Advances in Pure Mathematics, 2011, 1, 170183 doi:10.4236/apm.2011.14031 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Nonintegrability of Painlevé V Equations in the Liouville Sense and Stokes Phenomenon Tsvetana Stoyanova Tsvetana Stoyanova, Department of Mathematics and Informatics, Sofia University, Sofia, Bulgaria Email: cveti@fmi.unisofia.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 ZiglinRamisMoralesRuizSimó method yield to the nonintegrable results. Keywords: Differential Galois theory, Painlevé V equation, Hamiltonian Systems, Stokes Phenome nonAsymptotic Theory 1. Introduction The six Painlevé equations ( 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 socalled Painlevé transcendents or Painlevé functions. Although the Painlevé equations were discovered from strictly mathematical considerations 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 twopoint correlation functions at zero temperature for the onedimensional 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 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 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 casetwo). In [7] MoralesRuiz raises the question about the integrability of the Painlevé transcendents as Hamil tonian systems. Later MoralesRuiz in [8], and Stoya nova and Christov in [9] obtain a nonintegrable result for Painlevé II family. Nonintegrability 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 nonintegrability 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 nonequilibrium 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 ZiglinRamisMoralesRuiz Simó theory of nonintegrability 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 nonintegrability of the fifth Painlevé equation (2) for 0 =0, = and (witt). In n1 we prove the nonintegrability 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. NonIntegrability and Differential Galois Theory In this section we briefly recall ZiglinRamisMorales RuizSimó theory of nonintegrability of Hamiltonian systems following [12] and [15]. Consider a Hamiltonian system =H Xx (3) with a Hamiltonian on a complex analytic 2 dimensional manifold n . Let = xt be a parti cular solution of (3), which is not an equilibrium point of the vector field . Denote by the phase curve corresponding to = xt . We can write the variational equations along VE =H 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 xHx h. The newobtained 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. MoralesRuiz 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 L of the coefficient field of NVE. The group of all the differential automorphisms of 1 which leave fixed the elements of L defines the differential Galois group L K 1 of 1 over Gal L (or of equations NVE). Then the main result of ZiglinMoralesRamis 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] MoralesRuiz suggested the quite natural conjecture that the higher Galois groups are also responsible for nonintegrability. 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 t of the Hamiltonian system (3). We write the general solution as , tz, where parametrizes the solutions near z t, with corresponding to it. Then we write (3) as 0 z H ,,= tz X xtz Denote the derivatives of , 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 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 xtzjk. One can show that the system of nonhomogeneous equations for () , k 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 LLL = 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 (MoralesRamisSimó) If the Hamiltonian system (3) is integrable then for each ,L,Gal LlKK 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 definitions, 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], MoralesRuiz [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 . 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 H, gln L is in Jordan form, is a nonsingular constant matrix and T 1n with (6) is 1 =diag,, n q q 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 . FGG ([18, 20]) The formal differential Galois group of equation (5) over 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 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 Y is a formal invariant of the differential equation (5). ET ([20]) The exponential torus relative to t solution and it he Y (6) is the gof differential roup  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 . mt ([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 . Now we turn to the Galois grr thoup ovee eld of co fi nvergent Laurent series . The general theory of summability ensures thattrix the ma H in (6) is multisummable along any nonsingular . In the case when all nonzero polynomials ray d 11 ij qxqx j have the same degree 1 this means that H is eith definitions and theoretical re on ([23]) 1. A sector is defined to be a set of the fo er convergent or 1summable. 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 qxQ 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 but they could be polynomials in a fractional power of 1 , [24]. Here we assume that the polynomials 1 j qx tor is a set of the form be monomials of degree 1 in 1 some of them being possibly zero. Then the matrix 1 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 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 = fx be analytic in some sector ,,Sd at =0x. We say that is asymptotic to =n n 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 nonnegative integer N and every 1 S one s ha 1 11 ! N =0 nN nSS n 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 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 th all power serie set ofes of Gevrey order ]) 1. Let 1. R(Ramis) ([22 1 x such that there exists an open sector whose opening >V and a holomorphic function 1 V such that is asymptotic to on V in Gevrey 1 sense. We will say that is 1summble inrection d (d being the bisecting line of V) and a the di is the 1sum of in the direction d. 2. If 1 x is 1 summable in all but a finite number of directions, we will say that it is 1summable. We will dnote e 1 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 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 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 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 1summable cri([18]) Let 1 x and d be a rection. Then the fllowing are equivalent: 1. f is 1summable in the direction d. 2. The convergent po let di o wer series 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 h this case, is its 1sum. 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 differential equation of order n with polynomial coefficients and 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 1summable 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 x in all but a fimber of directions, solution of the differential equation =Lyg such that asymptotic to nite nu x in Gevrey1 sense. Moreover x be obtained from can x by a BorelLaplace transform. s assume that the matrix Let u ) 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 yx . Next, series the j yx , 1jn lie at the first row of the matrix H. We remark that the behaviour of the elements of Copyright © 2011 SciRes. APM
174 the first row of the matrix T. STOYANOVA 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 x In this such that are 1summable 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 nonsingular direction d to any series j yx we associate a unique holomorphic function j equatio , b xsolutions of the equation (10) and 1sum of the corresponding series j yx along d suh that c j x is asymptotic to j yx in Gevrey1 sense (R). Replacing t series j yx (and their derivations) he in matrix H by theums we obtain a holomorphic matrix ir 1s H sector with opening isected by a nonar direction d, asymp totic to on a ngul>, bsi H in Gevrey1 sense on this sector and denotes the 1 sum of H. The newobtained matrix (1/ ) =e Qx xxxYH actual fundamental matrix quation (5) and denotes the 1sum of is an of e 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 Q) we define: 1. Eigenvalues =ij ij qq q 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 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 nonsingular neighboring direc tions of d. Letd Y ll d Y and denote the 1sums 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 generated 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 ( as a Hamiltonian System Painlevé equation, V P, is equient to th Hamiltonian system, [25], The fifthval e =,= H qp pq (11) with the Hamiltonian , 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). Tonautoousem (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. HF The symplectic structure is canonical in the variables ,,,qptF , i.e. =dpdqdFdt . In what follows we denote by the time variable. for 3. NonIntegrability of V , 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 nonequilibrium 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 , 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 . 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 variational 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 noncommu 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 . 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 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 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 sSimó 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. NonIntegrability for =1 To prove nonintegrability 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 zz and 212 212 2e,e 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 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 of the field z by the entries of the matrix (1 ) e Rz zz over z; 12 =e,e 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 be theot extenf ation (21). To determine the convergent Galois oup at =0z over PicardVessision 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 Gevrey1 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 Gevrey1 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 1sum 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 qqqz zzz (28) 2 23 2 11 11 =e 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 1summable and d z d are their 1sum. 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 z J zzz QH with z and 1 e 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 and 2 e . 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 ii . In the same manner, the difference between d and d is given by 2 =1 2Res e2=2 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 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 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 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 nX where = Id and is a unipotent matrix. Denote by S the Zariski closure of the subgroup St , nn then the elements , 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 and , 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 for any Gal L =0q, i.e. 1 e( zz ), which is an ction. In the same mannerobviously contradi =1c implies that 2 e 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 isomorphic to the extended fine Weyl group ofaf (1) 3 type, [5]. It is well known ted with the fifth Painlevé equation. In particular the Bäcklund trans formations remain the property nonintegrability. 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 nonauto 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 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,== 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 gnonintegrable 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 and 1 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 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 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 ( 1 1i from ). 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 . 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 wellkn 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 nonintegrability 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 ZiglinM Si e oralesRamis 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 II P and 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. 7. References [1] P. 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T. STOYANOVA Copyright © 2011 SciRes. APM 183 Divergent Power Series to Analytic for Ordinary Diff [22] J.P. Ramis, “Gevrey Asymptotics and Applications to Holomorphic Ordinary Differential Equations,” Differen ysis,” Cambridge University Press, Cambridge, 1989. [27] P. Berman and M. Singer, “Calculating the Galois group of 12 12 =0, ,LL yLL Completely Reducible Opera tors,” Journal of Pure and Applied Algebra, Vol. 139, No. 13, June 1999, pp. 323. doi:10.1016/S00224049(99)000031 tial Equations and Asymptotic Theory, Series in Analysis, World Scientific Singapore, Vol. 2, 2004, pp. 4499. [23] W. Balser, “From Functions,” Lecture Notes in Mathematics, SpringerVer lag, Berlin Heidelberg, Vol. 1582, 1994. [24] W. Wasov, “Asymtotic Expansions [28] L. Schlesinger, “Handbuch der Theorie der Linearen Differentialgleichungen,” Teubner, Leipzig, 1897. er ential Equations,” Dover, New York, 1965. [25] H. Watanabe, “Solutions of the Fifth Painlev’e Equatins I,” Hokkaido Mathematical Journal, Vol. 24, No. 2, 1995, pp. 231268. [29] T. Masuda, Y. Ohta and K. Kajiwara, “A Determinant Formula for a Class of Rational Solutions of Painlevé V Equation,” Nagoya Mathematical Journal, Vol. 168, 2002, pp. 125. [26] E. Whittaker and E. Watson, “A Course of Modern Anal
