u^{2}} | < R^{−1}, |\frac{u^{3}}{u′^{2}} − \frac{1}{4}| < ϵ\} (2.13)

If u(t) is a solution of (2.2) such that the set T_{u} = {(t, v(t), w(t)) ; |t − t_{1}| < ϵ′} is included in Δ(x_{0}), then the equality y(x) = u(t) defines a solution of (2.1) for x ∈ γ (T_{u}).

Remark 3. If p ∈ ℂ is a pole of u(t) then v(t) and w(t) have removable singularities at p, with v(p) = 0 and w(p) = 1/4.

As a consequence of the equivalence of Proposition 1, the Painlevé property of (2.1) follows naturally:

Proposition 4. The only singularities of the solutions of (2.1) are second order poles.

Remark 5. The following description follows from the proof of Proposition 4. If y(x) is a solution of (2.1), then whenever x is sufficiently close to a singularity of y(x), (x, y(x), y′(x)) falls in the equivalence region of (2.1) with (2.2). In fact, the complex plane is divided into a union A of nonintersecting balls – each containing a pole of y(x), where (x, y(x), y′(x)) ∈ Domain(Γ) – and the complement of A, which is a connected set, where y(x) is analytic.

3. Proofs

3.1. Proof of Proposition 1. (i) The proof proceeds in several steps.

Step 1: A straightforward calculation shows that Γ (cf. (2.7)) maps (2.5) to (2.6) iff γ satisfies the equation

v(L^{2}γ) + 6(Lγ) − (6 + w^{2}v^{4}γ) (Lγ)^{3} = 0 (3.14)

where L is the linear operator

L = \frac{∂}{∂t} + (\frac{2}{w} − 6) \frac{∂}{∂v} + \frac{12}{v} (w − \frac{1}{4}) \frac{∂}{∂w}

and η is given by

η = \frac{v}{Lγ} (3.15)

Step 2: There exists a formal series solution of equation (3.14) of the form (2.9) where γ_{k} have convergent power series at w = 1/4

γ_{k}(t, w) = ∑_{n≥0} γ_{k,n}(t) (w − \frac{1}{4})^{n} (3.16)

with γ_{k,n} polynomials in t.

Indeed, substitution of (2.9) in (3.14) leads to the recurrence P_{k}(w)γ_{k+1} = h_{k}(γ_{5}, ..., γ_{k}, t, w) (k ≥ 4) (3.17) where P_{k} is the linear operator

P_{k}(w) = 144 (w − \frac{1}{4})^{2} \frac{∂^{2}}{∂w^{2}} + (w − \frac{1}{4}) f_{k}(w)\frac{∂}{∂w} + g_{k}(w) (3.18)

with

f_{k}(w) = 12(2k + 1) (\frac{2}{w} − 6) (3.19)

g_{k}(w) = (k + 1) [\frac{2(2k + 3)}{w^{2}} − \frac{24(k + 2)}{w} + 36(k + 2)] (3.20)

Furthermore, h_{k}(γ_{5}, ..., γ_{k}, t, w) is a polynomial in γ_{5}, ..., γ_{k}

h_{k}(γ_{5}, ..., γ_{k}, t, w) = h_{k}^{0}(γ_{5}, ..., γ_{k}, t, w) − h_{k}^{1}(γ_{k−7}, ..., γ_{k}, w) (3.21) where

(3.22) h_{k}^{0}(γ_{5}, ..., γ_{k}, t, w) = w^{2}tδ_{k=4} − 12∂_{t}γ_{k}

+ w^{2}γ_{k−4} + 18 ∑_{i+j=k} L_{i}L_{j} + 6 ∑_{i+j+l=k} L_{i}L_{j}L_{l}

+ tw^{2} {3L_{k−4} + 3 ∑_{i+j=k−4} L_{i}L_{j} + ∑_{i+j+l=k−4} L_{i}L_{j}L_{l}}

+ w^{2} {3 ∑_{i+p=k−4} γ_{p}L_{i} + 3 ∑_{i+j+p=k−4} γ_{p}L_{i}L_{j}

+ ∑_{i+j+l+p=k−4} γ_{p}L_{i}L_{j}L_{l}}

with the convention that the summation indices satisfy i, j, l ≥ 4 and p ≥ 5, where

Lγ := 1 + ∑_{k≥4} L_{k}v^{k} (3.23)

L_{k} = ∂_{t}γ_{k}δ_{k≥5} + 12 (w − \frac{1}{4}) ∂_{w}γ_{k+1} + (k + 1) (\frac{2}{w} − 6) γ_{k+1} (3.24)

and

h_{k}^{1}(γ_{k−7}, ..., γ_{k}, w) = −\frac{12}{w} (w − \frac{1}{4}) k∂_{t}γ_{k} + 24 (w − \frac{1}{4}) ∂_{tw}γ_{k} + ∂_{tt}γ_{k−1}

(3.25)

We note that (3.17) gives γ_{k+1} in terms of γ_{5}, ..., γ_{k} as a solution of a second order linear inhomogeneous ODE. We are looking for functions γ_{k+1} holomorphic at (t, w) = (x_{0}, 1/4). The point w = 1/4 is a regular singular

point of (3.17) and we need to show there exist analytic solutions there.

Substituting the formal series (3.16) and the expansions of f_{k}, g_{k} and h_{k}

f_{k}(w) = ∑_{n≥0} f_{k,n} (w − \frac{1}{4})^{n} , g_{k}(w) = ∑_{n≥0} g_{k,n} (w − \frac{1}{4})^{n}

h_{k}(γ_{5}(t, w), ..., γ_{k}(t, w), t, w) = ∑_{n≥0} h_{k,n}(t) (w − \frac{1}{4})^{n}

in (3.17), we get, for n ≥ 0,

[144n(n − 1) + nf_{k,0} + g_{k,0}] γ_{k+1,n} = h_{k,n} − ∑_{p+j=n,p<n} (pf_{k,j} + g_{k,j}) γ_{k+1,p} (3.26)

If for some k it is true that

144n(n − 1) + nf_{k,0} + g_{k,0} ≠ 0 (3.27)

for all n ≥ 0, then the system (3.26) can be solved for γ_{k+1,n} recursively, for any values of the right side of each equation. This is the case if k ≠ 6. Indeed,

f_{k,0} = f_{k}(1/4) = 24(2k + 1) , g_{k,0} = g_{k}(1/4) = 4(k + 1)(k − 6) (3.28)

and the solvability conditions (3.27) reads

144n(n − 1) + 24n(2k + 1) + 4(k + 1)(k − 6) ≠ 0

which holds if k ≠ 6.

We must look at the case k = 6 separately. The equations for γ_{5}, γ_{6}, γ_{7} are

P_{4}(w)γ_{5} − tw^{2} = 0 (3.29)

P_{5}(w)γ_{6} + 24 (w − \frac{1}{4}) \frac{∂^{2}}{∂t∂w}γ_{5} + (\frac{20}{w} − 72) \frac{∂}{∂t}γ_{5} = 0 (3.30)

P_{6}(w)γ_{7} + 24 (w − \frac{1}{4}) \frac{∂^{2}}{∂t∂w}γ_{6} + (\frac{24}{w} − 84) \frac{∂}{∂t}γ_{6} + \frac{∂^{2}}{∂t^{2}} γ_{5} = 0

(3.31)

Direct substitution shows that the power series of γ_{5,6} have the form

γ_{5} = −\frac{t}{640} − \frac{7t}{352} (w − \frac{1}{4}) + O ((w − \frac{1}{4})^{2}) (3.32)

γ_{6} = −\frac{1}{1920} + O (w − \frac{1}{4}) (3.33)

Using (3.32) and (3.33) it follows that (3.31) also has power series series solutions γ_{7} indexed by the arbitrary coefficient γ_{7,0} (there are infinitely many such solutions because of the potential obstruction at k = 6, see also the note below).

By Frobenius’ theory of regular singularities, the series (3.16) converge. Induction shows that the coefficients depend polynomially on the parameter t. In steps 3 to 8 we show that the series γ_{k} (cf. (3.16)) converge in fact on a common polydisk. Note. The special form of P1 is essential in overcoming the obstruction at k = 6. Generic perturbations of P1 lead to a k = 6 equation without solutions, implying that no (integer) power series for γ exists. Step 3: A space of analytic functions. Consider the class H_{p}(D) of holomorphic functions in the polydisk

D = {(t, w) ∈ ℂ^{2} : |t − x_{0}| < 1 , |w − 1/4| < ϵ} (for ϵ ∈ (0, 1/4))

which depend polynomially on t and are continuous in the closure \overline{D} of D. If

ψ(t, w) = ∑_{j=0}^{n} ψ_{j}(w) (t − x_{0})^{j} ∈ H_{p}(D)

then we define

(3.34) |||ψ||| := ∑_{j=0}^{n} ||ψ_{j}(w)|| , where ||ψ_{j}|| = sup_{|w−1/4|<ϵ} |ψ_{j}(w)|

We note the following estimate of t–derivatives

|||∂_{t}ψ||| = ∑_{j=0}^{n} j||ψ_{j}(w)|| ≤ n|||ψ||| (3.35)

and that multiplication is continuous:

|||ψ_{1}ψ_{2}||| ≤ |||ψ_{1}||| |||ψ_{2}|||

8 OVIDIU COSTIN AND RODICA D. COSTIN

The usual sup norm on \overline{D} is estimated in terms of the norm (3.34) by:

sup_{(t,w)∈D} |ψ(t, w)| ≤ |||ψ||| (3.36)

Thus the closure \overline{H_{p}(D)} of the space H_{p}(D) in the norm ||| · ||| is contained in the space of the holomorphic functions on D, continuous on \overline{D}.

We now denote by Const. constants which are independent of k, C, ϵ, R and x_{0}. Step 4: For k ≥ 7, (3.17) can be written in the form

γ_{k+1} = N_{k}γ_{k+1} + b_{k} (3.37) where b_{k} depends on γ_{5}, …, γ_{k}, t and w and with N_{k} a linear operator satisfying

N_{k} H_{p}(D) ⊂ H_{p}(D) and

|||N_{k}ψ||| ≤ Const.ϵa_{ϵ}^{2}|||ψ||| (3.38) for all ψ ∈ H_{p}(D), where a_{ϵ} = (1/4 − ϵ)^{−1}. It is convenient to rewrite (3.18), (cf. (3.19), (3.20), (3.28)) as

P_{k} = Λ_{k} + S_{k}

where

Λ_{k} = 144 (w − \frac{1}{4})^{2} \frac{∂^{2}}{∂w^{2}} + 24(2k + 1) (w − \frac{1}{4}) \frac{∂}{∂w} + 4(k + 1)(k − 6)

and

S_{k} = (w − \frac{1}{4}) [f_{k}(w) − f_{k}(1/4)] \frac{∂}{∂w} + [g_{k}(w) − g_{k}(1/4)]

Equation (3.17) becomes Λ_{k}γ_{k+1} = h_{k} − S_{k}γ_{k+1} (3.39) The equation Λ_{k}ψ = 0 has two independent solutions (w − 1/4)^{α_{1,2}} with

α_{1} = 1 − k/6 and α_{2} = −1/6 − k/6

In order to write (3.39) in integral form, we interpret it as a linear inhomogeneous equation, with the inhomogeneous part h_{k} − S_{k}γ_{k+1}. Then the solutions γ_{k+1} satisfy

γ_{k+1} = B_{k}(h_{k} − S_{k}γ_{k+1}) (3.40) where B_{k} is the operator

B_{k}(ψ)(t, w) = \frac{1}{168} ∫_{0}^{1} (s^{−1−α_{1}} − s^{−1−α_{2}}) ψ(t, w_{s}) ds (3.41)

where

w_{s} = s (w − \frac{1}{4}) + \frac{1}{4} (3.42)

The solutions of (3.39) satisfy the equation (3.37) where N_{k}ψ = −B_{k} (S_{k}ψ) (3.43) and b_{k}, as a function of (t, w), is given by b_{k} = B_{k} (h_{k}) (3.44) Let ψ ∈ H_{p}(D). Integrating in (3.43) by parts the term containing ∂_{w}γ_{k+1} we get

(N_{k}ψ)(t, w) = −\frac{1}{168} ∫_{0}^{1} ϕ_{k}(s, w)ψ(t, w_{s}) ds (3.45)

where

ϕ_{k}(s, w) = − (α_{1}s^{−1−α_{1}} − α_{2}s^{−1−α_{2}}) [f_{k}(w_{s}) − f_{k}(1/4)]

+ (s^{−1−α_{1}} − s^{−1−α_{2}}) [(w − 1/4)sf′_{k}(w_{s}) + g_{k}(w_{s}) − g_{k}(1/4)]

Direct estimates on (3.19), (3.20) give for |w − 1/4| ≤ ϵ < 1/4 |f_{k}(w) − f_{k}(1/4)| ≤ Const.kϵa_{ϵ} (3.46) |f′_{k}(w)| ≤ Const.ka_{ϵ}^{2} (3.47) |g_{k}(w) − g_{k}(1/4)| ≤ Const.k^{2}ϵa_{ϵ}^{2} (3.48) Also,

∫_{0}^{1} |α_{1}s^{−1−α_{1}} − α_{2}s^{−1−α_{2}}| ds < Const.k^{−1} (3.49)

Therefore, from (3.46) to (3.49) we get

∫_{0}^{1} |ϕ_{k}(s, w)| ds ≤Const.ϵa_{ϵ}^{2} (3.50)

which proves (3.38). Step 5: For k ≥ 7 the functions ∂_{w}γ_{k+1} satisfy an equation of the form ∂_{w}γ_{k+1} = N_{k}^{1}(∂_{w}γ_{k+1}) + N_{k}^{2}(γ_{k+1}) + b_{k}^{1} (3.51) where N_{k}^{j} are linear operators such that

N_{k}^{j} H_{p}(D) ⊂ H_{p}(D) (j = 1, 2) (3.52) |||N_{k}^{1}ψ||| ≤ Const.ϵa_{ϵ}^{2}|||ψ||| (3.53) |||N_{k}^{2}ψ||| ≤ Const.a_{ϵ}^{3}|||ψ||| (3.54) for all ψ ∈ H_{p}(D).

Indeed, differentiation with respect to w in (3.40) yields

∂_{w}γ_{k+1}(t, w) = \frac{1}{168} ∫_{0}^{1} (s^{−α_{1}} − s^{−α_{2}}) ∂_{w} (h_{k} − S_{k}γ_{k+1}) (t, w_{s}) ds (3.55)

and (3.51) follows with

(N_{k}^{1}ψ)(t, w) = \frac{1}{168} ∫_{0}^{1} ϕ_{k}^{1}(s, w)ψ(t, w_{s})ds (3.56)

where

ϕ_{k}^{1}(s, w) = (α_{2}s^{−1−α_{2}} − α_{1}s^{−1−α_{1}}) (f_{k}(w_{s}) − f_{k}(1/4))

− (s^{−α_{1}} − s^{−α_{2}}) (g_{k}(w_{s}) − g_{k}(1/4))

and

(N_{k}^{2}ψ)(t, w) = −\frac{1}{168} ∫_{0}^{1} (s^{−α_{1}} − s^{−α_{2}}) g′_{k}(w_{s})ψ(t, w_{s})ds

From (3.55) we get

b_{k}^{1} = B_{k}^{1} (h_{k}) (3.57)

where B_{k}^{1} is the linear operator

B_{k}^{1}(ψ)(t, w) = \frac{1}{168} (w − \frac{1}{4})^{−1} ∫_{0}^{1} (α_{1}s^{−1−α_{1}} − α_{2}s^{−1−α_{2}}) ψ(t, w_{s}) ds (3.58)

The estimates (3.53) and (3.54) are straightforward. Step 6: γ_{k}(t, w) is a polynomial in t, of degree at most (k − 1)/4. The proof is by induction on k. Note first that the degree in t of any holomorphic solution of (3.17) cannot exceed the degree of the inhomogeneous term h_{k}. Hence (cf. (3.29)) the degree in t of γ_{5} is at most 1.

Assume by induction that deg_{t}γ_{j} ≤ (j − 1)/4 if 5 ≤ j ≤ k. Then, (cf. (3.23)) deg_{t}L_{j} ≤ deg_{t}γ_{j+1} ≤ j/4 for j ≤ k − 1 and we have (cf. (3.21)) deg_{t}h_{k} ≤ k/4. Step 7: Estimates of sums. Given β < 1 there exist constants C_{1}, C_{2} > 0 such that for all integers n ≥ 1

n^{−β} < C_{2}(1 − β)(2 − β)...(n − β)/n! < C_{1}n^{−β} (3.59)

and (β = 3/2)

n^{−3/2} < −C_{2}(1 − 3/2)(2 − 3/2)...(n − 3/2)/n! < C_{1}n^{−3/2} (3.60)

These elementary Gamma function inequalities imply that given β < 1, there is a constant C_{0} > 0 such that for all k

∑_{i+j=k,i,j≥4} i^{−β}j^{−β} ≤ C_{0}k^{1−2β} (3.61)

∑_{i+j+l=k,i,j,l≥4} i^{−β}j^{−β}l^{−β} ≤ C_{0}k^{2−3β} (3.62)

∑_{q+j=k,j≥4,q≥5} q^{−3/2}j^{−1/2} ≤ C_{0}k^{−1/2} (3.63)

∑_{q+i+j=k,i,j≥4,q≥5} q^{−3/2}i^{−1/2}j^{−1/2} ≤ C_{0} (3.64)

∑_{q+i+j+l=k,i,j,l≥4,q≥5} q^{−3/2}i^{−1/2}j^{−1/2}l^{−1/2} ≤ C_{0}k^{1/2} (3.65)

Step 8: There exist constants ϵ ∈ (0, 1/4), C, R > 1 such that γ_{k} ∈ H_{p}(D) and

|||γ_{j}||| < Cj^{−3/2}R^{j−5} (3.66)

|||∂_{w}γ_{j}||| < Cj^{−1/2}R^{j−5} (3.67)

for all j ≥ 5.

The proof is by induction. Suppose that γ_{5}, ..., γ_{k} ∈ H_{p}(D) and that (3.66) and (3.67) hold for j = 5, ..., k; we prove these properties for j = k + 1. The plan of the proof is as follows:

We estimate the functions: (1) h_{k} (defined in (3.21), (3.22) and (3.25)); (2) b_{k} (defined by (3.44)); and (3) b_{k}^{1} (defined by (3.57)). (4) We obtain inequalities for γ_{k+1} and ∂_{w}γ_{k+1}; and finally (5) we choose the constants C, R and ϵ.

(1) Estimating the terms of h_{k}^{0} and h_{k}^{1}:

(a) From the induction hypothesis, (3.35), (3.23)) and the fact that |w| < 1/4, we see that

|||L_{i}||| ≤ deg_{t}γ_{i} |||γ_{i}||| + 12ϵ|||∂_{w}γ_{i+1}||| + (i + 1)(2a_{ϵ} + 6)|||γ_{i+1}|||

≤ Const.K_{ϵ} Ci^{−1/2}R^{i−4} (3.68)

for i ≤ k − 1, where

K_{ϵ} = 1 + a_{ϵ} (3.69)

(b) By (a) and (3.61)

||| ∑_{i+j=k} L_{i}L_{j}||| ≤ Const.K_{ϵ}^{2} ∑_{i+j=k} C^{2}i^{−1/2}j^{−1/2}R^{k−8} ≤ Const.K_{ϵ}^{2}C^{2}R^{k−8}

(c) By (a) and (3.62)

||| ∑_{i+j+l=k} L_{i}L_{j}L_{l}||| ≤ Const.K_{ϵ}^{3} ∑_{i+j+l=k,i,j,l≥4} C^{3}i^{−1/2}j^{−1/2}l^{−1/2}R^{k−12} ≤

Const.K_{ϵ}^{3}C^{3}k^{1/2}R^{k−12}

We finally have

(d) |||w^{2} ∑_{i+p=k−4} γ_{p}L_{i}||| ≤ Const.K_{ϵ} C^{2}k^{−1/2}R^{k−13}

(e) |||w^{2} ∑_{i+j+p=k−4} γ_{p}L_{i}L_{j}||| ≤ Const.K_{ϵ}^{2}C^{3}R^{k−17}

and

(f) |||w^{2} ∑_{i+j+l+p=k−4} γ_{p}L_{i}L_{j}L_{l}||| ≤ Const.K_{ϵ}^{3}C^{4}k^{1/2}R^{k−21}

Combining (a)...(f) we get

|||h_{k}||| ≤ Const.ΦC(k + 1)^{1/2}R^{k−4} (3.70)

where, using that R, K_{ϵ}, C and k are at least one and ϵ < 1/4 we get

Φ = (1 + |x_{0}|R^{−4}) (K_{ϵ}R^{−1} + K_{ϵ}^{2}CR^{−4} + K_{ϵ}^{3}C^{2}R^{−8} + K_{ϵ}^{3}C^{3}R^{−12})

(2) In view of (3.70)

(3.71) |||b_{k}||| = |||B_{k}(h_{k})||| ≤ \frac{1}{168}|||h_{k}||| ∫_{0}^{1} (s^{−1−α_{1}} − s^{−1−α_{2}}) ds

≤ Const.ΦC(k + 1)^{−3/2}R^{k−4}

(3) The function B_{k}^{1}(h_{k})(t, w) is regular at w = 1/4 since

∫_{0}^{1} (α_{1}s^{−1−α_{1}} − α_{2}s^{−1−α_{2}}) ds = 0

Its maximum over the disk |w − 1/4| ≤ ϵ is then attained for |w − 1/4| = ϵ and

|||b_{k}^{1}||| = |||B_{k}^{1}(h_{k})||| ≤ Const. |||h_{k}||| ∫_{0}^{1} |α_{1}s^{−1−α_{1}} − α_{2}s^{−1−α_{2}}| ds

≤ Const.Φϵ^{−1}C(k + 1)^{−1/2}R^{k−4}

(4) In this part of the proof we restrict the values of the parameters ϵ, R and C (conditions 1 to 4); the consistency of the conditions is shown in part (5). If λ_{ϵ} := Const.ϵa_{ϵ}^{2} < 1 (Condition 1) then, using (3.38), it follows that the operator N_{k} extends continuously on \overline{H_{p}(D)} and is a contraction there. Thus I − N_{k} is invertible. The function γ_{k+1} = (I − N_{k})^{−1}b_{k} (cf. (3.37)) is analytic in D, continuous on D̄. Since (by step 6) γ_{k+1} is a polynomial in t, it follows that γ_{k+1} ∈ H_{p}(D).

In view of (3.38) and (3.71) it follows that

(3.72) |||γ_{k+1}||| = |||(I − N_{k})^{−1}b_{k}||| < (1 − λ_{ϵ})^{−1}|||b_{k}||| ≤

Const.Φ(1 − λ_{ϵ})^{−1}C(k + 1)^{−3/2}R^{k−4}

The induction step for γ_{k+1} follows provided that Const.Φ(1 − λ_{ϵ})^{−1} ≤ 1 (Condition 2). The estimate of ∂_{w}γ_{k+1} is similar: step 5 and (3.72) imply (cf. (3.56))

|||N_{k}^{2}γ_{k+1}||| ≤ Const.a_{ϵ}^{3}C(k + 1)^{−3/2}R^{k−4}

If λ_{ϵ}^{1} ≡ Const.ϵa_{ϵ}^{2} < 1 (Condition 3 ) then (cf. step 5)

|||∂_{w}γ_{k+1}||| ≤ Const.(1 − λ_{ϵ}^{1})^{−1}Φ^{1} C(k + 1)^{−1/2}R^{k−4}

where

Φ^{1} = a_{ϵ}^{3}k^{−1} + ϵ^{−1}Φ

The induction step follows if Const.(1 − λ_{ϵ}^{1})^{−1}Φ^{1} ≤ 1 (Condition 4).

(5) Proving that conditions 1 through 4 can be satisfied. Let ϵ be small enough so that conditions 1 and 3 hold. It is convenient to impose CR^{−4} ≤ 1 (Condition 5).

Then conditions 2 and 4 are implied by an inequality of the form

C_{1}(ϵ)(k^{−1} + (R^{−1} + CR^{−4}) (1 + |x_{0}|R^{−4})) ≤ 1 (3.73)

with C_{1}(ϵ) depending on ϵ only. If k is larger than some k_{1}(ϵ), then C_{1}(ϵ)k^{−1} < 1/2. C can now be chosen so that the induction hypothesis ((3.66) and (3.67)) holds for j = 5, ..., k_{1}(ϵ). Finally, for large enough R, condition 5 and (3.73) are satisfied. Step 9: The series (2.9) converges for (t, v, w) ∈ Δ(x_{0}) (cf. (2.8)). This is an immediate consequence of (3.66) and (3.36).

Part (ii) follows from the fact that Γ is a small perturbation of the identity: Γ = Id + v^{4}Γ_{1} with Γ_{1} holomorphic on Δ(x_{0}). Therefore (possibly taking a larger R) we have sup_{Δ(x_{0})} |v^{4}Γ_{1}| < 1 , hence Γ is invertible. Part (iii) follows similarly.

3.2. Proof of Proposition 4. Let y(x) be a solution of (2.1), analytic at some point a ∈ ℂ. The idea of the proof is that if y, y′ are not too large, regularity of (2.1) implies y is analytic in a sufficiently wide region, whereas when y, y′ → ∞ equivalence with (2.2) applies, thus 1/y is locally analytic; uniform estimates of y outside the equivalence regions preclude the accumulation of poles of y.

Consider the open disk B ⊂ ℂ centered at a, of radius r, such that y(x) is meromorphic in B. We show that y(x) extends as a meromorphic function across ∂B, which implies y(x) is meromorphic on ℂ. Let

Δ_{1}(x_{0}) = \{(x, v_{1}, w_{1}) ∈ ℂ^{3} : |x − x_{0}| < ξ , |v_{1}| < S^{−1} , |w_{1} − \frac{1}{4}| < δ \}

where ξ, S and δ are small enough so that the closure of Δ_{1}(x_{0}) is contained in Γ (Δ(x_{0})). Let Δ̃_{1}(x_{0}) be its representation in (x, y, y′).

Let x^{∗} ∈ ∂B. Let x_{0} ∈ B such that |x^{∗} − x_{0}| < ξ and y(x) is analytic at x_{0}.

We show that either (x, y(x), y′(x)) ∈ Δ̃_{1}(x_{0}), for all x in a neighborhood of x^{∗}, thus y(x) is meromorphic at x^{∗} (cf. corollary 2) or else there is a path in B, ending at x^{∗}, on which y(x) is uniformly bounded. In the latter case, y(x) is analytic at x^{∗} as follows from lemma 6.

Lemma 6. Let y(x) be a solution of (2.1). Let l : [0, 1] → ℂ be a path, continuous on [0, 1], smooth on [0, 1), of finite length, and such that y(x) is analytic at each point on l[0, 1) and uniformly bounded on l[0, 1). Then y(x) is analytic at l(1).

The proof is given in §3.3.

Consider the ray R_{θ} starting at x_{0} through x^{∗}. If x, x′ ∈ R_{θ} we write x ≺ x′ when |x − x_{0}| < |x′ − x_{0}|.

Integrating (2.1) we get

(y′)^{2} = 4y^{3} + 2xy − 2 ∫_{x_{0}}^{x} y(s)ds + C (3.74)

Let ϵ_{1} ∈ (0, 1) be small.

In the following, Const. denotes positive constants, which may depend only on x_{0}, y(x_{0}), y′(x_{0}), r, ξ, S and δ but not on ϵ_{1}.

If |y(x)| ≤ ϵ_{1}^{−1} in R_{θ} ∩ B we define l to be the segment [x_{0}, x^{∗}]. Otherwise, let x_{1} be the least x in R_{θ} ∩ B with respect to ≺ such that |y(x_{1})| = ϵ_{1}^{−1}. We have

(y′)^{2} = 4y^{3} + 2xy − 2 ∫_{x_{1}}^{x} y(s)ds + C_{1} (3.75)

where

|C_{1}| ≤ |C| + 2|x_{0} − x_{1}| sup_{x∈[x_{0},x_{1}]} |y| ≤ |C| + 2ξϵ_{1}^{−1} ≤ Const.ϵ_{1}^{−1}

for small ϵ_{1}. Let y_{1} = y(x_{1}) and y′_{1} = y′(x_{1}). Then

|\frac{y′_{1}^{2}}{4y_{1}^{3}} − 1| = |\frac{2x_{1}y_{1} + C_{1}}{4y_{1}^{3}} | ≤ Const.ϵ_{1}^{2} (3.76)

so that

|\frac{y_{1}^{3}}{y′_{1}^{2}} − \frac{1}{4}| < Const.ϵ_{1}

Also, from (3.75),

|\frac{y′_{1}}{y_{1}^{2}} | < Const.ϵ_{1}^{1/2}

Thus (x_{1}, y_{1}, y′_{1}) ∈ Δ̃_{1}(x_{0}) if ϵ_{1} is small enough.

Remark 7. y(x) is meromorphic in a neighborhood of the closure of the disk D_{x_{1}} = {x; |x − x_{1}| < 4ϵ_{1}^{1/2}}.

Consider the solution of (2.2), which corresponds to y(x) through Γ: u(t) = y(x) (cf. corollary 2), defined in a neighborhood of t_{1}, where (t_{1}, u_{1}, u′_{1}) = Γ̃^{−1}(x_{1}, y_{1}, y′_{1}).

The change of variables u = 1/Q^{2} in (2.2) gives, after one integration,

(Q′)^{2} = 1 + K_{1}Q^{6} (3.77)

To estimate the constant K_{1} we use the fact that the map Γ̃^{−1} is close to the identity:

u′ = y′ [1 + (\frac{y′}{y^{2}} )^{4} θ_{1} (x, −\frac{y′}{y^{2}} , \frac{y^{3}}{y′^{2}} )]

with θ_{1} holomorphic for (x, y, y′) ∈ Γ̃ (Δ̃(x_{0})) (thus bounded on Δ̃_{1}(x_{0})). Then

|K_{1}| = \frac{1}{4} |u′_{1}^{2} − 4u_{1}^{3}| < Const.ϵ_{1}^{−1}

With the substitution Q(t) = Q(t_{1}) + (t − t_{1}) + U(t), (3.77) can be written in integral form as

U(t) = ∫_{t_{1}}^{t} \frac{K_{1}Q(s)^{6}}{1 + \sqrt{1 + K_{1}Q(s)^{6}}} ds := J(U)(t) (3.78)

A straightforward calculation shows that the operator J in (3.78) is a contraction in the sup norm over the closed disk D′_{t_{1}} = {t : |t − t_{1}| ≤ 8ϵ_{1}^{1/2}}, in the ball ∥U∥_{∞} ≤ ϵ_{1}^{2} (if ϵ_{1} is small).

Furthermore, (t, u(t), u′(t)) ∈ Δ̃(x_{0}) for t ∈ D′_{t_{1}} (and small ϵ_{1}). Hence, from Proposition 1 (iii), y(x) := u(t) is a solution of (2.1) if t ∈ D′_{t_{1}} (the same solution as in the beginning of § 3.2).

To estimate the domain in the x-plane where y(x) is defined by the equivalence, we rely on the fact that Γ is close to the identity:

|x − x_{1}| = |γ (t, −\frac{u′(t)}{u(t)^{2}} , \frac{u(t)^{3}}{u′(t)^{2}} ) − γ (t_{1}, −\frac{u′_{1}}{u_{1}^{2}} , \frac{u_{1}^{3}}{u′_{1}^{2}} )| ≥

|t − t_{1}| − M_{2} max |\frac{u′(t)}{u(t)^{2}} |^{5} ≥ |t − t_{1}| − Const.ϵ_{1}^{5/2}

Thus, taking ϵ_{1} small enough, remark 7 follows.

Remark 8. y(x) satisfies the estimate

|y(x)| < ϵ_{1}^{−1} for |x − x_{1}| = 4ϵ_{1}^{1/2}

Proof. Indeed, since y(x) = u(t) = [Q(t_{1}) + (t − t_{1}) + U(t)]^{−2} (note that Q(t_{1})^{−2} = u_{1}), we have y(x) = [Q(t_{1}) + (x − x_{1}) + Y (x)]^{−2} where

|Y (x)| = |U(t) − [\frac{u′(t)}{u^{2}(t)}]^{5} γ_{1}(t, v(t), w(t)) + [\frac{u_{1}}{u_{1}^{2}} ]^{5} γ_{1}(t_{1}, v(t_{1}), w(t_{1}))|

≤ ϵ_{1}^{2} + Const.ϵ_{1}^{5/2}

since γ_{1} is analytic on Δ(x_{0}), therefore bounded on Γ^{−1} (Δ_{1}(x_{0})).

Thus, for x such that |x − x_{1}| = 4ϵ_{1}^{−1}, and small ϵ_{1},

|y(x)| ≤ |Q_{1} + (x − x_{1})|^{−2} (1 − |\frac{Y (x)}{Q_{1} + (x − x_{1})}|)^{−2} < ϵ_{1}^{−1}

which proves remark 8.

If x^{∗} ∈ D_{x_{1}} then, by remark 7, proposition 4 is proved. Otherwise, let x′_{1}, x_{2}, with x′_{1} ≺ x_{2}, be the two points of intersections of the circle ∂D_{x_{1}} with R_{θ}. Thus x′_{1} ≺ x_{1} ≺ x_{2} ≺ x^{∗} and |y(x_{2})| < ϵ_{1}^{−1}. The construction in step 1 is now repeated with x_{2} instead of x_{0} and the same ϵ_{1}: if |y(x)| ≤ ϵ_{1}^{−1} on [x_{2}, x^{∗}) then l is defined as the segment [x_{2}, x^{∗}]. Otherwise, let x_{3} be the least point w.r.t. ≺ on (x_{2}, x^{∗}) such that |y(x_{3})| = ϵ_{1}^{−1}. As before, y(x) is meromorphic in a neighborhood of D_{x_{3}} and satisfies |y(x)| < ϵ_{1}^{−1} on ∂D_{x_{3}}.

Then, either x^{∗} ∈ D_{x_{3}}, in which case the Proposition is proved, or else, if x′_{3} ≺ x_{4} are the two points of intersection of ∂D_{x_{3}} with R_{θ} then x′_{1} ≺ x_{1} ≺ x_{2} ≺ x_{3} ≺ x_{4} ≺ x^{∗}. We take the path l going along R_{θ}, from x_{0} towards x^{∗}, avoiding the disks D_{x_{2j+1}} by going on the upper semicircle of ∂D_{x_{2j+1}}.

After a finite number of steps the construction stops since |x_{2j+1} − x_{2j−1}| > |x_{2j} − x_{2j−1}| = 4ϵ_{1}^{1/2}.

Lemma 9. Let y(x) be a solution of (2.1), which is analytic at x_{0}. Then the radius of analyticity is at least

min {|y(x_{0})|^{−1/2}, |y′(x_{0})/2|^{−1/3}, |y(x_{0})^{2} + x_{0}/6|^{−1/4}}

Proof. Straightforward estimates of Taylor series coefficients; for details see [1].

3.3. Proof of Lemma 6. Suppose |y(x)| < M for x ∈ l[0, 1). Let x = l(s), s ∈ [0, 1). Equation (2.1) can be integrated once and it yields, for the solution y(x)

y′(x)^{2} = 4y(x)^{3} + 2xy(x) − 2 ∫_{l[0,s]} y(x′) dx′ + C_{0} (3.79)

Thus sup{|y′(x)| : x ∈ l[0, 1)} = 2M_{1} < ∞. By Lemma 9, for any x ∈ l[0, 1), y(x) is holomorphic in a disk centered at x of radius at least

ρ(x) = min {M^{−1/2}, M_{1}^{−1/3}, (M^{2} + max_{x∈l[0,1]} |x|/6)^{−1/4}}

Therefore y(x) is analytic at l(1). Acknowledgments The authors would like to thank Professor Martin Kruskal for very interesting discussions.

References

[1] Costin O. Correlation between pole location and asymptotic behavior for Painlevé I solutions MSRI Preprint No. 1997-094 (1997).

[2] Flashka H.; Newell A. C. Monodromy and spectrum preserving deformations. I Commun. Math. Phys., v. 76 pp. 67–116 (1980)

[3] Ince E. L. Ordinary differential equations. Dover (1944)

[4] Its A. R.; Novokshenov, V. Yu. Isomonodromic deformation method in the theory of Painlevé equations. Lecture notes in mathematics 1191, Springer-Verlag New York (1986)

[5] Joshi, N; Kruskal, M. D. A direct proof that solutions of the first Painlevé equation have no movable singularities except poles. Nonlinear evolution equations and dynamical systems (Baia Verde, 1991), 310–317, World Sci. Publishing, River Edge, NJ, 1992.

[6] Joshi, N; Kruskal, M. D. A direct proof that solutions of the six Painlevé equations have no movable singularities except poles. Stud. Appl. Math. 93 no. 3, 187–207 (1994).

[7] Mehta M. L. Random matrices 2nd ed. Academic Press, Boston (1991).

[8] Jimbo M.; Miwa T.; Ueno K. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I- Preprint RIMS-319 Kyoto (1980)

[9] Jimbo M.; Miwa T.; Ueno K. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II- Preprint RIMS-327 Kyoto (1980)

[10] Painlevé P. Sur les équations differentielles du second ordre et d’ordre superieur dont l’intégrale génerale est uniforme. Acta Mathematica v. 25 pp. 1–86 (1902).

Ovidiu Costin, Mathematics Department, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637

Current address: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070

E-mail address: costin@math.uchicago.edu; costin@msri.org

Rodica D. Costin, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070

E-mail address: rcostin@msri.org