On the Lax pairs of the sixth Painlev´e equation∗ Robert Conte †Service de physique de l’´etat condens´e (URA 2464), CEA–Saclay F–91191 Gif-sur-Yvette Cedex, France E-mail: [email protected] 7 0 25 October 2006 0 2 Abstract n a Thedependenceof thesixth equation of Painlev´eon itsfour parameters (2α,−2β,2γ,1− J 2δ)=(θ∞2 ,θ02,θ12,θx2) is holomorphic, therefore one expects all its Lax pairs to display such a 4 dependence. This is indeed the case of the second order scalar “Lax” pair of Fuchs, but the 2 second order matrix Lax pair of Jimbo and Miwa presentsa meromorphic dependenceon θ∞ (andaholomorphicdependenceonthethreeotherθj). Weanalyzethereasonforthisfeature ] and make suggestions to suppressit. I S Keywords: Sixth Painlev´e function, Fuchsian system, Lax pair, apparent singularity, isomon- . n odromic deformation. i MSC 2000 35Q58,35Q99 l n PACS 1995 02.20.Qs 11.10.Lm [ 1 Contents v 9 4 1 Introduction 1 0 1 2 Holomorphic Lax pair by scalar isomonodromy 2 0 7 3 Meromorphic Lax pair by matrix isomonodromy 3 0 / n 4 Towards a holomorphic matrix Lax pair 4 i l n 5 A Kimura-type Lax pair 5 : v i 6 Conclusion 6 X r a 1 Introduction Consider a second order linear ordinary differential equation for ψ(t) with five Fuchsian singula- rities, one of them t = u being apparent (i.e. the ratio of two linearly independent solutions remains single valuedaroundit) and the four othershaving acrossratiox. The conditionthat the ratio ψ1/ψ2 of two linearly independent solutions be singlevalued when t goes aroundany of these singularitiesresultsinoneconstraintbetweenuandx,whichis[2]thattheapparentsingularityu, consideredasafunctionofthecrossratiox,obeysthesixthPainlev´eequationP6. Initsnormalized form (choice (∞,0,1,x) of the four nonapparent Fuchsian singularities), this ODE is [2] 1 1 1 1 1 1 1 E(u) ≡ −u′′+ + + u′2− + + u′ 2(cid:20)u u−1 u−x(cid:21) (cid:20)x x−1 u−x(cid:21) u(u−1)(u−x) x x−1 x(x−1) + α+β +γ +δ =0, x2(x−1)2 (cid:20) u2 (u−1)2 (u−x)2(cid:21) ∗RIMS Kˆokyuˆroku Bessatsu,toappear (2007). Kyoto,15–20May2006. PreprintS2006/074. nlin.SI/0701049 1 its four parameters α,β,γ,δ representing the differences θ of the two Fuchs indices at the four j nonapparent singularities t=∞,0,1,x, (2α,−2β,2γ,1−2δ)=(θ2 ,θ2,θ2,θ2). (1) ∞ 0 1 x TheproofbyPoincar´e[11]oftheimpossibilitytoremovetheapparentsingularityinthesecond orderscalarisomonodromicdeformationcertainlymotivatedJimboandMiwatoconsider,inplace of the scalar isomonodromy problem, the matrix isomonodromy problem of the same order (two), ∂ ψ =Lψ, ∂ ψ =Mψ, [∂ −L,∂ −M]=0. (2) x t x t There indeed exists a choice [5] of second order matrices (L,M) whose isomonodromy condition also yields P6, in which the singularities of the monodromy matrix M in the t complex plane are four Fuchsian points of crossratio x, without the need for an apparent singularity. Thisbeautifulresulthoweverpresentsthedrawbacktohaveameromorphicdependenceonone ′′ of the four monodromy exponents θ , while u in P6 has a holomorphic such dependence. The j purpose of this work is to explore severaldirections in order to removethis drawbackfrom matrix Lax pairs. ApossibilitytoachievethatistoconsidersomesimplephysicalsystemadmittingaLaxpairand a reduction to P6. The corresponding reduction of its Lax pair could then provide a holomorphic Lax pair of P6. One such system if the three-wave resonant interaction, but the resulting Lax pair has third order, and its reduction to second order still encounters some obstacles [1]. The Maxwell-Bloch system [12] could be a better candidate because its Lax pair is second order. The paper is organized as follows. In section 2, we recall the scalar “Lax” pair of Richard Fuchs, because its expression is required later on. In section 3, we point out the meromorphic dependence in the second order Lax pair obtained by matrix monodromy. In section 4, we define in some detail the small amount of required computations in order to obtain a holomorphic Lax pair. In section 5, we explore the simplest possibility beyond the assumption of Jimbo and Miwa. The resulting Lax pair is linked to a type studied by Kimura [6] and the matrix elements are ′ algebraic functions of u,u,x while in the JM case they are rational functions. 2 Holomorphic Lax pair by scalar isomonodromy This pair [2, 3], as more nicely written in Ref. [4], is characterized by the two homographic invariants (S,C), ∂2ψ+(S/2)ψ =0, (3) t ∂ ψ+C∂ ψ−(1/2)C ψ =0, (4) x t t with the commutativity condition, X ≡S +C +CS +2C S =0, (5) x ttt t t where t(t−1)(u−x) −C = , (6) (t−u)x(x−1) u−x S 3/4 β1u′+β0 [(β1u′)2−β02]u(u−1) +fG(u) − 2 = (t−u)2 + (t−u)t(t−1) + t(t−1)(t−x) +fG(t), (7) x(x−1) 1 β1 =− , β0 =−u+ , (8) 2(u−x) 2 a b c d f (z)= + + + , (9) G z2 (z−1)2 (z−x)2 z(z−1) (2α,−2β,2γ,1−2δ)=(4(a+b+c+d+1),4a+1,4b+1,4c+1). (10) ′′ Like u in the definition of P6, this scalar Lax pair depends holomorphically on the four θ , j and also on their squares. Its singularities in the complex plane of t are the five Fuchsian points t=∞,0,1,x,u, among which t=u is apparent. 2 3 Meromorphic Lax pair by matrix isomonodromy Let us introduce the Pauli matrices σ k 0 1 0 −i 1 0 σ1 =(cid:18)1 0(cid:19), σ2 =(cid:18)i 0 (cid:19), σ3 =(cid:18)0 −1(cid:19), σjσk =δjk+iεjklσl, (11) σ+ = 0 1 , σ− = 0 0 . (cid:18)0 0(cid:19) (cid:18)1 0(cid:19) As provenin[5], the apparentsingularityofthe scalarLax paircanbe removedby considering a second order matrix Lax pair, ∂ Ψ=LΨ, ∂ Ψ=MΨ, (12) x t and defining the monodromy matrix M as the sum of four Fuchsian singularities t=∞,0,1,x, M0(x) M1(x) Mx(x) M = + + , M∞+M0+M1+Mx =0. (13) t t−1 t−x However, in order to integrate the differential system of the monodromy conditions, ∀t: L −M +LM −ML=0. (14) t x the choice of L is not unique and the type of dependence of L(x,t) on t must be an input. With the very convenient choice [5] of a simple pole at the crossratio t=x, Mx M0(x) M1(x) Mx(x) L=− , M = + + , M∞+M0+M1+Mx =0, (15) t−x t t−1 t−x and after minor transformations[10] mainly aimed at making all entries (L ,M ) algebraic(not jk jk only with algebraic logarithmic derivatives), one obtains the traceless, algebraic Lax pair, Mx M0 M1 Mx L = − +L∞, M = + + , (16) t−x t t−1 t−x (Θ∞−1)(u−x) L∞ = − σ3, (17) 2x(x−1) 2M∞ = Θ∞σ3, (18) u x 2M0 = z0σ3− σ++(z02−θ02) σ−, (19) x u u−1 x−1 2M1 = z1σ3+ σ+−(z12−θ12) σ−, (20) x−1 u−1 x x−1 u−x 2M = (θ2−z2) −(θ2−z2) σ−− σ+ x (cid:18) 0 0 u 1 1 u−1(cid:19) x(x−1) −(Θ∞+z0+z1)σ3, (21) 1 ′ 2 z0 = (x(x−1)u −(u−1)(u−Θ∞(u−x))) 2Θ∞x(u−1)(u−x)(cid:20) −(Θ2 +θ2)x(u−1)(u−x)+θ2(x−1)u(u−x)−θ2x(x−1)u(u−1) , ∞ 0 1 x (cid:21) −1 ′ 2 z1 = (x(x−1)u −u(u−1−Θ∞(u−x))) 2Θ∞(x−1)u(u−x)(cid:20) +(Θ2 +θ2)(x−1)u(u−x)−θ2x(u−1)(u−x)−θ2x(x−1)u(u−1) , ∞ 1 0 x (cid:21) (2α,−2β,2γ,1−2δ)=((Θ∞−1)2,θ02,θ12,θx2). The origin of the meromorphic dependence in (16), as displayed in z0 and z1, seems to be the simplifying assumption [5] that the residue M∞ can be chosen diagonal, Θ∞ M∞ = σ3. (22) 2 3 Indeed,whenΘ∞ vanishes,theresiduealsovanishesandonesingularpointislost,thuspreventing to obtain P6 which requires four nonapparent singular points. As an additional motivation of the present work, this meromorphic feature is also present in many discrete Lax pairs ofdiscrete P6 equations,for instance in the Lax pairfound by Jimbo and Sakai [8], as an output to the matrix discrete isomonodromy problem Y(x,qt)=A(x,t)Y(x,t), (23) A=A0(x)+A1(x)t+A2(x)t2, (24) where x is the independent variable, t is the spectral parameter, and the matrix A defines four singular points in the t complex plane. If the residue A2 at t=∞ is chosen diagonal[8, Eq. (10)], A2 =diag(κ1,κ2), (25) thentheLaxpaircontainsthedenominatorκ1−κ2and,whenκ1 =κ2,theisomonodromyproblem cannot yield a q−P6 equation. 4 Towards a holomorphic matrix Lax pair In order to get rid of this unwanted meromorphic dependence, let us change the assumptions on the matrix Lax pair (L,M) along the lines explored in Ref. [9]. For the assumption (13) on M, which must be kept, we adopt the convention θ2 j trM =0, detM =− =constant, j =∞,0,1,x, (26) j j 4 and we represent the four residues so as to preserve the invariance under permutation, 1 z (θ −z )u Mj = 2(cid:18)(θj +zjj)u−j1 j −zjj j(cid:19), j =∞,0,1,x, (27) in which z ,u are functions of x. j j After an assumption has been chosen for the dependence of L(x,t) on t, there is no need to integrate the monodromy conditions (14). Indeed, one a priori knows that their general solution is expressed in terms of a P6 function. Therefore a “lazy” method to perform the integration is to first convertthe matrix Lax pair (12) to scalar form, then to identify the result with the scalar Lax pair (3)–(4). Let us denote Ψ = t(ψ1 ψ2) the base vectors of the matrix Lax pair after rotation by an arbitrary constant angle ϕ, P = cosϕ sinϕ , ∂ Ψ=P−1LPΨ, ∂ Ψ=P−1MPΨ. (28) (cid:18)−sinϕ cosϕ(cid:19) x t ′ After elimination of ψ2 and removal of the first derivative ψ1 in the resulting second order linear ODE for ψ1, the identification of the two sets of coefficients (S,C) will provideL and M in terms of a solution u of P6. Whatever be the assumption for L, the three scalar conditions of zero sum for the residues, M∞+M0+M1+Mx =0, (29) under the condition that u0,u1,ux are all different, are first solved for z0,z1,zx,  uu1x−−JJ uux0zz01 ==++zzθθ∞∞01(cid:0)(cid:0)(cid:0)uuuu−1−x01uu11−1−x++11uuuu−x−0−x−01111(cid:1)(cid:1)−−−+uu−0−1((θθ11∞∞(cid:1)++−−θθzz1x∞∞(cid:0)u))uuu−x−0∞∞11uu−−−1−xuu11−1uu−x−x−011(cid:1)11++−−θθ((x0θθ∞∞(cid:0)uu++−x−111zz−−∞∞))uuuu−0−x−∞−∞1111(cid:1),, (30)  u0−J u1zx =+zθ∞x(cid:0)(cid:0)(cid:0)uu−0xu1−0+1uu−1−111(cid:1)−+u(−xθ1∞(cid:1)(cid:1)+−θz0∞(cid:0)(cid:0))uu−1∞1u−−0u1u−0−11(cid:1)(cid:1)1+−θ(θ1∞(cid:0)(cid:0)u+−01z−∞)uu−1−∞11(cid:1)(cid:1), 4 in which J denotes the Jacobian D(M∞,11,M∞,12,M∞,21) (u0−u1)(u1−ux)(ux−u0) J ≡ =− . (31) D(z0,z1,zx) u0u1ux 5 A Kimura-type Lax pair Following (16) and [9, Eq. (4.18)], let us assume M x L=− +L∞, L∞ =m(x)M∞, (32) t−x which defines the differential system ′ [Mx,M0]  M0 = x −m[M∞,M0],  M1′ = [Mxx−,M11] −m[M∞,M1], (33) ′ [Mx,M0] [Mx,M1] Such a choice ensures thatMM0+x =M−1+Mxx is a−firstxi−nte1gra−l,man[dMt∞he,rMefxo]r.e M∞ a constant. The system (33) is equivalent to P (u ,z ,θ ,m) Q (u ,z ,θ ,m) ′ j k k k ′ j k k k z = , u = , j ∈{0,1,x}, k ∈{∞,0,1,x}, (34) j x(x−1)u0u1ux j x(x−1)u0u1ux ′ ′ in which P ,Q denote polynomials of their arguments, and the closure conditions z = (z ) j j j j between the systems (34) and (30) are identically satisfied. The identification of the two C’s of the two scalar Lax pairs of the type (3)–(4) is equivalent to the two relations u−x (cos2ϕ+1)u∞+(cos2ϕ−1)u−∞1  (cid:20)m+ x(x−1)(cid:21)(cid:20)z∞−θ∞(cos2ϕ+1)u∞−(cos2ϕ−1)u−∞1−2sin2ϕ(cid:21)=0, (35)  when ϕ=0: (z∞−θ∞)u∞u(u−x)+(z0−θ0)u0(u−x)−(zx−θx)ux(x−1)u=0, in which, for brevity, the rotation angle ϕ has been set to 0 in the second relation. Solving the first equation in (35) for the second factor would result in the vanishing of M∞ with θ∞, hence in the samesingularityof the Lax pairat θ∞ =0than in(16). Thereforethis first equation is solved for m, and in the second one can eliminate z0,z1,zx with (30), u−x m=− ,  x(x−1) (36)  F(z∞,u∞,θ∞,u0,θ0,ux,θx,u,x,eiϕ)=0, in which F is a polynomial of its arguments, of degree two in u and each u . j Before transformation to the normalized form (3), the second order ODE for ψ1 is then 2 d ψ1 p4(t) dψ1 p6(t) (t−u)p1(t) dt2 + t(t−1)(t−x) dt + [t(t−1)(t−x)]2ψ1 =0, (37) in which p denotes polynomials of degree j whose dependence on x has been omitted. The j condition that (t − u)p1,p4,p6 have a common zero t (otherwise there would be two apparent singularities) results in (when ϕ=0), x x−1 x(1−x) (z∞−θ∞)u∞ =(z0−θ0)u0u2 =(z1−θ1)u1(u−1)2 =(zx−θx)ux(u−x)2, (38) andtheserelationsimplyp1(t)=t−uandamultiplicitytwoforthezerot=uoftheelementM12 of the monodromy matrix M. As proven in [7], this results in a difference of 3 between the two Fuchs indices at the apparent singularity t = u, not 2 like in (7). The Schwarzian associated to (37), which cannot be identified to (7), must then be identified to the Schwarzian of the equation labelled Ln in [6]. The resulting matrix Lax pair will probably be holomorphic in the four θ but VI j surely not rational in u,u′,x, since the transformation between the apparent singularities of Ln VI and (3) is not birational [7]. Therefore its explicit expression will not be given. 5 6 Conclusion In order to build a second order matrix Lax pair of P6(u,x) at the same time holomorphic in θ j ′ and rational in u(x),u(x),x, it is necessary to make an assumption for L which is different from (32), probably by adding to L a term linear in t like in [6, §6] and [9, Eq. (4.18)]. This will be the subject of future research. Acknowledgments RC thanks the Japanese organisers for their warm hospitality. References [1] R. Conte, A. M. 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