Advances in Pure Mathematics, 2011, 1, 170-183
doi:10.4236/apm.2011.14031 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Non-integrability of Painlevé V Equations in the Liouville
Sense and Stokes Phenomenon
Tsvetana Stoyanova
Tsvetana Stoyanova, Department of Mathematics and Informatics, Sofia University, Sofia, Bulgaria
E-mail: cveti@fmi.uni-sofia.bg
Received March 15, 2011; revised May 2, 2011; accepted May 15, 2011
Abstract
In this paper we are concerned with the integrability of the fifth Painlevé equation () from the point of
view of the Hamiltonian dynamics. We prove that the Painlevé equation (2) with parameters
for arbitrary complex
V
P
V
0
=0, =

(and more generally with parameters related by Bäclund
transformations) is non integrable by means of meromorphic first integrals. We explicitly compute formal
and analytic invariants of the second variational equations which generate topologically the differential
Galois group. In this way our calculations and Ziglin-Ramis-Morales-Ruiz-Simó method yield to the
non-integrable results.
Keywords: Differential Galois theory, Painlevé V equation, Hamiltonian Systems, Stokes Phenome-
nonAsymptotic Theory
1. Introduction
The six Painlevé equations (
I
VI ) were introduced
and first studied by Paul Painlevé [1] and his student B.
Gambier [2] who classified all the rational differential
equations of the second order
PP
2
2
dd
=,,
d
d
yy
Rtyt
t



free of movable critical points. The solutions of these
equations define some new functions, the so-called
Painlevé transcendents or Painlevé functions.
Although the Painlevé equations were discovered from
strictly mathematical considera-tions they have recently
appeared in several physical applications. Among field-
theoretical problems which can be solved in terms of the
considered below Painlevé V transcendent we mention
the two-point correlation functions at zero temperature
for the one-dimensional impenetrable Bose gas (Jimbo,
Miwa, Mori, Sato [3]).
In the present article we deal with the fifth Painlevé
equation, PV,

 
22
22
0
2
1
11 1
=21 2
1
11
y
yyyy
y
yt
t
yy
y
ty


 


 
 
 
where t
and 0
,,,

are arbitrary complex
parameters. It is well known that when =0,= 1
equation (1) can be solved by quadratures, [4]. If
1
and =0
the fifth Painlevé equation (1) is
equivalent to the third Painlevé equation
I
II . In this
paper we investigate the generic case of V when
P
P
0
. Hence, by replacing by t2t
one can
normalize the constant as =12
[5], and so we
consider

 

22
22
0
2
1
11 1
=21 2
1
121
y
yyyy
yy ty
t
yy
y
ty


 


 
 
 
(2)
As the Painlevé equations can be written as time-
dependent Hamiltonian systems of 112 degrees of
freedom (see Malnquist [6] and Okamoto [5]) their
integrability should be considered in the context of
Hamiltonian systems. We recall that by this we mean the
existence of enough meromorphic first integrals (in our
case-two). In [7] Morales-Ruiz raises the question about
the integrability of the Painlevé transcendents as Hamil-
tonian systems. Later Morales-Ruiz in [8], and Stoya-
nova and Christov in [9] obtain a non-integrable result
for Painlevé II family. Non-integrability of the Painlevé
y
(1)
T. STOYANOVA
171
VI equation for some particular values of the parameters
is proved by Horozov and Stoyanova in [10] and by
Stoyanova in [11]. In the present note we continue the
study of Painlevé transcendents with the fifth Painlevé
equation and obtain an analogous result for one family of
the parameters. Our method uses the differential Galois
approach to non-integrability of Hamiltonian systems [12]
which is an extension of the Ziglin theory [13, 14]. In
particular, studying the differential Galois group of the
first and second variational equations along a particular
rational non-equilibrium solution we can find non-
integrable results. It appears that the corresponding
variational equations have an irregular singularity and
new difficulty have to be overcome.
Our main result is the following theorem:
main Assume that 0 where =0, =

is an
arbitrary complex parameter. Then the fifth Painlevé
equation (2) is not integrable. We chose to investigate
V (2) for these values of the parameters because then
the Hamiltonian system (11) possesses a simple rational
solution. The point is our method requires one single-
valued solution.
P
By Bäcklund transformations of V we can extend
the result of main for an infinite subfamily of V (2):
genm Assume that where m is even
and at least one 0 is integer. Then the fifth
Painlevé equation (2) is not integrable.
P
P
0=m
 

,

The paper is organized as follows. In section 2 we
recall the main results of the Ziglin-Ramis-Morales-Ruiz
-Simó theory of non-integrability of the Hamiltonian
systems, of the differential Galois theory and asymtotic
theory for ordinary differential equations needed in the
proofs. In section 3 we prove the non-integrability of the
fifth Painlevé equation (2) for 0
=0, =

and
(witt). In n1 we prove the non-integrability of
V (2) for 0. In section 4 using
Bäcklund transformations of Painlevé equation we
extend the results of section 3 to the entire orbit of the
parameters and establish the main results of this article.
*

P=0,==1

V
2. Preliminaries
2.1. Non-Integrability and Differential Galois
Theory
In this section we briefly recall Ziglin-Ramis-Morales-
Ruiz-Simó theory of non-integrability of Hamiltonian
systems following [12] and [15].
Consider a Hamiltonian system

=H
x
Xx
(3)
with a Hamiltonian
H
on a complex analytic 2
-dimensional manifold
n
M
. Let

=
x
xt be a parti-
cular solution of (3), which is not an equilibrium point of
the vector field
H
X
. Denote by the phase curve
corresponding to
=
x
xt . We can write the variational
equations along
VE


=H
X
x
t
x

We can always reduce the order of this system by one
restricting to the normal bundle of on the level
variety
VE
=| =
h
M
xHx h. The new-obtained system
is called normal variational equation (NVE). In his
papers [13], [14], Ziglin showed that if system (3) has a
meromorphic first integral, then the monodromy group
of the normal variational equations has a rational
integral.
Morales-Ruiz and Ramis generalized the Ziglin
approach replacing the monodromy group by the
differential Galois group of NVE, [12]. Solutions of
NVE define the an extension 1
K
L of the coefficient
field
K
of NVE. The group of all the differential
automorphisms of 1 which leave fixed the elements of L
K
defines the differential Galois group
L K
1
of 1 over
Gal
L
K
(or of equations NVE). Then the main
result of Ziglin-Morales-Ramis theory is 0.1 (Morales-
Ramis) Suppose that the Hamiltonian system (3)
possesses independent first integrals in involution.
Then the connected component of the unit element
of the Galois group
n
0
G
K
1
The opposite is not true in general, that is, if the
connected component of the unit element of the
Galois group is Abelian it is not sure that the
Hamiltonian system is integrable. This means that we
need other obstruction to integrability. Already in [12]
Morales-Ruiz suggested the quite natural conjecture that
the higher Galois groups are also responsible for
non-integrability. In [7] he announced this result and
recently in a joint paper of Morales, Ramis and Simó [15]
it was proved. Let us recall the corresponding notions
and results.
Gal L
0
G
is Abelian.
Again we take a solution

x
t of the Hamiltonian
system (3). We write the general solution as
,
x
tz,
where parametrizes the solutions near z
x
t, with
corresponding to it. Then we write (3) as
0
z

H

,,=
x
tz
X xtz
Denote the derivatives of
,
x
tz with respect to
by
z
,tz
(1) 2)
xt x
(
,,z
 , etc. Let us differentiate the
last equation with respect to and evaluate it at 0.
The functions
z z
() ,
k
x
tz satisfy equations of the type


 

1)
, ,
,, ,
xtztz
tz tz

()
xt
(1)
(1)
H
X
Px
()
(
,
kk
k
x
x
,=z (4)
Copyright © 2011 SciRes. APM
172 T. STOYANOVA
where denotes polynomial term in the components
of its arguments. The coefficients depend on through
P
t


() ,,<
j
H
X
xtzjk. One can show that the system of
non-homogene-ous equations for

() ,
k
x
tz
K
, (4), can be
arranged to a homogeneous linear system of higher
dimension. These recently built systems define succe-
ssive extensions of the main field of the coefficients
the NVE, i.e. we have 12k
K
LLL
= 2
k
where 1
is as above, 2 is the extension obtained by adjoining
the solutions of (4) for , etc. We can define the
Galois groups
L
L

12
. Then the
theorem from [15] asserts that 0.2 (Morales-Ramis-Simó)
If the Hamiltonian system (3) is integrable then for each
,L,Gal LlKK Ga
*
m the connected component of the unit

0
m
G
element of the Galois group
m
Gal LK
is commu-
tative.
2.2. Galois Group and Irregular Singularities
In this section we review some definiti-ons, facts and
notations from the theory of the differential equations
with an irregular point, as well as, from the differential
Galois theory of such equations which is required to
prove our main theorem. For the basic facts on differential
Galois group at the irregular points we refer to Martinet
and Ramis [16,17], van der Put and Singer [18],
Morales-Ruiz [12], Mitschi [19], Singer [20]. For the
basic facts on the analytic theory (formal solutions,
formal power series, asymptotic and summability) we
refer to Ramis [21, 22], Balser [23], Wasov [24].
We consider a linear homogeneous differential equa-
tion
 
()( 1)
10
=0
nn
n
yaxy axy
 (5)
with coefficients in

j
ax

x
. From now on we
shall assume that equation (5) admits over one
irregular point of rank one at zero and one or more
regular points. That is enough for our purpose. Classical
theory says, [24], that in this case equation (5) has a
formal fundamental matrix at 0
1

 

1
1eQx
L
xxx
THTY (6)
where



GL n
x
H,
gln
L is in Jordan
form, is a non-singular constant matrix and
T
 

1n with
(6) is 1
=diag,, n
q
q
x
x

Q with , not necessary
distinct.
j
q
We now turn to the Galois group of equation (5) over
the field of formal Laurent series


x
. FGG ([18,
20]) The formal differential Galois group of equation (5)
over
x
is the Zariski closure of the group
generated by the formal monodromy and the exponential
torus. FM ([17]) The formal monodromy matrix
GLnγ relative to in (6) is defined by

ˆ
Yx

exxY2i
γ
and the formal monodromy group is the closed subgroup
of the corresponding Galois group topologically gene-
rated by . The formal monodromy group is inde-
pendent on the choice of the fundamental solution
γ
x
Y
is a formal invariant of the differential equation (5).
ET ([20]) The exponential torus relative to t
solution
and it
he
x
Y (6) is the gof differential roup
x
-
automorphism


=GalEx where


1
=e,,e
qx
x may iden-
tify with the subgroup of Next lemma gives
rmal
=e qx
Qn
Ex . We

*n
.
the relationship between the fomonodromy γ and
the exponential torus . m-t ([19]) The formal ono-
dromy γ acts by conation on the exponential torus
. Hee is a normal subgroup of the differential
lois group equation (5) over


m
jug
nc
Ga of
x
.
Now we turn to the Galois grr thoup ovee eld of
co
fi
nvergent Laurent series


x
. The general theory
of summability ensures thattrix

the ma
x
H in (6) is
multisummable along any non-singular . In the
case when all non-zero polynomials
ray d

11
ij
qxqx j
have the same degree 1 this means that

x
H is eith
definitions and theoretical
re
on ([23]) 1. A sector is defined to be a set of the
fo
er
convergent or 1-summable.
We need to recall some
sults. All angular directions and sectors are to be con-
sidered on the Riemann surface of the (natural) loga-
rithm.
Secti
rm

=,,==e0<<, <<
22
i
SSdxrrdd
 


where is an arbitrary real number (bisecting direc-
d
Stion of ),
is a positive real (the opening of S),
and
eir a positive real number or is the
(the
radius of S).
2. A closed sec
=diag1,,1qx qxQ
1
j
qx
-polynomials.
In general case of rank one at zero the polynomials
1
j
qx
are of maximal degree 1 with respect to 1
x
but they could be polynomials in a fractional power of
1
x
, [24]. Here we assume that the polynomials
1
j
qx
tor is a set of the form
be monomials of degree 1 in 1
x
some of
them being possibly zero. Then the matrix
1
x
Q in

=,,==e0<<, <<
22
i
dxrr d
SS d
 


Copyright © 2011 SciRes. APM
T. STOYANOVA
173
with andd
as before, but
is a positive real
num i.e. n ver equal to ber (e
). sd ([19]) A singular
n quatio
directiofor en (5) relative to

x
Y in (6) is a
bisecting ray of any maximal angular sector where
<0
ij
qq
Re x



for some ,=1,,ijn.
Following Balser [efine a Gevrey function
and a Gevrey st
23] we d
eries. ga Le

=
f
fx be analytic in
some sector

,,Sd
at =0x. We say that
f
is
asymptotic to

=n
n
f
xfx x
Gevrey order1
sense, if for sed sctor

in
every cloubse1
S of S there
exist positive cuch that for every onstants 0
s
11
,>
SS
CA
non-negative integer N and every 1
x
S one s

ha
1
11
!
N
=0
N
nN
nSS
n
f
xf Nx

xCA
We, according to standard, denote by the ring
of all Gevrey functions of order in
rey fu
m pow

1S
. 1S
Corresponding to the notion of a Gevnction is
the notion of a Gevrey series. ga1 A foral er series
n
0n
n
f
fx
is said to be of Gevrey order 1 if there
exist two positive constants ,>0CA such that
!
n
n
fCAnforevery n
We denote by


1
x
th all power serie set ofes of
Gevrey order
]) 1. Let
1.
R(Ramis) ([22

1
f
x such that there
exists an open sector whose opening >V
and a
holomorphic function

1
f
V such that
f
is
asymptotic to
f
on V in Gevrey 1 sense. We will
say that
f
is 1-summble inrection d (d
being the bisecting line of V) and
a the di
f
is the 1-sum of
f
in the direction d. 2. If

1
f
x is 1-
summable in all but a finite number of directions, we will
say that it is 1-summable. We will dnote e
1
f
x.
This summability definition is very useful but it does not
say how to compute the sum. Another definition of -
summability is gives in terms of Borel and Laplace
transforms. In the next two definitions we follow Balser
[23], van der Put [18] and Singer [20].
Borel The formal Borel transform of order 1 to a
formal power series

=0
n
n
n
1
f
tfx
is called the
formal series

=0
=!
n
n
n
n
f
fx n
n

(7)
Then




11
fx xf
 
 (i.e.
convergent) [20].
verse of the Borel transform is the LaThe inplace
transform: laplac Let
f
be analytic and of exponential
size 1, i.e.


expfAB

along the ray d
r
om 0 to 0 in direction d. Then the integral
 
=expd
d
dr
fx f
x
x

8
h lorder
in the direction of The followin
ves useful criterir avrey series of or to be



()
is said to be te Laplace compex transform of
d
g pr
der
1
gi
d
a fo
f.
Ge
eposition
1
1-summable cri([18]) Let


1
f
x and d be a
rection. Then the fllowing are equivalent:
1. f is 1-summable in the direction d.
2. The convergent po

let
di o
wer series
f
has an
analytic continuation h in a full sector
fr

|0<<,arg<d
 
. In addn, this
analytic continuation as exponential g
itio
hrowth of order
1
at

on this sector, i.e. expABh

. In
=d
f
h this case, is its 1-sum. T
Tfiiers relative to the s
e need th
ho
he good is that the
two definitions R and cri are in fact equivalent [22].
o dene the Stokes multiplolu-
tion (6) we following fundamental result of
Ramis sum(Ramis, [22]) Let =Lyg be a linear non
mogeneous ordinary differenti-al equation of order n
with polynomial coefficients and

g
xx. We
suppose the Newton polygon arigin admits only
one strictly positive slope 1. If the series
t the o
at
fx
is a formal power series solution of f is
1-summable or convergent. In this paper we will not
apply the theory of the Newton polygon. W
when
=g then
e no
Ly
te th
1= ,=1,,
jj
qxqxj n with j
q
the
ewton polygon of equation (5) admits only one strictly
positive slope 1. sum says that there exists a unique
holomorp
N
hic function
f
x in all but a fimber
of directions, solution of the differential equation
=Lyg such that asymptotic to

nite nu
f
x in Gevrey-1
sense. Moreover
f
x be obtained from can
f
x
by a Borel-Laplace transform.
s assume that the matrix Let u
L
) is in a diago-
nal form, i.e. there are no logarithms in the solutiof
equation (5). Due to our assum
in (6
ns o
ption the fundamental set
ofn solutions of equation (5) is spaned by the functions


=e, =1,,
qj
ljx
jj
yx xyxjn (9)
,
jj
lq

j
yx
are
act that ea
on of a
where and . The formal
series “in nt. I
lynom


j
yx xx
general” diverge
hese divergent se
t is well
known fch of tries

j
yx is
a soluti linear homogeneous differential equation
with poial coefficients

()( 1)=0
nn
bxybxybxy
 10)
10
nn
(
whose other solutions are

e
qq
ij
ll
ij xj
x
yx
. Next,
series
the

j
yx
, 1jn
lie at the first row of the matrix
x
H. We remark that the behaviour of the elements of
Copyright © 2011 SciRes. APM
174
the first row of the matrix
T. STOYANOVA
x
H
for ou
(and from here of
hr purpo
the important r
their derivations) is enougse. This is
based onesult that the set

1,d
of all

f
x
In this
such that are 1-summable in a direction d is a
differential algebra over (see Balser [23], Chapter
3.3, Theorem 2). We we will consider the equations (10)
(not equation (5)) and we will apply sum ton (10).
way for any open sector V with vertex 0 with
opening > and bisectedy non-singular direction d
to any series

j
yx we associate a unique holomorphic
function

j
equatio
,
b
f
xsolutions of the equation (10) and 1-sum
of the corresponding series

j
yx
along d suh that c
j
f
x is asymptotic to

j
yx in Gevrey-1 sense (R).
Replacing t series

j
yx (and their derivations)
he
in matrix

x
H by theums we obtain a
holomorphic matrix
ir 1-s
x
H sector with opening
isected by a non-ar direction d, asymp-
totic to
on a
ngul>, bsi
x
H in Gevrey-1 sense on this sector and
denotes the 1

-sum of
x
H. The new-obtained matrix
 
(1/ )
=e
L
Qx
xxxYH actual fundamental matrix
quation (5) and denotes the 1-sum of
is an
of e
x
Y along
any nonsi ray d.
Further, following [1 [20], relative to equations
lution (6) (the matrix
ngu
and t
lar
o the so
8] and
(5), (10)
1
x
Q) we
define:
1. Eigenvalues =ij
ij
qq
q
x
of equation (10), where
,
i
qx j
qx are the eigenvalues of equation
2. A Stoke
(5);
ion is a
th
co
say thi
direction suc
s direct
12 e
d
nsecut
for ij
q
iv
h
r if
at

=0
ij
Re q;
3. Let ,dd be Stokes directions. We
apair

12
,dd is a negative Stokes pa
t the

<0
ij
q for
 
12
arg ,Re
x
dd
4. Adirection is the bisector of a negative
St
;
(5) and (10) a
sd).
sing
.
rresp
ular
0)
to
okes pair
sd1 Equations (1e at least one zero eigenvalue,
th onding

j. In this way the
hav
is co th
si
re exact
e series yx
ngular directions corresponding to the series

j
yx,
from here to equations
de
these
) a
fined by (4) (this defined more exactly the expre-
ssion “for some ,=1,,ij n ” in
Let d be a singular direction for equations t
=0x and let =dd
(10
and =dd
where
>0
is smabe two non-singular neighboring direc-
tions of d. Letd
Y
ll
d
Y and
denote the 1-sums of
Y
a gi
Sto
alond and d respectively. st Wit0h respect to
n formal fu solution (6) the
s matrix (or multiplier)

St GL
dn
corres-
g
=S
dd

ve e as in
ke
fo
ndam
a
ntal
is)
Y
ith
ponding to the singr line at =0x is defined by
ul
t
d. R (Ram
d
([16])
YY W respect to a
rmal solution Ygiven as in (6) the analytic
differential Galois group of equation (5) at 0 is the
Zariski closure in
GL n of tenerated
by the formal monodromy
he subgroup g
, the eential torus
matrices Std for all singular rays d. It
is possible to generalize the above theorem to a global
linear differential equation: Mi ([19]) The global Galois
group G of equati topologically genera-ted in
xpon
and the Stokes
on (5) is
GL n by the local Gal is subgroup a
G where
runs over the set oingular points of 5).
2.3. V
o
f s
a
(
P
as a Hamiltonian System
Painlevé equation, V
P, is equient to th
Hamiltonian system, [25],
The fifthval e
=,=
H
H
qp
pq

(11)
with the Hamiltonian
H
,
 
23
Hpt qpq
t


1
1
=1q 1p qtp q
 
(12)
In fact, tting pu
10 32
,=0
=, =1
 
 

3
(13)
with
012
=1

 (14)
we see that equation for =1
st
o
c vari
w Ham
1yq is but
(2). Ton-autoousem (11) can
o ous onh degrees dom
g new dynamables—
juga iltonian
nothin
be t
of free
g V
urned into
P
by
he n
m
ucin
te to it—
nom
e wit
The ne
sy
tw
i
auton
introd t
be
and the con-
comes F.
H
HF
The symplectic structure
is canonical in the
variables
,,,qptF , i.e. =dpdqdFdt
 . In
what follows we denote by
s
the time variable.
for 3. Non-Integrability of V
P
,

0
=0 =
n this section we begin studing V
P with the foI
v
yllowing
alues of the constants: 0==0,

where
is
e
g
an arbitrary parameter. The rlts of thiesus part are th
base oe main th
orese e constants the correspondin
f th
th
eorem
values of t
s.
hF
autonomous Hamiltonian system becomes

1
=2 1qptqqq
t


1
=21 1pqppt tp
t
 
Copyright © 2011 SciRes. APM
T. STOYANOVA
175
=1t


2
1
=11
111
Fpqqq
t
pp tqqqtpq
t





We choose as our non-equilibrium particular solutions
As we plan to compute the second variational
equations it would be convenient to put
It is easy to see that as a normal to the phase curves of
these solutions we can pick the plane in the
hypersurface
=0,=, = ,=0qpstsF.
2
=0qqq

 
12
2
12
=psp p

 
2
12
=ts tt

 
2
12
=0FFF

 

,qp
Const
H
,
are just the eq
he equation
i.e. thes in normal
variationsuation -variables.
Because of t as an
in
equation
s for ,qp
we ca =1t
nuse t
dependent variable instead of
s
.
For the first variational equations we obin the system ta
111
=1,= 1qqp
tt


 
 

This system can be solved by uadratures and
1
p

q
is a fundamental matrix solution. Thus the differential
Gon is
Abelian, it is conjugated to the diagonal matrices
ob

=0e
t
tt



e0
t
t

alois group G of the first variatio-nal equati
*
1
0
=,
c
Gc





0c



and there is no struction to integrability. Next, for the
second variational equations we obtain
2
111
2
=1qqqpq
tt




2
11
1
=1 2ppppq
tt




1
where we have replaced
 
22
:,:qtq tptp t
s the Stokes multipliers we
o study the second variation
uation than a system. Further, as
does not depend on
.
why As our approach usefind
that it is more suitable tal
equations as a scalar eq
the equation for

qt
pt
o consider only the scalar
it is
us eorde
enough for our purpose t
homogeneous equation corresponding to this equation.
Furthermore, we find our problem will get more simple
if we write the equation for

qt as a fourth order
linear homogeneoquation, not as a third r (the
natural method). Then the equations of

qt will have
a Galois group contained in

4
GL (not in
3
GL ).
This apparent complication preserves the non-commu-
tativity of the unit element of torresponding diffe-
rential Galois group. The equation of

qt can be
written as a fourth order linear homogeneouquation

he c
s e

 

43
43
2
22
d53d
=5
dd
24 3
16 11d
8d
qq
Lq t
tt
q
ttt









23
1
0
tt

23
61 67 1
d
22 d
21
24
2=
q
tt
q
ttt

 


2
 




(15)
As it said from here on we will studfferential
Galois group only of Equation (15).
Equation (15) has over two sinints - the
point is a regular arity and tht
y the d
gular po
e po
i
in
1

singul
=0t =t
re,is an irregular singulk one. o
eq
arity of ranFurtherm
atio
this
uation is reducible 12
=LLL and it is solvable in
terms of second order linear differential equn
 
21
22
11
=,= 1Lft Ltt

q
 


)
The second equation
(16
1=0Lq is

1
2
22 1
()= 4
42 1 22 1
4
t





=
0
Lq qq
t
t





 
(17)
and the system
q
22 212
e, e
tt
tt

is a fundamental syste
uation
m
of solution of (17).
Let us consider the homogeneous eq
2=0Lq .
Changing the dependent variable
11
dq t
=e
xp1
2
yt



we transform equation
2=0Lq to the reduced form
2
2
11 1
=0
42 4
yy
tt



 


 (18)
Equation (18) is known as the Whittaker equation [26]
Copyright © 2011 SciRes. APM
176 T. STOYANOVA
2
2
141

 =0
44
yy
tt




with parameters 1
=2
and =2
. Then we have
w
group of equation (18) is Abelian if and only if
.
own from the work of Martinet and Ramis
o) that they com
ois group Whittaker equ
itt The identity component of the differential Galois
*

It is well kn
[16] (see [12] to identitponent 0
G of
the differential Galof the ation is
Abelian if, and only if, 11
,
22
 

 


belong
to




**
  (i.e. 11
,
22
 
 
are integers, one of them being positive and the other
negative). In our case 11
=1, =1
22
 .
Gal ) is Abelian if,

Hence the identity component of the differential
ois group of equation (18, and only if
0
G
1
 . This proves the lemm
8) is spanned by the set
a.
We finish with same notes. The fundamental set of
solutions of equation (1
11t


1
t
22 2 2
120
=e,=eed
txt
yt ytxx
Stokes
di
group of
Inr, when
*
re Equation





, i.e. only one of
the multipliers of the Whittaker equation (18) is
fferent from zero when *
.
Observe that the identity component 0
G, as well the
Galois group G of equation (18) is a sub

2
SL . particula*

0*
==, ,
0
GG
 







1
and when *

0
1
0
== ,
0
GG





2=0Lq has a solution space that is
y funconsx
t
.
Therefore the identity component of its Galois group is
not a subgroup ofi.e. equations
spanned bti 1
12
0
=e, =ed
t
t
qt qtxx

 

2
SL ,
2=0Lq
ponents of the
ly
2=0Lq i
and (18) have difentity com
g Galo
of equation
ferent id
is gr
iers
correspondinoups. As in equation (18) on

one of the Stokes multipls
different of zero when *
. Furthermore, the
(resp. not
identity components of the Galois group of both of the
equations

2=0Lq 8) are Abelian
Abelian) in the same way, as both of them depend on the
same integral 1ed
txt
and (1
0
x
x
 
. In particular, when
*
for the Galois group of equation
2=Lq 0
we have
0*
0
= ,
01
=GG




When
*
he identity componen
Ga
to define tt
of the ven
0
G
in thelois group is a complicated task. But e
worst case when
0*
=,,
01
G
 






0
G is elian group. However, as witt gives
necessary and sufficient condition for Abelian diffe-
rential Galois group of equation (18) (and from above
remark of equation
not Abus
2=0Lq )
e following theorem
*. Then the Painlevé
the Liouville sense.
, thus as a corollary we
h. witt Assume that
ation (2)
D
have t
0
=0,=


is not integrable in
V equ
ue to the reducibility of 12
,=LL LL, with equation
(15) one can associate a matrix equation
1
=0
0

AC
QQ
A
(19)
where
2
21
=,QQQ
, the matrix equation 111
=0AQ
is completely reducible and C may be
Q
matrix
equation and the matrix
taken to be the
matrix 00
=10


C, [27]. In this way the

111
=0AQQ 1
A
may be taken
to be the correspond
ution of th
mn vector
ing matrix equation and the
n the standard
is a sole scalar if and only if
corresponding matrix to the equation (17) i
sense. Namely, if we put 12
, =qqqq
then a function =
equation q (17)
the colu

12
1=,qq
Q is a solution of the
following matrix equation
111
=0
QAQ

1
01
=421221 221
44
tt
 




2
t



A
We will not fix on the equation but
will note that its differential
Hence, there is no obstruction to in
the equation and the m may
be taken to be ng equatior
ponding matri
111
=0AQQ
Galois group is Abelian.
tegrability. Note that
atrix 2
A
n and the cor
222
=0QAQ
the correspondi
x to equation
es-
2
Lq=0 in the standard
sense, i.e.
2
01
=11
1t





A
Next, the matrix
t
Copyright © 2011 SciRes. APM
T. STOYANOVA
177
is a fundamental matrix solution of equation (19) if and
only if satisfies
21

Q (20)
1
=0
U


Q
QQ
U

21
=U 
A
UUA C
2
Q and 1
Q in (20) are t
lutions of equations
respectively.
y to see that the iden
, [27]. Fur-
thermomatrices he fun-
da
and
the differential Galois group of the equation (15) is not
up of equation (19). T
gai
at
re, the
22
QA
Now, it is
mental matrix so111
=0QAQ
2
=0Q
eastity component of
Abelian for *
. Indeed, let us denote G to be the
differential Galois grohen we have
that

,G

Q is an a fundamental matrix solu-
tion of equation (19), and a calculation show

1
Qherefore

s th

=0
Q. T

=R
Q for some Q
4
GLR
. Expressing any such

R
in block
notation,

23
41
=GG
RGG



let us write the equation
=R
Q Qlicitly exp
The last two equations imply and
1
. Hence one can i the matrix
entation of tial Galois group
n . Ne
results that
e Ga
ot Abel the M
s-Simó theorem
Hamiltonian sy

 

14
1
2214 211
14 11
1
=0
=
(0)
GG
GUG GUG
GG
U





QQ
QQQQ
QQ
Q
Q

21
=


Q
2123
UG
G

QQ
3
4=0G
dentify
the differen
xt, from
one can identify the m
witt and
nent

11
=G
QQ
with the repres
equation (17) i
that

=
QQ

22
GLG
equation 2
Lq
fo
group 2
G is
1
G
of
follows
atrix
of
lois

2
GL
2
G and
with the
=0. Now from
4=0G
differential Galois group
re it
22

r *
 the identity compo0
2
G of th
not commutative and as a corollary the
identity component of the differential Galois group of
equation (21) is nian. Thus fromorales -
Rami [15] and why the corresponding
stem is not integrable. This proves witt.
3.1. Non-Integrability for
=1
To prove non-integrability for *
 we will study
the matrix 1
UQ in the matrix solution (20). Let for
simplicity =1
. In the last section from this particular
case, =1
, we will extend the present results ove
*
r
nd transform
to apply th irregular points
ndent vble
As we ar
bylu Bäck
e going
ations.
e theory of
at =0t we change the depearia =1tz
.
hTen for =1
equations (15) and (17) ,

2=0Lq
become

432
232342
d45d 238d
=d d
284d 28
qq q
Lq zzzzzzz
q
 
 
 
 

 
(21)
4
d
4
z
456 567
=0
dq
z
zzz zzz



23
11
==0,Lq qqq'
(22)
2
d
=
dz
zz


12234
442 4 4
==Lq qqq
zzzzz
 
 
 
 
 
0
(23)
Equations (22) and (23) have solution spaces spa
by the sets
nned
1
,e
z
zz and
212 212
2e,e
z
z
zz zz

 respe
formal fundamental set of solutions around the irregular
singularity at 0 of the equation (21) we can
fundamental set of solutions
ctively. As a
take the

1
12
=e , =z of
z
qzq
equation
2=0Lq and
 
22
34
=e =e
zz
qzqz z

where
,z
and
z
z
are two formal series
 
234 1
=2!3!=1!
nn
zzzzznz
=0n
 


234
234
1
1
=0
=2
n
n
z
111
=2!3!
2222
1!
n
n
zzzz z
n
1

Then a formal fundamental matrix of equation (21) is

(1 )
e
J
Rz
zzQH (24)
where

122
=diag,0,,,=diag1,1, 2,2RJ
zzz

 

 (25)
and


33
23 24
2
22
11
111
00 21
24 2
001 1
zz zz
hh
zz
z
zzz
zzz












H (26)
The precise form of and are not important
for our purpose but we do note that they are elements of
23
h24
h
Copyright © 2011 SciRes. APM
178
. We remark that the formal series
T. STOYANOVA


z
z
and
are the result of formal operations that have
to do with sectors.
We now turn to the Galois group of equation (
. The formal Galois group over is the
osure of the group generatedformal
romy

z
nothing
5) over


z
Zariski cl
monod


z
by the
and the exponential torulast
re
s . The
ones on the other hand present differential automor-
phisms of the extension
F
of the field

z by the
entries of the matrix


(1 )
e
J
Rz
zz over
z;



12
=e,e
z
z
Fz in our case. Furthermore, the
formal monodromy
is trivial. Therefore



Gal Fz is equal to the exponential torus



*
2
2
00 0
01 00
==00 0
00 0
c
Gal Fzc
c
c













 (27)
Let now
F
be theot extenf
ation (21). To determine the
convergent Galois oup at =0z over
Picard-Vessision o
for the equ
gr


z
z
pute the Stokes matrices at =0z

Qz, (24).
1n
zaz
we
o rela-
tive to the solution
s cus no
will need t
Let u
com
fow on

0
=n
n
s
is the so called Euler seri
and
z. The first serie
zes,
the second is a modified Euler series. The properties of
th er
phenome
gIn the
serie

1
0
=n
n
n
zb
 
=1
n
zn
1
=0 !
n
n
e Euler sies and the corresponding Stokes
non are well studied, for example in a paper of
Ramis [22] and in a paper of Siner [20]. next
following these two papers we will compute the Stokes
matrix relative to thes
z
and
Gevrey-1 growth e can consider the
formal Borel transforms

z
.
These series are divergent and they obviously satisfy a
condition. Then w
of
z
and

z
  
1
1
0
1
== =log1
1
nn
zn

 
 
1
11
0
1
==log2ln2
21
nn
n
zn
 

The Gevrey-1 growth condition ensures that

and


are analytic in the neighborhood of the
origin of the
plane. For any ray d
the
functions

log 1
and

2log
h
cont . For ray the
transforms (see Example 1.4.22 in [20])
ave
i
analytic
r Laplace
inuations alg d
on such a



1,
=log1
1
=log1 ed=ed
1
dd
zz
dd
zz
z




1,
=log2ln2
log2ed
=ln2ed
dd
z
z
d
zz
z






=
1
=e
d
2
d
z
d
z
define the corresponding 1-sum of the series
z
and
z
in the direction. We note that funct dions
 
2
3=e
zd
qz zz
and
 
2
4=e
zd
qz zz
ed, one can complete th
again
satisfy equation (21). Indee set
1
12
=e ,=
z
qz qz
to the fundamental set olutions
of equation (21) by the particular solutions of equations
f so
2
23 2
11 21
=e
z
qqqz
zzz

 
 


(28)
2
23 2
11 11
=e
z
qqqz
zzz

 
 


Looking for ssolutions of the uch form
2
ez
qz zz
and

2
ez
qz zz
respectively,
we obtain the folowing non homogeneous differential
nomial coefficients
l
equations with poly
4322
23 2=2zzzzz

 
 
(29)
432
23 2=zzz z

 2
z

respectively with unique formal solutions
z
 
1
=1!
nn
zn
and
 
1
1
== 2n
1!
n
n
n
zz
her, so (29) by the
variation of constants we get a particular solution
respectively. Furtlving equations

1/
ex
z
1
0
=e d
z
zx
x
of the first equation and a particular solution

2/
2e
=e d
u
z
z
zu
ion, w
0u
of the second equathere for convenience the
integrals are taken in the direction
. Next, let

11
e
=d
xz
z
zx

and define a new variable
0x





by
11zx
z
  gives

0
e
=d
1
z
z


. Ihe
same manner let
n t

22
euz
z
0
=dzu
u
and setting
Copyright © 2011 SciRes. APM
T. STOYANOVA
179
22zuz
 gives

0
e
=d
2
z
z

. More
ic
associate unique fun
general, these integrals exist if d
. Therefore,
applying to equations (29) Ramis’ sum, for any ray d
except the negative real axis, we able to canonre aally
ctions
dz
and

dz
, anlytic
d
a
in a large sector around thnt se
an
is ray, with the divergeries

1
1!
nn
nz
1
1
1!
ly and so
these series are

z and
2
n
n
n
nz
respective
1-summable and d
z
d
are their 1-sum. Furthermore the functions
 
2
3=e
zd
qz zz
and
 
2
4=e
zd
qll satisfy z zz
wi
here equations (28) respectively and fromthe equation
(21). In this way the matrix =
 

1
=e
R
z
J
zzz
QH
with
J
z and

1
e
R
z as before but


33
23 24
2
=002
hh
zz
z
zz

H
22
1
111
1
24 2
001 1
d
z
z
zzz

 


1d
z zz



where
 
3
23 2
d
12
=d
d
z
hz z
z
z






and
dz
 
3
24 d
d
zz
z




2
d
12
=dz
hz z



is an actual funda-
ntal matrix of equation (21). The matrix
zH
=0z
is
holomorphic in an open angular secto at of
opening angle ()
r
and 2<arg <z 
zH
Gevrey
is
asymptotic to sector in -1
sense.
We are now in a position to describe the analytic
elements which, together with the formal Galois group
(27) determine the analytic Galois group of equation (21).
Stokes multipliers depend on the fun

zH (26) on this
As the ctions
z
der
poly-
and
we consi
equation (21) by

z
ollowi
al coefficients
changing
(resp. on and
the fng homogeODEn with
mi from
to

dz
neous
obtained


dz
)
equatio

qz

qz
12
=e
z
zz




6(4)5 44 3
3
83 1462
(44 )( )( )
zqzzz qzzzzz
zzqzqz

 

(*)
2
4= 0
q

The functions

z
and

z
are its formal
solutions. The other entries of the fundamental set of
solutions of equation (*) are 1
e
z
and 2
e
z
. So the
eigens of this equare valuen atio1
0, zz



.
Now we are almost ready to compute the Stokes
constants sponda
2
0, ,
corre to the singulr direction ing
(=d
). Relative to tion we define:
- eigenvalues of en
equa
quatio
(*)
(*): 12
0,0, ,
zz



;
- the Stokes direction 3
=,
2

2



such that
d
21
=2 =0Re Re
 
 
 ;
zz
- the negative Stokes pair 3
,
22

suchat
th
2<0Re z


 ;
ngular direction as the bisector of - the sithe
negat
=d
ive Stokes pair 3
,
22



.
tions ach are
respectively slightly to th to the right of the
critical direction
We select two direcnd d whid
e left and
. Let
==d
11
=ed =ed
zz
dd
11
dd
and








11
=ed =ed
2
zz
dd
dd
and

2







betransf the associated Laplace orms of
and
of
rections d
the di
and d
. The
rata path comirm ity
aloht till th
difference between
them amounts to intege on ng fo
orig
nfini
ng the critical line on the rige in and then
doing to infinity by following the critical line on the left.
As there is no singularity between 0 and 1, and no
other between and
1
, Cauchy’s fa ie-
diately implies that the difference
ormul mm
dd

is given by
1
=1
2Res e1=2e
z
z
ii
. In the same manner,
the difference between d
and d
is given by
2
=1
2Res e2=2
z
e
z
ii
.
Next, we must have
 
11
==
e=eSt
Rz Rz
JJ
dd
Hz Hz

 
where
4
St GL
is the Stokes matrix in the
direction =d
. Furthermore,

21
33 1
=e=2e=2
zz
dd
qqziz iqz

 
 . Then
for the Stokes matrix St
we have

1,3 =2
s
ti
. In
the same manner,
Copyright © 2011 SciRes. APM
180 T. STOYANOVA


2
2iqz. Then we
have that

2,4 =2
2
44
=e=2=
z
dd
qqz iz

 
 
s
ti
.
e From the above reasoning for thStokes matrix St
w
(30)
ume that
e obtain
102 0
01
St =
i
0 2
00 10
00 01
i


Thus we have n1 Ass0
=0,=1,= 1

.
) is not integrable in the Then the PainlevéV equation (2
Liouville sense.
rem of Schlnger [28] the local differential
oup of equatio (21) at infinity is generated
to ote
monodromy around anound 0 of
equation (21) are the same. There the diffe-
rential Galois group of equation (2nity can be
interpreted as a subgroup of the loif
group at the origin. Next, observe t thfferential
Galois group of equation (21) is a cnnecroup. As
th
By a theo esi
Galois grn
pologically by the monodromy group at . We n
that the actual
fore
1) at i
cal d
hat
o
d ar
local
fin
e di
ted g
ferential Galois
e formal monodromy
is trivial th’
theorem the Galois group is topologically generated by
the exponential tous and the Stokes matrix St
en be Ramis
r
y th
.
The Zariski closure of the subgroup ,n is the
same and the elements c
n
f
of is
*
2
2
00 0
01 00
=00 0
00 0
c
c
fwherec
c
c






The matrix

St ,
nn
is of the kind
I
nX
where =
I
Id and
X
is a unipotent matrix. Denote by
S the Zariski closure of the subgroup

St ,
nn
then the elements ,
s
of S are
1020
01 0 2
i
ere



,=00 1 0
00 01
i
sw
h



When
is different from zero and 1c
the
commutat or ofc
f
and ,
s
is
(31)
which is not identically equal to
The case implies that

1
11
,,
1 0
01 02
=
0 1
cc
c
fsf s





2
102
1
0 0
00 01
i
ic





Id .
=1c

11
e=e
z
z

for any
Gal L
=0q, i.e. 1
e(
zz
), which is an
ction. In the same mannerobviously contradi =1c
implies that 2
e
z
is invariant under any m orphism
=0Gal, i.e. L q

2
ezz
. Fhese
remarks, formula (31) anhe fact
rom
d tthat
t
=0q
ponent
1) is not
that for
Gal L
ntity com
ation (2
e have
is a connected group follow that the ide
is group of equ
why and 0.2 w
of the differential Galo
Abelian. Thus from
0
=0,=1,= 1

the correspondi
ss two meromorphic
is We will not de
group of equatio
is not finite and
f thun
roup of
conjugate to the follatrices
ng Ham
termine p
n (21). Bu
it coincid
it elemen
equatio
iltonian
first integrals.
recisely
t we
es
t. Fur-
system does not posse
This proves n1. galo
the differential Galo
can say that this gro
therm
is
upwith
its connected component 0
G oe
ore the differential Galois gn (21) is
owing m


2
2
00
01 0
=0 =0,
000
000
cd
GalLqca d
c
c
,
a







(32)
4. Generalization
In this paragraph we will extend the results of the
previous section to the entire orbits of the parameters
using the Bäcklund transformations of the Painlevé fifth
equation, given by the following list of restriction of this
group on the parameter space, [29]
=,= ,(
iiii jji
ss
 

=1)
=,(,1)
ij j
j
sjii

,i

 (33)




1001
22 31
=,=,=
=, =
jj 3
,

 
We note that the Bäckl
the fifth Painlevé equatio
und transformations group of
n is isomor-phic to the extended
fine Weyl group ofaf (1)
3
A
type, [5]. It is well known
ted with the fifth
Painlevé equation. In particular the Bäcklund trans-
formations remain the property non-integrability.
We define (following Masuda et al. [29]) th
lation operators ) by
(see Okamoto [5]) that the group of Bäcklund
transformations of the V
P equation is represented as the
group of birational canonical transformations of the
Painlevé system (that is the corresponding non-auto-
nomous Hamiltonian system) associa
e trans-
i
T (=0,1,2,3i
1 321213232130321
=,=,=,=TsssTsssTsssTsss
 (34)
Copyright © 2011 SciRes. APM
T. STOYANOVA
181
These operators acts on parameters i
as
,)i
 

11
=1, =1,=(1
iiiiiiijj
TTTj


 i
(35)
4.1. Generalization of the Results of the
Paragraph 3
In n1 we have proved that for 0
=0,==1

the
Painlevé V equation (2) is not integrable. Let us recall
that these values of the parameters are particular case of
the family 0
=0,=

taken at =1
. In the
following proposition using the operators
j
T, (34), (35),
we will show that the result of n1 can be extended for
*
. p1 Assume that 0
=0,==mere
*
m. Then th

wh
V equation (2) is
e Painlevé not
uile appro-
priate transformations which extend the initial parameter
family
integrable.
We will prove the statement by bding th
0
=0,==
1

or 01 23
(,,,) = (0,0,0,1)

to this
The Bäcklund transformation (starting from
of the proposition.
210
TTT
0
T) maps the parameter family 01
(,,
23
,)

12 3
(,, 1, 1)
to
0
 
. Now, applying 1m ti
ter fa
mes TT
,)mm
210
T
to the initial paramemily we obtain
,0,1
012 3
(,,1, 1)=(0mm
 
  re-
call that 20
=1
. If we

 then we obtain that =m
.
The proof follows from the fact that the Bäcklund
transformatio l canonical rrmations
[5] and n1.
As a corollary from witt and p1 we have the following
gnon-integrable result:
0=

where
ns are birationat ansfo
eneric g1 Assume that
=0,
is n arbitrary complex
parameter. Then the Painlevé equation (2) is not
in
a
V
tegrable.
The next lemma describes the orbit of the vector
0=( ,,1,
0123
, , )=(0,0)

transformation group of the fift
Le
 
h Pain
al
Ham
0=(0,0,1, )
under the Bäclund
levé equation. orbit
s
eters
t (, )=(0,)qpt be a rationolution of the
iltonian system (11), (12) with param

. Then beginning h 0
wit
by
Bäcklund transformations (33) we ob
,2,3
the
tairational n a
1solution of (11), (12) with new ,=0,
jj
as at
least two teger and at least one e
integer is 1
of them are inof thes
or 3
. Furthermore the parameters satisfy
either 1232
12

 or
0132
12


 relations.
Let 01 23
=( ,,,)
iiiii

be the vector of parameters
obtained by i successive transformations ,,
i
s
from 0
. We w thill provee statement inductively. At
first for 0
the statement is true
Let =1i,
.
pplied oni.e. we have a 0
som
tra
e of the
nsformations (33). Under 0
s
and 1
s
the vector 0
10
=
does not change, i.e. and the statement is true.
Let us denote 11230
=1
iiiii
S

 and
20132
=1
i i
Si
 
ii

 . Under 0
2,s
becomes
1= (0,1,1,1)

 and 1
1er
=0 2S. Und0
3,s
becomes 1=(,0,1,)

and 1
2=0 2S. Under
0
,
becomes 1=(0,1,,0)

 and 1
1=0 2S.
Under 0
,
becomes 1=(0,

1
,1
true for
,0)
and
1=0 2S. Hence for =1i the statement is true.
Supposeat the statement is true for i. We will
prove t
th
hat it is 1i
. Let us recall (14
i
), that is
01
i
2 3
=1()ii
A
forevery i
Observe that the conditions

12,
i
SA imp
0
i


ly that
. In the same manner

22,
i
SA imply that
2
i
der 0
. Un
s
the vector i
becoes
1
01 023 0
=(,, ,)
iiiiii
m
i
  
  and fr the sums 1
So1i
and 1
2
i
S
we obtain: 1
2
2,=22
ii i
S
1
021
=
i
S
. So if
2
i
S then 2 then 2
1
1
i
S2
i
S
1
Next, if
2 and if 12.
i
S
 

1
2
ii
11
,2
ii

3
SS, then
1i
1i
 ly, if
13
. Similar
12 3
2
ii i

  1
3
i
2
2
i
SS then,,
from
A
( 1
1i
from
A
). Hence the statement is
true.
We leave the proof statement for 1i
of the
app-
lying on i
ea
the e transformatio
f 0
rest of thns of (33) as an
sy exercise similar to the case o
s
.
rtue of
g1. g2 For
ollows t,
sformati
forma p and
all that 20
=1
The following corollary, by viturns out to
tural generalization o values of th
orbit
u
e bi
,
be a na
parameters satisfyin
is not in
The proof f
canonic
f
from orbi
tran
tion
e
tion (2)
own
0
,=
g orbit the Painlevé V eq
tegrable.
g1 and the well-kn
fact that the Bäcklundrational
al transs on ,qt [5].
a
1
=
ons ar
If we rec 3


20
iii
 
,
(13), then 3
1
1, ,
i

 show how the initial
0
,,

ch
group (33).
that 0
ange cklund
Henry we
=
under the Bä transfor
: M Assu
mation
me ce as a corolla
m
obtain

 m is
,0
where even and at least
one ,=
jj
le.
p
. Then te fifth Painlevé equation (2)
is not in
h
licitly compu
tegrab
theory. We ex
5. Concluding Remarks
We prove non-integrability of one parameters’ family of
the fifth Painlevé equation as a Hamiltonian system. The
main tool to identify obstruction to complte integrability
of this Hamiltonian system is Ziglin-M
Si
e
orales-Ramis-
mó theory reducing the question to differential Galois
te formal and analytic
Copyright © 2011 SciRes. APM
T. STOYANOVA
182
basmptotic
an singularity at zero. From thes
mpute the Galois
equation
invariants of the second variational equation (in fact of
the part of it) by a methoded on the asy
alysis of its irregulare
results we co group of our differential
.
We consider here only the case 0
=0,=

with
anex paramer. It is tempting to use
the mis theory for V
P with other values
of the parameters
arbitrary complte
ethods of Galo
, as well as, fvé
nscedents
or other Painle
tra
I
II
P and
I
V
equations along each particular
P, where the variati
solution will hav
onal
e an
regular singularity at zero. We can hope that in the case
Theported by the
to
is
ity.
l’Intégrale Générale est a
cta Mathematica, Vol. 33, No.
5. doi:10.1007/BF02393211
ir
of Abelian differential Galois group of the first
variational equation and one irregular singularity at 0, the
reducibility of the second variational equation, consi-
dered as a linear homogeneous scalar differential equa-
tion, could be an efficient tool to write down the
corresponding solution space expressly and therefore to
compute formal monodromies, exponential tori and
Stokes multipliers.
6. Acknowledgements
e results of this paper has been rauthor
the 7th DEDS Conference 2010 at University of South
Florida. This work, as well as the vit of Ts. Stoyanova
to Tampa are partially supported by Grant 225/2010 of
Sofia Univers
I thank the referees for the careful reading of the ma-
nuscript and helpful remarks.
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