Truncated Solutions of Painlevé Equation Pv
We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering...
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Інститут математики НАН України
2018
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| Цитувати: | Truncated Solutions of Painlevé Equation Pv / R.D. Costin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 38 назв. — англ. |
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| author | Costin, R.D. |
| author_facet | Costin, R.D. |
| citation_txt | Truncated Solutions of Painlevé Equation Pv / R.D. Costin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 38 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter, they represent tri-truncated solutions, analytic in almost the full complex plane, for a large independent variable. A brief historical note and references on truncated solutions of the other Painlevé equations are also included.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 117, 14 pages
Truncated Solutions of Painlevé Equation PV
Rodica D. COSTIN
The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA
E-mail: costin.10@osu.edu
URL: https://people.math.osu.edu/costin.10/
Received May 01, 2018, in final form October 25, 2018; Published online October 31, 2018
https://doi.org/10.3842/SIGMA.2018.117
Abstract. We obtain convergent representations (as Borel summed transseries) for the five
one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero
parameters, valid in half planes, for large independent variable. We also find the position
of the first array of poles, bordering the region of analyticity. For a special value of this
parameter they represent tri-truncated solutions, analytic in almost the full complex plane,
for large independent variable. A brief historical note, and references on truncated solutions
of the other Painlevé equations are also included.
Key words: Painlevé trascendents; the fifth Painlevé equation; truncated solutions; poles of
truncated solutions
2010 Mathematics Subject Classification: 33E17; 34M30; 34M25
1 Introduction
1.1 Historical notes
By the middle of the 19th century it became apparent that solutions of many linear differential
equations should be considered new, “special” functions. The natural question then arose: can
nonlinear differential equations have solutions that could be thought of being special functions?
Fuch’s intuition was that the answer is affirmative if “solutions have only fixed branch points
none of which depend on the initial conditions” [15]. This is now called the Painlevé property,
the fact that solutions of an equation are meromorphic on a common Riemann surface. Fuchs
studied first order equations having this property, concluding in 1884 that all such equations can
be solved in terms of previously known functions; his results were later extended by Poincaré [15].
The idea that absence of movable branch points would mean integrability was then used by
Sofie Kowalevski in the study of the rotation of a solid about a fixed point, for which she disco-
vered a third integrable case [29] (previous two cases being discovered by Euler and Lagrange),
a discovery for which she was awarded the Prix Bordin of the French Academy of Science in 1888.
Around the turn of the 20th century, Painlevé, Picard and Gambier studied nonlinear second
order differential equations, rational in y and y′, discovering that those possessing the Painlevé
property can be brought to fifty canonical forms. Of these fifty equations, all but six could be
solved in terms of earlier known functions. Painlevé went on to show that generic solutions of
the remaining six equations cannot be expressed in terms of earlier known functions, or in terms
of each other [32]. These equations are now known as the Painlevé equations, denoted PI up
to PVI, and their solutions as the Painlevé transcendents.
Having been discovered as a result of a purely theoretical inquiry, the Painlevé transcendents
have appeared later in modern geometry, integrable systems [1], statistical mechanics [21, 36, 37],
and recently in quantum field theory. Their practical importance makes it necessary to study
their properties in detail and to develop good numerical methods for their calculation [4].
This paper is a contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei
Kapaev. The full collection is available at https://www.emis.de/journals/SIGMA/Kapaev.html
mailto:costin.10@osu.edu
https://people.math.osu.edu/costin.10/
https://doi.org/10.3842/SIGMA.2018.117
https://www.emis.de/journals/SIGMA/Kapaev.html
2 R.D. Costin
1.2 Truncated solutions of Painlevé equations
It is known that generic Painlevé transcendents have poles in any sector towards infinity. But
there are special solutions, called tronquée (or truncated) which are free of poles in some sectors,
at least for large values of the independent variable. It was later conjectured that truncated solu-
tions have no poles whatsoever in these sectors, the Dubrovin–Novokshenov conjecture [12, 31].
For PI, Boutroux showed that there are five special sectors in the complex plane, each of
opening 2π/5, where solutions may lack poles. There are truncated solutions free of poles in
two adjacent such sectors, for large x. Among these, there are solutions free of poles in four
sectors – tri-truncated solutions [3].
In each sector, there is a one parameter family of truncated solutions, all asymptotic to
the same power series; these solutions can be distinguished by a parameter multiplying a small
exponential; a complete formal solution can be obtained as a transseries, mixing negative powers
of the variable and exponentially small terms. The form of the exponentially small terms as
well as the location of the first array of poles of truncated solutions were found in [9]. The
Dubrovin–Novokshenov conjecture was proved for PI [11].
The Stokes constant (which can be viewed as a complex number, attached to the equation,
controlling the relation between the exponentially small terms associated to different sectors)
was calculated for the tritronquée solution for PI for the first time by Kapaev [24], see also [28],
using the isomonodromy method. Kapaev then obtained the complete description of the global
asymptotic behavior of the tronquée solutions of PI and connection formulae [26]. The Stokes
constant was re-calculated later using WKB methods [35], and again later by continuation of
a tri-truncated solution through the sector with poles [10].
More recently, Kapaev considered the study of the tronquée solution of the PI2 equation (the
second member of the PI hierarchy); a detailed global asymptotic analysis of the trintronqués
solutions is found in [14].
For PII Boutroux showed that there are six special sectors, with truncated solutions free of
poles in two adjacent sectors, and tri-truncated solutions free of poles in four sectors. Using
the Riemann–Hilbert approach, Kapaev gave a complete description of the global asymptotic
behavior of the tronquée solutions of PII, together with all relevant connection formulae [20, 27],
see also Chapter 11 of the monograph [13]. Existence of tri-truncated solutions of the PII
hierarchy was shown in [23]. The Dubrovin–Novokshenov conjecture was established for the
Hastings–McLeod solution (tri-truncated in pairs of non-adjacent sectors) [18].
For PIV, a quite detailed global asymptotic analysis of its tronquée solutions was obtained in
Kapaev’s work [25]. The connection formulae were found by Its and Kapaev [19], based on the
Riemann–Hilbert isomonodromy method. Existence of truncated solution fo PIII and PIV was
re-established in [30] following methods in [22], and using a different method in [38], where the
location of the first array of poles was also found. An overview of PVI is contained in [17]; see
also [8]. The truncated solutions of the fifth Painlevé equation are the subject of the present
article. It is to be noted that fixed singularities of PV can only be located at 0 and ∞.
1.3 Truncated solutions of Painlevé equation PV
The fifth Painlevé equation
PV(α, β, γ, δ) :
w′′ =
(
1
2w
+
1
w − 1
)
w′
2 − w′
x
+
(w − 1)2
x2
(
αw +
β
w
)
+
γw
x
+
δw(w + 1)
w − 1
(1.1)
is known to be reducible to PIII if δ = 0 [16].
Truncated Solutions of Painlevé Equation PV 3
We assume αβδ 6= 0. The following algebraic behaviors towards infinity are then possible for
solutions of (1.1) [33]:
I. w = ±
√
β
δ
x−1 +O
(
x−2
)
, x→∞,
II. w = ±
√
δ
−α
x+O(1), x→∞,
III. w = −1 +O
(
x−1
)
, x→∞.
It is known that these five families represent asymptotic behaviors of truncated solutions,
analytic in (almost) a half plane for large |x|; the position of the half plane is determined by the
exponentially small terms [2, 34].
In the present paper we express the exponentially small terms using a full formal solution
(transseries), Borel summable to actual solutions. This yields one-parameter families of trun-
cated solutions as series which converge in appropriate (almost) half-planes and large |x|. For
special values of the parameter we obtain tri-truncated transcendents. Moreover, we find the
location of the first array of poles beyond the sector of analyticity.
We use techniques used before in [9]. For completeness, the statements of the theorems used
here are included in Appendix A.
1.4 Relations between different truncated solutions
Remark 1.1 ([16]).
(i) If w(x) satisfies PV(α, β, γ, δ), then 1/w(x) satisfies PV(−β,−α,−γ, δ).
(ii) If w(x) solves PV(α, β, γ, δ) then w(x/λ) solves PV
(
α, β, γλ, δλ2
)
, for any λ 6= 0.
By Remark 1.1(i), truncated solutions in the family II. are obtained as reciprocals of truncated
of the family I.
By Remark 1.1(ii), if w(x) solves PV(α, β, γ, δ) with w(x) ∼
√
β
δ x
−1, x → ∞, then w(−x)
solves PV(α, β,−γ, δ) and satisfies w(−x) ∼ −
√
β
δ x
−1, x→∞.
We can assume any nonzero value for δ, by rescaling x and using Remark 1.1(ii).
It is interesting to note that there are other Bäcklund transformations as explained in [16,
Theorem 39.2]. These transform truncated solutions of the first two families into truncated
solutions of the first two families, and truncated solutions in the family III into solutions in the
same family.
Due to these relations, it suffices to obtain results for the truncated Painlevé transcendents
satisfying
I0 w =
√
β/δx−1 +O
(
x−2
)
, x→∞, for δ = −1/2
and for
III0 w = −1 +O
(
x−1
)
, x→∞ for δ = 2.
2 Truncated solutions in the family I0
In this section we state the main results; their proofs are found in Section 4.
4 R.D. Costin
Theorem 2.1. Assume αβγ 6= 0. Let w(x) be a solution of PV
(
α, β, γ,−1
2
)
such that there
exists some φ ∈
(
−π
2 ,
π
2
)
so that
w(x) =
√
−2βx−1(1 + o(1)) as x→∞ along arg x = −φ. (2.1)
Then (2.1) holds along any φ ∈
(
−π
2 ,
π
2
)
.
(i) Furthermore, w(x) is asymptotic to a unique power series solution:
w(x) ∼ w̃0(x) =
∞∑
n=1
w0;nx
−n as x→∞ along any arg x = −φ,
where |φ| < π
2 , w0;1 =
√
−2β.
(ii) The complete formal solution (transseries) along any half-line arg x = −φ with 0 < |φ| < π
2
is
w̃(x) = w̃0(x) +
∞∑
k=1
Cke−kxx−qkw̃k(x) where w̃k(x) =
∞∑
n=0
wn;kx
−n, (2.2)
C is an arbitrary constant and
q = γ + 2
√
−2β.
(iii) w(x) has a Borel summed transseries representation: there exist C±, where C+ − C− is
a multiple of the Stokes constant, such that for any ε > 0 there is Rε > 0
w(x) =
w0(x) +
∞∑
k=1
Ck+e−kxx−qkw+;k(x) for arg x ∈
(
0, π2 − ε
)
, |x| > Rε,
w0(x) +
∞∑
k=1
Ck−e−kxx−qkw−;k(x) for arg x ∈
(
−π
2 + ε, 0
)
, |x| > Rε,
(2.3)
where w0(x), w±;k(x) are the Borel sums of w̃0(x), w̃k(x) along half-lines eiφR+ (where
φ = − arg x). For arg x = 0 the transseries is Écalle–Borel summable.1 The function
series (2.3) converge for x satisfying |x| > R and |C±e−xx−q| < µ for suitable µ > 0 and
R > 0. In particular, if Re q > 0 then w(x) is analytic in the right half plane for |x| > R
(for some R > 0).
(iv) If Re q > 0 the constants C± can be determined from
w(x) ∼
bRe qc∑
n=1
w0;nx
−n + C+x
−qe−x as x→ +i∞
and
w(x) ∼
bRe qc∑
n=1
w0;nx
−n + C−x
−qe−x as x→ −i∞.
Corollary 2.2 (existence of tri-truncated solutions). Consider the unique truncated solution
w(x) as in Theorem 2.1 with C+ = 0. Then w(x) is analytic for large |x| in the left-half plane,
for arg x ∈
(
−π
2 + ε, 3π2 − ε
)
, |x| > Rε (for any ε > 0).
Similarly, the unique truncated solution with C− = 0 is analytic for |x| > Rε with arg x ∈(
−3π
2 + ε, π2 − ε
)
. If Re q > 0 then we can take ε = 0.
1It is obtained by special averages of Laplace transforms in the upper and the lower half plane [5].
Truncated Solutions of Painlevé Equation PV 5
Remark 2.3. Truncated solutions in the left half-plane, satisfying
w(x) = −
√
−2βx−1(1 + o(1))
(
x→∞ along arg x = −φ, |φ+ π| < π
2
)
form a one-parameter family, which, by Remark 1.1 are obtained by replacing (x, γ) with
(−x,−γ) in the representation given by Theorem 2.1.
Truncated solutions w(x) as in Theorem 2.1 with C+ 6= 0 develop arrays of poles. The
following result shows the position of the array of poles closest to iR+.
Theorem 2.4. Assume αβγ 6= 0 and
2α 6=
(√
−2β − q − 1
)2
.
Let w(x) be as in Theorem 2.1 with nonzero constant C+ in (2.3). Them w(x) has two arrays
of poles located at
xn;1,2 = 2nπi +
(
γ + 2
√
−2β + 2
)
ln(2nπi) + lnC+ − ln ζ1,2 + o(1), n→ +∞,
where
ζ1,2 =
2√
−2β
1
√
−2β − q − 1±
√
2α
. (2.4)
3 Truncated solutions in the family III0
In this section we state the main results; their proofs are found in Section 5.
Theorem 3.1. Assume αβγ 6= 0. Let w(x) be a solution of PV(α, β, γ, 2) such that there exists
some φ ∈
(
−π
2 ,
π
2
)
so that
w(x) = −1 + o(1) as x→∞ along arg x = −φ. (3.1)
Then (3.1) holds along any φ ∈
(
−π
2 ,
π
2
)
.
(i) Furthermore, w(x) is asymptotic to a unique power series solution:
w(x) ∼ −1 + w̃0(x) = −1 +
∞∑
n=1
w0;nx
−n x→∞ along any arg x = −φ,
where |φ| < π
2 .
(ii) The complete formal series solution (transseries) along any half-line in the right half-plane
has the form
w̃(x) = w̃0(x) +
∞∑
k=1
Cke−kxx−k/2w̃k(x),
where C is an arbitrary constant, w̃k(x) are series in x−n, n ∈ N.
(iii) There are unique constants C± so that
w(x) ∼ −1 + C+x
−1/2e−x as x→ +i∞ (3.2)
and
w(x) ∼ −1 + C−x
−1/2e−x as x→ −i∞
and C+ − C− is a constant which does not depend on the particular solution (and it is
a multiple of the Stokes constant of the equation).
6 R.D. Costin
(iv) w(x) is analytic in the right half plane for |x| > R (for some R > 0) and has the Borel
summed transseries representation
w(x) =
w0(x) +
∞∑
k=1
Ck+e−kxx−k/2w+;k(x) for arg x ∈
(
0, π2
]
,
w0(x) +
∞∑
k=1
Ck−e−kxx−k/2w−;k(x) for arg x ∈
[
−π
2 , 0
)
,
(3.3)
where w0(x), w±;k(x) are Laplace transforms on the half-lines eiφR+ (where φ = − arg x
is between 0 and ±π
2 ) of the Borel transforms of the series w̃0(x), w̃k(x). The domain of
convergence of the series in (3.3) is the set of all x with |x| > R and
∣∣C±e−xx−1/2
∣∣ < µ
for some suitable R,µ > 0.
Corollary 3.2 (existence of tri-truncated solutions). Consider the unique truncated solution
w(x) as in Theorem 3.1 with C+ = 0. Then w(x) is analytic for large |x| also in the left-half
plane, for arg x ∈
(
−π
2 + ε, 3π2 − ε
)
.
Similarly, the unique truncated solution with C− = 0 is analytic for large |x| with arg x ∈(
−3π
2 + ε, π2 − ε
)
.
Remark 3.3. Truncated solutions in the left half-plane, satisfying
w(x) = −1 + o(1)
(
x→∞ along arg x = −φ with 0 < |φ+ π| < π
2
)
form a one-parameter family, which, by Remark 1.1 are obtained by replacing (x, γ) with
(−x,−γ) in the representation given by Theorem 3.1.
Truncated solutions w(x) as in Theorem 3.1 with C+ 6= 0 have arrays of poles near iR+. The
following result shows the position of the closest array of poles.
Theorem 3.4. Assume αβγ 6= 0. Let w(x) be as in Theorem 3.1 with nonzero constant C+
in (3.2) and (3.3). Them w(x) has an array of poles located at
xn = 2nπi− 1
2
ln(2nπi)− iπ
2
− ln(−C+/2) + o(1), n→∞.
We conjecture that there are, in fact, two poles near each xn.
4 Proofs for family I0
4.1 Proof of Theorem 2.1
Denote
m =
√
−2β, q = γ + 2m.
Then w(x) satisfies
w′′ =
(3w − 1)w′2
2w(w − 1)
− w′
x
+
αw(w − 1)2
x2
− m2
2
(w − 1)2
x2w
+
(q − 2m)w
x
− 1
2
w(w + 1)
w − 1
(4.1)
for which we study solutions satisfying w = mx−1 +O
(
x−2
)
as x→∞.
To normalize the equation, substitute
w(x) =
m
x
(
1− q
x
+ u(x)
)
, (4.2)
Truncated Solutions of Painlevé Equation PV 7
which transforms (4.1) into
u′′ =
(
1 +
2q
x
)
u+ f
(
x−1, u, u′
)
, (4.3)
where f is analytic at (0, 0, 0) and f = O
(
x−2
)
+ O
(
u2
)
+ O
(
u′2
)
+ O(uu′). To show that this
a normal form for second order equations, we turn it into a first order system by substituting
u(x) = y1(x) + y2(x), u′(x) =
(
−1− q
x
)
y1(x) +
(
1 +
q
x
)
y2(x), (4.4)
upon which (4.3) is turned into a first order system in normal form (A.1):
d
dx
[
y1
y2
]
=
−1− q
x
0
0 1 +
q
x
[y1
y2
]
+ g
(
x−1, y1, y2
) [ 1
−1
]
, (4.5)
where
g
(
x−1, y1, y2
)
=
1
2
x
x+ q
[
−f
(
x−1, y1 + y2,
(
1 +
q
x
)
(y2 − y1)
)
+
q(q + 1)
x2
y1 +
q(q − 1)
x2
y2
]
. (4.6)
There are general theorems that can be applied for differential equations in normal form;
these are presented, for convenience, in the Appendix A: applying Theorem A.3 to the system
(4.5), (4.6) and then reverting the substitutions (4.4), (4.2), Theorem 2.1 follows.
4.2 Proof of Theorem 2.4
Searching for asymptotic expansions of the form (A.8) setting ξ = Ce−xx−q, plugging in an
asymptotic series u(x) ∼ F0(ξ) + 1
xF1(ξ) + 1
x2
F2(ξ) + · · · in (4.3) and expanding under the
assumption that ξ � x−k for all k we obtain that all Fn are polynomials:
F0(ξ) = ξ, F1(ξ) = c1ξ,
F2(ξ) = m(m− q − 1)ξ2 + c2ξ − 1
2m
2 + 3
2q
2 − a+ 1
2 , . . . ,
where c1, c2 are uniquely determined in terms of the parameters. By Theorem A.4 we have
u(x) ∼
∞∑
m=0
x−mFm(ξ(x)) for x→∞ with arg x ∈
[
−π
2 + δ, π2 − δ
]
, |ξ(x)| < δ1 (4.7)
for some δ1 > 0.
On the other hand, F0 has no singularities, so Theorem A.6 yields no additional information
on the position of the first array of poles, beyond the right half plane.
A direct calculation of the first few terms of the transseries (2.2) unveils a non-generic struc-
ture, namely that wn,k = 0 for all k = 0, 1, . . . , n − 1 (at least for n = 1, . . . , 4). This peculiar
structure suggests (by formal re-arrangement of the transseries) to use instead the second scale
ζ = Ce−xx−q−2
and look for an expansion of the form
u(x) ∼ x2Φ(ζ) +
∞∑
n=−1
1
xn
Φn(ζ), ζ(x)� x−k for all k, (4.8)
8 R.D. Costin
where Φ(ζ) = ζ + O
(
ζ2
)
when ζ → 0 and Φn are analytic at ζ = 0, and for x so that |ζ| =∣∣Ce−xx−q−2
∣∣ < µ, where µ is small enough (to be determined).
Introducing the formal expansion (4.8) in (4.3) and expanding in powers of x−1, we obtain
that Φ(ζ) must satisfy
ζ2Φ′′ + ζΦ′ +
1
2
Φ = αm2Φ3 +
3
2
ζ2Φ′2
Φ
having the general solution
Φ(ζ) =
2ζ/C1
8αm2/C2
1 − (ζ − C2)2
, (4.9)
where C1,2 are determined from the condition that Φ(ζ) = ζ+O
(
ζ2
)
and that Φ−1(ζ) be analytic
at ζ = 0, yielding
C1 = −2β
[
2α− (m− q − 1)2
]
, C2 = − 2
m
m− q − 1
2α− (m− q − 1)2
. (4.10)
With the notation (2.4) (recall that m =
√
−2β) and using (4.9), then (4.10) becomes
Φ(ζ) =
ζζ1ζ2
(ζ − ζ1)(ζ − ζ2)
.
Letting µ < min |ζ1,2|, the next term in the expansion is
Φ−1(ζ) =
ζ(2ζ − ζ1 − ζ2)C3
(ζ − ζ2)2(ζ − ζ1)2
+
ζ
(
ζ2 − ζ1ζ2
)
C4
(ζ − ζ2)2(ζ − ζ1)2
(4.11)
− 1
2a
ζ
(
ζ1ζ2
√
2(ζ1 + ζ2)a
3/2 + (ζ1 − ζ2)
(
− 1√
2
(ζ1 − ζ2)
√
a+ aζ2ζ1
))
(ζ1 − ζ2)
(ζ − ζ2)2(ζ − ζ1)2
,
where C3,4 are determined from the condition that the next term, Φ0, be analytic at ζ = 0.
To justify that the actual truncated solution w(x) is also singular near ζ1 and near ζ2, we
first note that, since ζ = ξx−2 and u(x) ∼ x2Φ then
x2Φ(ζ) =
ξ(
1− x−2ξζ−11
)(
1− x−2ξζ−12
) = ξ +
1
x2
ξ
(
ζ−11 + ζ−12
)
+ · · · ,
which is a convergent series in powers of x−2. Also xΦ−1(ζ) has a similar convergent expansion.
Therefore x2Φ(ζ) = F0(ξ)+O
(
x−2
)
and, in view of (4.11), xΦ−1(ζ) = O
(
x−1
)
therefore (4.7)
implies that
u(x) ∼ x2Φ(ζ) +O
(
x−1
)
for x → ∞ in the same region where (4.7) holds (and |x| > 1, |ζ| < µ). The same argument
as in [7, Section 4.6] implies that u(x) has singularities within o(1) distance of the singularities
of Φ(ζ).
5 Proofs for the family III0
The existence of a unique power series formal solution w̃0(x) = −1 + o(1) is established by
standard techniques.
Truncated Solutions of Painlevé Equation PV 9
It is also relatively algorithmic to obtain the form of the transseries. Since the procedure
may not be well known some details are provided here, also illustrating why δ = 2 is a natural
choice.
Plugging in a formal solution of the type w̃(x) = w̃0(x) + εg(x) +O
(
ε2
)
in the equation (1.1)
and expanding in power series in ε, the coefficient of ε0 vanishes, since w̃0(x) is already a formal
solution. Next, the coefficient of ε is a linear second order differential equation for g(x), with
coefficients given in terms of w̃0(x) and its derivatives. It has the form
g′′(x) +
1
x
g′(x)− δ
2
g(x) =
1
x2
R
(
x−1, g, g′
)
with two independent solutions
g±(x) = x−1/2 exp
(
± x
√
δ/2
)(
1 +O
(
x−1
))
,
where we see that it is convenient to take δ = 2.
Let w(x) satisfy the assumptions of Theorem 3.1. To normalize the equation, substitute
w(x) = −1 +
γ
x
+ u(x),
so that u(x) = O
(
x−2
)
and satisfies an equation of the form
u′′ =
(
1− 1
x
)
u+ f
(
x−1, u, u′
)
,
where f is analytic at (0, 0, 0) and f = O
(
x−2
)
+O
(
u2
)
+O
(
u′2
)
+O(uu′). This is a normal form
for a second order equation, by the argument in Section 4.1: the normalizing transformation
is (4.4) with q = −1/2 and the normal form as a first order system is of the type (4.5) with
q = −1/2. Theorem A.3 applies, and reverting the substitutions, Theorem 3.1 follows.
5.1 The first array of poles: proof of Theorem 3.4
Setting ξ = Ce−xx−1/2, plugging in P(α, β, γ, 2) an expansion (A.8) and expanding, it follows
that F0(ξ) must satisfy
ξ2F ′′0 + ξF ′0 =
ξ2(3F0 − 4)F ′20
2(F0 − 2)(F0 − 1)
+ 2
F0(F0 − 1)
F0 − 2
,
whose unique solution analytic at ξ = 0 with F0(ξ) = ξ +O
(
ξ2
)
, ξ → 0, is
F0(ξ) =
ξ
(ξ/4 + 1)2
.
Let xn be solutions of ξ(x)/4 + 1 = 0. By Theorem A.6, if xn solve Ce−xx−1/2 = −4, then
solutions u(x) have singularities located at xn + o(1) for large n and Theorem 3.4 follows.
Remark 5.1. While F0(ξ) has a pole of order two, it is known that the poles of PV have order
one if α 6= 0. This suggests that there are two array of poles within O
(
n−1
)
distance from
each other, as the expansion in x−1 would collapse them into a double pole. The author has
no rigorous argument at this time that this is indeed the case, and it is formulated here as
a conjecture.
10 R.D. Costin
A Results used
A.1 Summation of transseries formal solutions
Theorem A.1 ([5]). Consider the (nonlinear) system of first order differential equations:
y′ +
(
Λ− 1
x
A
)
y = g(x−1,y), y ∈ Rd, (A.1)
with g = O
(
x−2
)
+O
(
|y|2
)
for x→∞, |y| → 0 and Λ = diag(λ1, . . . , λd), A = diag(α1, . . . , αd).
Assume the following non-resonance condition: any collection among λ1, . . . , λd which are in the
same open half-plane are linearly independent over Z.
Assume, for simplicity, that the independent variable, x, is scaled so that λ1 = 1.
(i) Then (A.1) has formal solutions
ỹ = ỹ(x;C) = ỹ0(x) +
∑
k∈Nd\0
Cke−λ·kxỹk(x) with ỹk(x) = xα·ks̃k(x), (A.2)
where s̃k(x) are integer power series in x−1. A formal solution (A.2) is a transseries for
x → ∞ along any direction along which all the exponentials present are decaying, i.e.,
along any direction in the sector
Strans = {x ∈ C | Re(λjx) > 0 for all j with Cj 6= 0}. (A.3)
(ii) A transseries is Borel summable for large x along any direction of argument φ, where the
sector a1 < arg x < a2 contains only one Stokes line, arg x = 0. The Borel sum is an
actual solution: for any ε > 0 there is Rε > 0 so that
y(x) =
LφY0(x) +
∑
k∈Nd\0
C+
ke−λ·kxLφYk(x), −φ = arg x ∈ (0, a2−ε), |x| > Rε,
LφY0(x) +
∑
k∈Nd\0
C−
ke−λ·kxLφYk(x), −φ = arg x ∈ (a1+ε, 0), |x| > Rε,
(A.4)
where Yk = Bφỹk (the analytic continuation of the Borel transform of ỹk along the di-
rection of argument φ). Along arg x = 0 balanced averages of LφYk sum to the solu-
tion y(x) (Écalle–Borel summation). The solution y(x) is analytic for |x| > Rε with
arg x ∈ (a1 + ε, a2 − ε). Conversely, any solution asymptotic to ỹ0 for x → +∞ has
a representation (A.4).
(iii) Only the first component C1 of the constant beyond all orders in (A.4) changes when arg x
crosses the Stokes line arg x = 0, corresponding to λ1 = 1. The change of this constant
depends only on the equation: it is a multiple of the first Stokes constant.
Note A.2. More recent results showed that the region of convergence of (A.4) is in fact given
by conditions of the form |x| > R and
∣∣Cie−λixxαi
∣∣ < µi, i = 1, . . . , d for suitable constants µi
and R (Ci = 0 if Reλi > 0) [7], see also [6].
We use Theorem A.1 in the particular case when d = 2, λ1 = 1, λ2 = −1, in which case the
sectors (A.3) become the right (respectively left) half plane and the constants have the form
C = (C, 0) (respectively C = (0, C)); we also assume that solutions have only poles as moving
singularities, as it is the case for the Painlevé equations (otherwise domains of analyticity would
have to extend on Riemann sheets). Theorem A.1 takes the following simpler formulation:
Truncated Solutions of Painlevé Equation PV 11
Theorem A.3. Consider the two-dimensional system of first order differential equations (A.1)
with Λ = diag(1,−1), A = diag(α1, α2). Then:
(i) The system has a one parameter family of transseries solutions
ỹ = ỹ(x;C) = ỹ0(x) +
∑
k≥1
Cke−kxỹk(x), C ∈ C, along arg x = −φ, |φ| < π
2 , (A.5)
where ỹk(x) = xkα1 s̃k(x) = xkα1
∞∑
n=0
s̃k,nx
−n and ỹ0(x) = O
(
x−2
)
.
(ii) For any C the formal solution (A.5) is Borel summable for large x along any direction
of argument φ = − arg x if 0 < |φ| < π
2 . (Along arg x = 0 it is Écalle–Borel summable.)
The Borel sum is an actual solution: for any constant C+, and any ε > 0 there exists C−
(C+ − C− is a multiple of the Stokes constant, [9]) and Rε > 0 such that
y(x) =
LφY0(x) +
∑
k≥1
Ck+e−kxLφYk(x), −φ = arg x ∈
(
0, π2 − ε
)
, |x| > Rε,
LφY0(x) +
∑
k≥1
Ck−e−kxLφYk(x), −φ = arg x ∈
(
−π
2 + ε, 0
)
, |x| > Rε.
(A.6)
The domain of convergence of the function series (A.6) given by{
x | |x| > R,
∣∣C±e−xxα1
∣∣ < µ
}
for µ > 0 small enough and R > 0 large enough. Conversely, any solution asymptotic
to ỹ0 for x→ +∞ has a representation (A.6) for some C±.
(ii) Similarly, the system has a transseries solution
ỹ = ỹ(x;C) = ỹ0(x) +
∑
k≥1
Ckekxỹk(x) for arg x = −φ, 0 < |φ+ π| < π
2
, (A.7)
where ỹk(x) = xkα2 s̃k(x) with s̃k(x) are integer power series.
Statements similar to those in (i) hold regarding Borel summability (A.7) along any direction
of argument arg x 6= π in the left half plane, and Écalle–Borel summability along arg x = π, the
sum being an actual solution, analytic in a domain |x| > R and
∣∣C±exxα2
∣∣ < µ for some µ > 0
and R > 0. Conversely, any solutions asymptotic to ỹ0 for x → −∞ is the Borel sum of such
a transseries.
A.2 Arrays of singularities bordering the sector of analyticity
Theorem A.1 establishes existence of solutions, in one-to-one correspondence with formal transs̄e-
ries solutions, and which are analytic for large x in the sector (A.3) where these formal solutions
are defined (i.e., they are well-ordered with respect to �). In [7] it is further shown that on the
boundary of the sector of analyticity, these solutions develop arrays of singularities.
In the particular case when d = 2, λ1 = 1, λ2 = −1 of this paper the results in [7] are as
follows. Denote
ξ = ξ(x) = Ce−xxα1 .
For x near iR+, i.e., when ξ � x−k for all k > 0, the transseries (A.5) can be formally reordered
as ∑
k≥0
ξks̃k,0(x) +
1
x
∑
k≥0
ξks̃k,1 +
1
x2
∑
k≥0
ξks̃k,2 + · · · .
It turns out that the series in ξ are convergent, and the resulting expansion is asymptotic to
the solution the original transseries summed to:
12 R.D. Costin
Theorem A.4 ([7]). Let δ, c > 0. There exists δ1 > 0 so that for |ξ| < δ1 the power series
Fm(ξ) =
∞∑
k=0
ξks̃k,m, m = 0, 1, 2, . . .
converge. Furthermore
y(x) ∼
∞∑
m=0
x−mFm(ξ(x)) for x→∞ with arg x ∈
[
−π
2 + δ, π2 − δ
]
, |ξ(x)| < δ1. (A.8)
The asymptotic series is uniform, it is differentiable and satisfies Gevery-like estimates.
Note A.5. F0(0) = 0 and F′0(0) = 1.
In fact (A.8) is valid in a larger domain, up to distance o(1) of the singularities of F0 (Fm with
m > 0 can have no other singularities). In general F0 has branch point singularities, and
a Riemann surface needs to be considered. But Painlevé equations have no movable branch
points, so the singularities of F0 can only be poles. Then for simplicity we state here the general
result of [7] in this case only.
Let ρ1,2 so that the small term g in (A.1) is analytic in the polydisk
∣∣x−1∣∣ < ρ1, |y| < ρ2.
Let Ξ be a finite set (possibly empty) of poles of F0. Let D ⊂ C \ Ξ be open, connected,
relatively compact, containing |ξ| < δ1, so that F0 is analytic in an ε-neighborhood of D (ε > 0),
so that sup
D
|F0(ξ)| = ρ3 < ρ2.
We need the counterpart of D in the x-plane. Let
X = ξ−1(D) ∩
{
|x| > R, arg x ∈
[
−π
2 + δ, π2 − δ
]}
.
Theorem A.6 ([7]). All Fm with m ≥ 1 are analytic on D and for R large enough the asymptotic
expansion (A.8) holds for x→∞ with x ∈ X.
Furthermore, assume that F0 is singular at ξs ∈ D. Then y(x) is singular near points x with
ξ(x) = ξs, more precisely at
xn = 2nπi + α1 ln(2nπi) + lnC − ln ξs + o(1) as n→∞.
Acknowledgements
The author is grateful to the editors for valuable references and information, and to the referees
careful reading of the manuscript and for their helpful comments.
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1 Introduction
1.1 Historical notes
1.2 Truncated solutions of Painlevé equations
1.3 Truncated solutions of Painlevé equation PV
1.4 Relations between different truncated solutions
2 Truncated solutions in the family I0
3 Truncated solutions in the family III0
4 Proofs for family I0
4.1 Proof of Theorem 2.1
4.2 Proof of Theorem 2.4
5 Proofs for the family III0
5.1 The first array of poles: proof of Theorem 3.4
A Results used
A.1 Summation of transseries formal solutions
A.2 Arrays of singularities bordering the sector of analyticity
References
|
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| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Costin, R.D. 2025-11-27T14:47:47Z 2018 Truncated Solutions of Painlevé Equation Pv / R.D. Costin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 38 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E17; 34M30; 34M25 arXiv: 1804.11273 https://nasplib.isofts.kiev.ua/handle/123456789/209840 https://doi.org/10.3842/SIGMA.2018.117 We obtain convergent representations (as Borel summed transseries) for the five one-parameter families of truncated solutions of the fifth Painlevé equation with nonzero parameters, valid in half planes, for large independent variable. We also find the position of the first array of poles, bordering the region of analyticity. For a special value of this parameter, they represent tri-truncated solutions, analytic in almost the full complex plane, for a large independent variable. A brief historical note and references on truncated solutions of the other Painlevé equations are also included. The author is grateful to the editors for valuable references and information, and to the referees careful reading of the manuscript and for their helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Truncated Solutions of Painlevé Equation Pv Article published earlier |
| spellingShingle | Truncated Solutions of Painlevé Equation Pv Costin, R.D. |
| title | Truncated Solutions of Painlevé Equation Pv |
| title_full | Truncated Solutions of Painlevé Equation Pv |
| title_fullStr | Truncated Solutions of Painlevé Equation Pv |
| title_full_unstemmed | Truncated Solutions of Painlevé Equation Pv |
| title_short | Truncated Solutions of Painlevé Equation Pv |
| title_sort | truncated solutions of painlevé equation pv |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209840 |
| work_keys_str_mv | AT costinrd truncatedsolutionsofpainleveequationpv |