Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations
aa r X i v : . [ n li n . S I] J a n Gauge Symmetry Origin of B¨acklund Transformations forPainlev´e Equations
V.C.C. Alves , H. Aratyn , J.F. Gomes , and A.H. Zimerman Instituto de F´ısica Te´orica-UNESP, Rua Dr Bento Teobaldo Ferraz 271,Bloco II, 01140-070 S˜ao Paulo, Brazil Department of Physics, University of Illinois at Chicago, 845 W. Taylor St.Chicago, Illinois 60607-7059January 18, 2021
Abstract
We identify the self-similarity limit of the second flow of sl ( N ) mKdV hierarchy withthe periodic dressing chain thus establishing a connection to A (1) N − invariant Painlev´eequations. The A (1) N − B¨acklund symmetries of dressing equations and Painlev´e equationsare obtained in the self-similarity limit of gauge transformations of the mKdV hierarchyrealized as zero-curvature equations on the loop algebra b sl ( N ) endowed with a principalgradation. We study the integrable mKdV hierarchy on the loop algebra of sl ( N ) = A N − endowedwith a principal gradation. We show that the second t = t flow of this hierarchy real-ized as a zero-curvature equation becomes in the self-similarity limit the dressing chaininvariant under the extended affine Weyl group A (1) N − that gives rise to A (1) N − invariantPainlev´e equations for all N after imposing the periodicity condition. For N = 3 and N = 4 cases the corresponding Painlev´e equations are well-known as Painlev´e IV and Vequations and their relations to the self-similarity limit of the sl ( N ) mKdV equation wasestablished in [17].In this paper we present an explicit and straightforward construction of the dressingchain and Painlev´e equations in terms of mKdV objects in the self-similarity limit. Oneadvantage of the zero-curvature approach is that it naturally allows for formulation of theB¨acklund symmetries as gauge transformations that preserve the algebraic structure ofthe gauge potentials, see [9] and references therein. Here we utilize gauge symmetries ofthe zero-curvature equations defined on the loop algebra of sl ( N ) to derive the B¨acklundsymmetries of the extended affine Weyl group A (1) N − in the context of Painlev´e equa-tions including higher-order Painlev´e equations that follow from the associated integrablestructure.In order to obtain our results we connect several observations made by various re-searchers coincidentally around the same time. In 1993, the periodic dressing chain andits Hamilton formalism appeared as the main focus of study [21] in the context of thespectral theory of the Schr¨odinger operators. Its symmetry group was shown in [1] to beisomorphic to the extended affine Weyl group A (1) N − thus clearing the way to establishing quivalence of periodic dressing chains [19, 22, 20, 18] with the corresponding Painlev´esystems invariant under the same A (1) N − symmetry groups [16]. There were other con-temporary observations that now can be understood as related to these breakthroughsand remarkably made in the same year. In [17] the first non-trivial equation of sl ( N )mKdV model was given and shown to give rise to Painlev´e equations for N = 3 , A (1) N invariant Painlev´e systems [6] defined for integrable systems with a different grada-tion than the principal gradation used throughout this paper.Here we are able to explicitly tie several of these observations into an unified approachby realizing the self-similarity limit of the second flow of sl ( N ) mKdV hierarchy as N -periodic dressing chain with its parameter α = P Ni =1 α i taking values N/ A (1) N − from gauge symmetry of the second flow inthe self-similarity limit.The paper is organized as follows. In Section 2 we present the zero curvature approachto the sl ( N ) mKdV hierarchy and use it to explicitly derive its second flow that is identifiedin Section 3 with the dressing chain and A (1) N − invariant Painlev´e equations in the self-similarity limit. In Section 4 we formulate the B¨acklund symmetries of hierarchy equationsas a class of gauge transformations that maintain the matrix form of the potentials A µ , µ = x, t that enter the zero-curvature equations of the integrable hierarchy. In Subsection 4.2these gauge transformations are shown to give rise to the extended affine Weyl symmetryof Painlev´e equations. The class of gauge symmetry generators we are working with issuch that each such generator can be factorized into a product of maximum N − A (1) N − . As discussed in thenext section 5 such construction further extends explicit invariance under A (1) N − obtainedfrom B¨acklund gauge symmetries to all higher-order Painlev´e equations [10] that appearin the self-similarity limits of higher flows of the mKdV integrable hierarchy. These flowsare conveniently labeled as t nN + k with k = 1 , ..., N − n = 0 , , ... (see formula (A.9)for an underlying graded structure behing this labeling). We illustrate this invariancefor the mKdV hierarchy with flows t n +1 for N = 2 , k = 1 and n = 1 , ,
3. Explicitlywe will work with t , t and t flows of the mKdV hierarchy and the associated Painlev´eequations of order order 2 , A (1)1 Weyl groupsymmetry as consequence of invariance of zero curvature representation under B¨acklundgauge transformation.In Section 6, that provides an outlook, we discuss the modification of the dressingchain that maintains the Kovalevskaya-Painlev´e property together with the part of itsB¨acklund symmetry. This relates to the program that we have pursued in several papers[3, 4, 5] attempting to connect integrability to the remaining symmetry of Painlev´e modelsthat survives the partial breaking of the extended affine Weyl group A (1) N − by deformationterms that preserve some notion of integrability.Some background material on the sl ( N ) algebra and technical details of deriving thePainlev´e equations out of the dressing chain are given in the two appendices to ensure thepaper being self-contained. We will be working with a loop algebra of sl ( N ) endowed with a principal gradation asdefined in (A.7) of Appendix A, where we provide brief account of sl ( N ) Lie algebra, itsroots, fundamental weights, and its loop algebra b sl ( N ) generalization, that all constitute lgebraic foundation of zero-curvature formulation of the sl ( N ) mKdV flows. The startingpoint of the zero-curvature construction is the semi-simple b sl ( N ) element of grade one E (1) = N − X k =1 E α k + λE − α − ... − α N − . (2.1)We refer the reader to Appendix A for definitions of basic matrices E α a , . . ., E α a + ... + α b , a, b =1 , . . ., N − t m flow of sl ( N ) mKdV hierarchy is given by solving the following zero-curvatureequation: [ ∂ x + E (1) + A , ∂ t m + D (0) + D (1) + · · · + D ( m ) ] = 0 , (2.2)where D ( i ) ∈ G i (see Appendix A for definition of G i ) are to be solved for. Further A = N − X k =1 v k h α k = J · · · J · · · · · · J N , is a diagonal element of grade zero that satisfies the trace zero condition P Nk =1 J k = 0.The elements h α k of the Cartan subalgebra are defined in Appendix A. The followingrelations: v i − v i − = J i , i = 1 , . . ., N − , v = 0 , v N = 0 , (2.3)hold between v k , k = 1 , . . ., N − J i , i = 1 , . . ., N . The relation v i = P ij =1 J j followsfrom (2.3) and is highly reminiscent of relation (A.5) between fundamental weights Λ i and e j vectors.With b sl ( N ) being endowed with the grading structure (A.8) the zero-curvature equa-tion (2.2) can be decomposed into separate equations according to their grade. The highestgrade component of equation (2.2), [ E (1) , D ( m ) ] = 0, is solved by the grade m element D ( m ) in the kernel of E (1) . From now on we will focus on the case of m = 2 for which itfollows that, D (2) = N − X k =1 E α k + α k +1 + λE − ( α + ... + α N − ) + λE − ( α + ... + α N − ) . (2.4)The lower grade matrices D ( i ) , i = 0 , D (1) = N − X k =1 F k E α k + F N λE − ( α + ... + α N − ) , (2.5)has coefficients F N , F i , i = 1 , . . ., N − F k = v k +1 − v k − = J k + J k +1 , k = 1 , . . ., N − , F N = v − v N − = J N + J , (2.6)as determined by the grade 2 component of the zero-curvature equation (2.2):[ E (1) , D (1) ] + [ A , D (2) ] + ∂ x D (2) = 0 . The grade 1 projection of the zero-curvature equation (2.2):[ E (1) , D (0) ] + [ A , D (1) ] + ∂ x D (1) = 0 , with D (0) = P N − a =1 d a h α a , can be cast as a vector equation ∂ x F + V F = Kd , (2.7) ith F = F ... F N − , d = d ... d N − , v = v ... v N − , representing key objects as N − V denotes the diagonal matrix V km = (2 v k − v k − − v k +1 ) δ km = ( J k − J k +1 ) δ km , (2.8)and K is the ( N − × ( N −
1) Cartan matrix given in (A.4).Solving (2.7) and inserting it into the grade 0 projection of the zero-curvature equation(2.2) determines the flow of the model through equation ∂ t A = ∂ x D (0) , with t ≡ t , thatreads in vector notation as : ∂ t v = ∂ x d . (2.9)After multiplying both sides of relation (2.9) by the Cartan matrix K and making use ofrelation (2.7) we obtain: ∂ t N − X m =1 ( K ) km v m = ∂ t (2 v k − v k +1 − v k − ) = ∂ x ( ∂ x F + VF ) k = ∂ x (cid:20) ∂ x ( v k +1 − v k − ) + V kk ( v k +1 − v k − ) (cid:21) , k = 1 , . . ., N − . (2.10)Recalling the identities (2.6) and (2.8) the above equation (2.10) can be rewritten as: ∂ t ( J k +1 − J k ) = − ∂ x (cid:20) ∂ x ( J k + J k +1 ) − ( J k +1 − J k ) ( J k + J k +1 ) (cid:21) , k = 1 , . . ., N − , (2.11)that defines t flows of sl ( N ) mKdV hierarchy in agreement with the result given earlierin [17]. From equation (2.11) one can derive a flow for an individual J m as ∂ t J m = ∂ x (cid:20) N − X k =1 N − kN J ′ k + m + J m − N N X k =1 J k (cid:21) , m = 1 , . . ., N − . (2.12) A (1) N − Painlev´e Equa-tions
Introducing the self-similarity limit through relations z = t − / x, J i ( x, t ) = t − / J i ( z ) , (3.1)one obtains in such limit from equations (2.11) after an integration :( J i + J i +1 ) z = − z J i − J i +1 ) − (cid:0) J i − J i +1 (cid:1) + γ i , i = 1 , . . ., N , (3.2)with γ i being integration constants that satisfy P Ni =1 γ i = 0 and where we introduced theperiodicity condition J N + i = J i .Define now: j i ( z ) = J i ( z ) + 14 z , (3.3)with the zero trace condition P Nk =1 J k = 0 replaced by N X i =1 j i = N z (3.4) nd impose periodicity condition: j i = j i + N . (3.5)In this notation and with α i = γ i + , such that P Ni =1 α i = N/
2, equations (3.2)simplify to: ( j i + j i +1 ) z = − ( j i − j i +1 ) ( j i + j i +1 ) + α i , i = 1 , . . ., N , (3.6)in which we recognize the dressing chain equation [21].There is also an alternative way of obtaining the dressing chain equation (3.6) from the t -flow (2.11) of the sl ( N ) mKdV hierarchy that employs the traveling wave reduction: J i ( x, t ) = j i ( z ) − C , z = x − Ct . (3.7)Here C is a constant representing speed of a wave. We find that such reduction yields thesame differential system (3.6), but with the major difference being that the summationof j i now yields a constant: N X i =1 j i = N C N equations (3.6) is:2 N X j i z = N X α i = 0 . (3.9)This is the case of α ≡ P Ni =1 α i = 0 for which the periodic dressing chain admits aparametric dependent Lax representation [21]. Equation (3.6) with condition (3.9) for N = 3 yields I (denoted as equation XXX in [15]) as shown in [3]. See reference [2] forrelation between equation (3.6) with condition (3.9) for N = 4 and I (equation XXXVIIIin [15]).Define now f i = j i + j i +1 , (3.10)then due to identities shown in Appendix B it follows that equation (3.6) becomes in thisnew notation df i dz = f i k X r =1 (cid:0) f i +2 r − − f i +2 r (cid:1) + α i , i = 1 , . . ., N , (3.11)for N = 2 k + 1 odd and with conditions P Ni =1 f i = N z/ P Ni =1 α i = N/
2. Equation(3.11) reproduces A (1) N − Painlev´e equations for N odd. For even N = 2 n we obtain from(3.6) nz df i dz = f i X ≤ r ≤ s ≤ n − f i +2 r − f i +2 s − X ≤ r ≤ s ≤ n − f i +2 r f i +2 s +1 + n X r =1 (1 − α i +2 r ) + nzα i , i = 1 , . . ., N (3.12)reproducing A (1)2 n − = A (1) N − Painlev´e equations with conditions n X r =1 f i +2 r = n X r =1 f i +2 r +1 = n z . (3.13)We notice that equations (3.6) are manifestly invariant under order- N automorphisms: π ( j i ) = j i +1 , π ( α i ) = α i +1 , (3.14) ρ ( j i ) = − j N +1 − i , ρ ( α i ) = − α N +1 − i , ρ ( z ) = − z . (3.15)In the next section we will obtain the remaining B¨acklund symmetries by employing thegauge transformations of the zero-curvature equations. Gauge transformations of sl ( N ) mKdV zero-curvature equations Below we will discuss symmetries of equations (3.6) derived from gauge transformation ofthe b sl ( N ) zero-curvature equation (2.2) compactly rewritten as[ ∂ x + A x , ∂ t + A t ] = 0 , (4.1)with A x = E (1) + A , A t = D (0) + D (1) + D (2) . The term D (2) is given in (2.4), D (1) in (2.5) and D (0) = P N − a =1 d a h α a is such that itsmatrix elements satisfy the second-flow equation ∂ t v i = ∂ x d i . We will now discuss thegauge symmetries of equation (4.1) that preserve the matrix form of potentials A x , A t and will be shown to reproduce the B¨acklund symmetries of Painlev´e equations in theself-similarity limit. In the setting of (4.1) we will explore invariance of (2.11) under thegauge transformation A µ ( J i ) → A µ ( ˜ J i ) = U A µ ( J i ) U − + U ∂ µ U − , µ = x, t (4.2)or U A µ ( J i ) = A µ ( ˜ J i ) U + ∂ µ U (4.3)where U = U ( ˜ J , J ) maps from J i configuration to another one denoted by ˜ J i , whilepreserving the zero-curvature equation.We will work with the gauge transformations that has been previously constructed toproduce B¨acklund transformations in the context of sl ( N ) mKdV hierarchy [9] (see also[14] for the associated relativistic Toda model) to be of the form, U = I + U − , where I is the identity matrix and U − is a grade − b sl ( N ) : U ( ˜ J , J ) ≡ U ( β , . . ., β N ) == . . . λ − β N ( x, t ) β ( x, t ) 1 · · ·
00 . . . . . . · · · ... β N − ( x, t ) 1 00 · · · β N − ( x, t ) 1 (4.4)where β i ( x, t ) , i = 1 , . . ., N are coefficients parameterizing grade − b sl ( N )and will be determined by imposing (4.3) as a graded equation.For convenience we employed above a notation that expresses explicitly the dependenceof gauge transformation U on the parameters β i which in turn, depend upon the originaland the transformed configurations, { J } and { ˜ J } respectively.We now consider matrix elements of both sides of relation (4.3) and derive differentialequations for the gauge parameters β i that follow from them. Plugging expression for thematrix U into equation (4.3) with µ = x and considering its diagonal ( i, i ) element weobtain an expression :( i, i ) : ˜ J i = J i − β i + β i − , i = 1 , . . ., N (4.5)for the transformed configuration of ˜ J i in terms of J i and the β i coefficients of U ( β , . . ., β N ).The ( i + 1 , i ) element of (4.3) with µ = x is given by( i + 1 , i ) : β i,x = J i β i − ˜ J i +1 β i = β i ( − β i + β i +1 + J i − J i +1 ) , i = 1 , . . ., N − ntegrating (4.6) yields : β = B ( t ) e R ( J − ˜ J ) dx , · · · , β i = B i ( t ) e R ( J i − ˜ J i +1 ) dx , · · · , β N = B N ( t ) e R J N − ˜ J dx (4.7)where we indicated explicitly that B i -coefficients do not depend on x but may depend onthe t variable. However considering the ( i + 1 , i ) element of (4.3) with µ = t we obtain :( i + 1 , i ) : ∂ t ∂ x ln β i = ∂ t ( J i − ˜ J i +1 ) → β i = B i ( x ) exp Z ( J i − ˜ J i +1 ) dx, i = 1 , . . ., N − B i are constants as is consequently the product ofall β i : β ( x, t ) · · · β N ( x, t ) = B · · · B N = constant (4.9)which shows that there are in fact only N − β − parameters.For diagonal elements of (4.3) for µ = t we obtain :( d i +1 − ˜ d i +1 ) − ( d i − ˜ d i ) + β i F i − β i +1 ˜ F i +1 = 0 , i = 1 , . . ., N − d − ˜ d + β N F N − β ˜ F = 0 d N − − ˜ d N − + β N ˜ F N − β N − F N − = 0 (4.10)Next we use that ∂ x d i = ∂ t v i and ˜ v i = v i + β N − β i with F i , F N defined in (2.6) andaccordingly transforming as ˜ F i = F i + β i − − β i +1 , to obtain from (4.10) that the quantitiesΓ i ≡ ∂ t β i − ∂ x ( β i F i − β i β i +1 ) , (4.11)are always equal to each other:Γ i = Γ i +1 , i = 1 , . . ., N − . (4.12)A composition law may be derived by acting successively with U ( ˜ J , J ) to transform from J to ˜ J , followed by ˜ J → ˜˜ J transformation, U ( ˜˜ J, ˜ J ). It therefore follows the compositegauge transformation U ( ˜˜ J, J ) = U ( ˜˜ J, ˜ J ) U ( ˜ J , J ) (4.13)We start by making a remark concerning matrix U ( β , . . ., β i − , , β i +1 , β i +2 , β i +3 , . . ., β N )with one parameter β i = 0 being zero. As a matrix it can be represented by an orderedproduct of matrices U i with only one non-zero parameter β i : U i ≡ U (0 , . . ., β i , . . ., . (4.14)In order to maintain the form of the gauge matrix as in equation (4.4) the product mustbe ordered starting from the right with U i +1 ending at U i − as can be explicitly verifiedto be, U ( β , . . ., β i − , , β i +1 , β i +2 , β i +3 , . . ., β N ) = U i +1 U i +2 · · · U N − U N U U · · · U i − . (4.15)In the next subsection it will be explained that the gauge matrices considered in thissection must, in the self-similarity limit, have at least one parameter equal to zero. Theyall can be represented by an ordered matrix composition in terms of a basis of singlegauge matrices U j since the right hand side of equation (4.15) only involves the matrixmultiplication of single U i matrices. However each gauge transformation not only acts asa matrix but also as a transformation that obeys the composition rule (4.13). To takeinto consideration those two actions we will introduce below a “star”-product of two gaugematrices (4.27). basis for the B¨acklund transformation can be proposed to consist of generators withonly one non-zero coefficient, β i = 0 , i = 1 , · · · N −
1, it holds from the transformationrule (4.5) that U i from (4.14) generates: U i : J i → J i − β i , J i +1 → J i +1 + β i , J j → J j , j = i, i + 1 (4.16)with β i that satisfies differential equations : β i, x = β i ( − β i + J i − J i +1 ) (4.17)and ∂ t β i = ( F i β i ) x = (( J i + J i +1 ) β i ) x (4.18)due to relations (4.6) and (4.12). Similar relations were derived in [17] by a differentmethod. Recall now the self-similarity limit (3.1) that in accordance with equations (4.5) and (4.6)needs to be augmented by β i ( x, t ) = t − / b i ( z ) . (4.19)The self-similarity limit can not be applied consistently to the constant in (4.9) as longit is different from zero since the self-similarity limits of the right and left hand side ofequations ∂ x ( β ( x, t ) · · · β N ( x, t )) = ∂ t ( β ( x, t ) · · · β N ( x, t )) = 0will lead to inconsistent limits if all β i ( x, t ) = 0. Thus at least one of the B i constants willneed to vanish for us to be able to take the self-similarity limit. Clearly in such case Γ i = 0in (4.12) and this equation yields in the scaling limit the following algebraic relations : b i b i +1 − b i ( j i + j i +1 ) + κ i = b i b i +1 − b i f i + κ i = 0 , i = 1 , . . ., N − κ i are integration constants. Furthermore equation (4.5) becomes in the self-similarity limit˜ j i = j i − b i + b i − → ˜ f i = f i − b i +1 + b i − , i = 1 , . . ., N (4.21)and equation (4.7) becomes b = b (0)1 e R j − ˜ j dz , · · · , b i = b (0) i e R j i − ˜ j i +1 dz , · · · , b N = b (0) N e R j N − ˜ j dz (4.22)In the self-similarity limit the choice (4.14) results in constants b (0) i = 0 and b (0) j = 0 , j = i ,and we obtain from relations (4.20) that κ i = b i f i → b i = κ i f i (4.23)Let us recall equation (4.18) describing the time flow of β i parameter. According to theself-similarity limits (3.1) and (4.19) we obtain from (4.18) :(( J i + J i +1 + z/ b i ) z = 0 → /b i ( z ) = ( J i ( z ) + J i +1 + 12 z ) /κ i = ( j i + j i +1 ( z )) /κ i , (4.24)where we have chosen the integration constant κ i to be in agreement with (4.23). We nowturn our attention to equation (4.17) that becomes in the self-similarity limit b i z = b i ( − b i + j i − j i +1 ) , → ∂ z ln b i + b i = j i − j i +1 (4.25) lugging relation b i = κ i /f i from equation (4.24) into equation (4.25) one obtains ∂ z ln (cid:18) κ i f i (cid:19) + κ i f i = j i − j i +1 that agrees with the dressing equation (3.6) for κ i = α i . Indeed it is easy to explicitlyverify that the dressing chain equation is invariant under the symmetry operations j i U i −→ ˜ j i = j i − b i = j i − κ i j i + j i +1 , j i +1 U i −→ ˜ j i +1 = j i +1 + b i = j i +1 + κ i j i + j i +1 ,j k U i −→ j k , k = i, k = i + 1 (4.26)generated according to transformation rule (4.16) by self-similarity limit counterpart ofquantity defined in (4.14) for κ i = α i , when it is accompanied by transformations ofcoefficients α i → − α i , α i ± → α i ± + α i . We will derive below these transformationsdirectly from actions of U i matrices (see equations (4.33) and (4.37)). Thus the abovetransformation induced by the gauge matrix U i from relation (4.14) agrees with knownB¨acklund transformations of A (1) N − Painlev´e equations [1].We will need to define the composition laws for the gauge transformations that willhold when acting with an additional gauge transformation U ( ˜˜ J, ˜ J ) following U ( ˜ J , J ) asshown in (4.13). The ordering of equation (4.13) will be important when defining belowproducts of U i from equation (4.14). We now define a “star”-product of two single U i and U k matrices that follows the composition rule (4.13) : U i ( b i ) ⋆ U k ( b k ) = U i (˜ b i ) U k ( b k ) . (4.27)Note that in addition to matrix multiplication between U i and U k (on the right handside of the above equation), U k simultaneously acts through transformation of all thequantities b i , j i to ˜ b i , ˜ j i according to j i U k −→ ˜ j i = U k ( j i ) = j i → j i i = k, i = k + 1 j i → j i − b i i = kj i → j i + b i − i = k + 1 (4.28)where we used transformation rule (4.21). The following commutation relation followsfrom (4.28) and from the corresponding rules of matrix multiplication for matrices U i and U k that do not have any neighboring matrix element b i , b k : U i ( b i ) ⋆ U k ( b k ) = U k ( b k ) ⋆ U i ( b i ) , i = k, i = k ± k = i meaning we let U i ( b i ) act twicewith each U i acting simultaneously through matrix multiplication and through transfor-mation of all the quantities b k , j k to ˜ b k , ˜ j k . Thus the result of performing the gaugetransformation twice is U i ( b i ) ⋆ U i ( b i ) or explicitly : U i ( b i ) ⋆ U i ( b i ) = U i (˜ b i ) U i ( b i ) , where ˜ b i = U i ( b i ) must satisfy equation˜ b i z = ˜ b i ( − ˜ b i + ( j i − b i ) − ( j i +1 + b i )) = ˜ b i ( − ˜ b i + j i − j i +1 − b i ) (4.30)obtained from equation (4.25) by acting with U i . Plugging in the above expression equa-tion (4.25) that amounts to setting j i − j i +1 = ∂ z ln b i + b i we get: ∂ z ln ˜ b i + ˜ b i = ∂ z ln b i − b i → ∂ z ln ˜ b i b i = − ˜ b i − b i (4.31) ince ˜ b i b i = ˜ κ i κ i because ˜ f i = f i , the left hand side of the last equation in (4.31) is zero andconsequently ˜ b i = − b i and ˜ κ i = − κ i or ˜ α i = − α i . It follows now by simple multiplication U i ⋆ U i = U (0 , . . ., , − b i , , . . . U (0 , . . ., , b i , , . . .
0) = 1 (4.32)with the identity matrix on the right hand side of the above equation. We thus obtainthat U i ⋆ U i = 1 for all i = 1 , , . . ., N . We also learn that α i U i −→ − α i (4.33)Next we will verify that U i i +1 ≡ U (0 , . . ., , b i , b i +1 , , . . .
0) = U ( b i ) ⋆ U ( b i +1 ) (4.34)according to the definition given in relations (4.27) and (4.28). Therefore we now considerthe U -matrix from (4.4) with two neighboring β i , β i +1 that are different from zero and allthe other β j = 0. The following equations hold for β ’s and their self-similarity limits : β i, x = β i ( − β i + β i +1 + J i − J i +1 ) → ∂ z ln b i + b i = b i +1 + j i − j i +1 ,β i +1 , x = β i +1 ( − β i +1 + J i +1 − J i +2 ) → ∂ z ln b i +1 + b i +1 = j i +1 − j i +2 ,∂ t β i = ( β i F i − β i β i +1 ) x → b i = κ i f i − κ i +1 /f i +1 ,∂ t β i +1 = ( β i +1 F i +1 ) x → b i +1 = κ i +1 f i +1 , (4.35)which agree with the dressing equations (3.6) for f i , f i +1 for κ i +1 = α i +1 , κ i = α i + α i +1 .Comparing expression for b i given in (4.35) with those in (4.28) that define the ⋆ -product(4.27) we find that the relation (4.34) holds.Further it follows that U i i +1 , as defined in (4.34), with the non-zero neighboring U -matrix elements b i , b i +1 will induce the following non-zero transformations f i → ˜ f i = f i + ✟✟✟ b i − − b i +1 = f i − α i +1 f i +1 ,f i +1 → ˜ f i +1 = f i +1 + b i − ✟✟✟ b i +2 = f i +1 + α i + α i +1 f i − α i +1 f i +1 ,f i +2 → ˜ f i +2 = f i +2 + b i +1 − ✟✟✟ b i +3 = f i +2 + α i +1 f i +1 ,f i − → ˜ f i − = f i − + ✟✟✟ b i − − b i = f i − − α i + α i +1 f i − α i +1 f i +1 , (4.36)where we crossed out those coefficients b i ’s that are absent in the U i i +1 matrix.If we now define the s i +1 transformation as : s i +1 : f i → ˜ f i = f i − α i +1 f i +1 , f i +1 → ˜ f i +1 = f i +1 , f i +2 → ˜ f i +2 = f i +2 + α i +1 f i +1 , (4.37)together with s i +1 : α i → ˜ α i = α i + α i +1 , then we can represent the transformation (4.36)as a result of action of s i +1 followed by s i transformation: s i : ˜ f i → ˜ f i = f i − α i +1 f i +1 , ˜ f i +1 → ˜ f i +1 + ˜ α i ˜ f i = f i +1 + α i + α i +1 f i − α i +1 f i +1 ˜ f i +2 → ˜ f i +2 = f i +2 + α i +1 f i +1 , ˜ f i − → ˜ f i − − ˜ α i ˜ f i = f i − − α i + α i +1 f i − α i +1 f i +1 ote also that as follows from relation (4.33) s i ( α i ) = − α i . Thus the gauge transformation U i i +1 agrees with the composite s i s i +1 B¨acklund transformation and it follows from thisidentification by repetitive action with s i s i +1 that( U i ⋆ U i +1 ) = 1 . (4.38)More generally the gauge transformation U (0 , . . ., , b i , b i +1 , . . ., b i + k , , . . .,
0) can befactorized into a star-product of one parameter gauge transformations U i that correspondsto the composite s i s i +1 . . .s i + k B¨acklund transformation with relations between b i ’s beingof the form (for k = 3) : b i = κ iκ i +1 κi +2 κi +3 bi − ji − ji +3 − ji +2 − ji +3 − j i +1 − j i +2 − j i − j i +1 and so on for higher k .The gauge matrices U (0 , . . ., b i , b i +1 , . . ., b i + n , . . .,
0) and U (0 , . . ., b j , b j +1 , . . ., b j + m , . . ., j + m < i − i + n < j − s i · · · s i + n with s j · · · s j + m under the same conditions.The key conclusion of this section is that the identities (4.29), (4.38) and (4.32) estab-lish equivalence of the group of gauge transformation endowed with the ⋆ -product to theextended affine Weyl group A (1) N − with its fundamental relations s i = 1 , ( s i s i +1 ) = 1 , s i s k = s k s i , k = i ± ∂ x ( β · · · β N ) → ∂ z ( b · · · b N ) (4.39) ∂ t ( β · · · β N ) → − C∂ z ( b · · · b N ) (4.40)and we conclude that ∂ z ( b · · · b N ) = 0 (4.41)Contrary to the case of the self-similarity limit, the condition b · · · b N = 0 is no longerthe only remaining option. From equations (4.17) we find b i, z = b i ( − b i + j i − j i +1 ) (4.42)and combined with relation (4.20) we find that the condition (4.41) has another validsolution apart from b · · · b N = 0, namely that κ · · · κ N = 0 while b i = 0 , i = 1 , ..., N . Letus set κ i − = 0, then from relations (4.20) we find a solution: b i − = f i − , b i +1 = f i − κ i f i − , b i +2 = f i +1 − κ i +1 f i − κ i f i − etc. Thus in this case we can obtain a solution that has all the coefficients b i = 0. Willsuch transformation still be equivalent to a finite product of B¨acklund transformations s i ?To answer this question let us consider for illustration the case of N = 3 and set κ = 0.Then the relevant 3 × U ( b , b , b ) has arguments: b = f , b = κ f + f , b = κ κ f + f + f (4.43)Recalling that the b i transform the j i according to relation (4.21) we find˜ f = π ( f + α f ) , ˜ f = π f − α f + α + α α f + f ! , ˜ f = π f − + α + α α f + f ! , (4.44) sing that κ = α , κ = α + α (4.45)as obtained from plugging the above relations directly back into (3.6). We also get thetransformation on the constants α i of (3.6) α → − α − α , α → α + α + α , α → α (4.46)We see that despite of having all the three b i -coefficients different from zero that thetransformation U ( b , b , b ) obtained in the traveling wave reduction is equivalent to thecomposition of B¨acklund transformations πs s that this time involves the automorphism π in addition to transformations s i . Thus this example confirms that the conclusionsobtained in our study of gauge transformations in the self-similarity limit extend mutatismutandis to the traveling wave reduction. In references [12, 13] it was observed that the B¨acklund symmetries obtained out of thegauge symmetries of the zero-curvature equations will hold for all flows of the integrablemodel. We can here extend this observation to all higher-order Painlev´e equations thatare derived from these flows. The point we are making here is that the affine extended A (1) N − Weyl group is the symmetry group for all Painlev´e equations that are derived byappropriate similarity transformations from the higher flows of soliton equations. Forsimplicity we will illustrate this point for N = 2 mKdV hierarchy with the commutingflows t n +1 , n = 1 , , , . . . . Here we focus at the first three higher times t , t , t . For allthe flows the A x matrix remains defined by: A x = (cid:18) v λ − v (cid:19) . (5.1)The gauge matrix U that interpolates between two solutions of the same mKdV model isgiven by U = (cid:18) β λ β (cid:19) . (5.2)In the self-similarity limit U generates B¨acklund transformations as described in the pre-vious section.Following the steps similar to those taken in Section 2 we obtain in the zero-curvatureframework the corresponding mKdV equations for the flows t , t , t :4 v t = v x − v v x , (5.3)16 v t = 30 v v x − v v x − vv x v x − v x + v x , (5.4)64 v t = − v v x + 70 v v x + 560 v v x v x + 420 v v x − v v x (5.5) − vv x v x − vv x v x − v x v x − v x v x + v x . The usual dimensional considerations [11] lead to the following self-similarity limits for t j = t , t , t : v ( x, t ) = v ( z ) t /jj , β i = b i ( z ) t /jj , z = xt /jj , j = 3 , , . (5.6)Taking such limit of equation (5.3) (and integrating once) we get the P II equation: v z ( z ) = α + 2 v ( z ) − zv ( z ) , (5.7) here v iz denotes the i -th derivative with respect to the argument z . From the gaugetransformation U we inherit two B¨acklund transformations. The first is given by v → v + b, b = 2 − α − v z ( z ) − v ( z ) + 2 z , α → − α , (5.8)and squares to identity as follows by inspection.Since v → − v, α → − α , (5.9)is an obvious symmetry of P II (and mKdV) one easily sees that ¯ b ( v, α ) = b ( − v, − α ) willbe the second B¨acklund transformation of P II as can also be derived directly from thegauge symmetry argument involving the matrix U (5.2). Such symmetry (5.9) is alsopresent for all higher times t n , therefore it will suffice for us to show from now on onlyone of the two B¨acklund transformations.For the t = t mKdV equation (5.4) after imposing the appropriate similarity limitfrom (5.6) and performing an integration, we obtain: v z = 10 v v z + 10 vv z − v − zv + σ (5.10)This equation can be identified with the F-XVII equation of reference [10] for the param-eters δ = 0 , λ = − , µ = 0 . As before we are able to derive two B¨acklund transformations from the gauge symmetryargument that are related by the symmetry operation (5.9). We show one of them here: v → v + b, b = 8 − α v − v z − vv z + 5 v z + 30 v v z + 8 z , α → − α . (5.11)Finally for the t = t and the corresponding mKdV flow (5.5) the appropriate self-similarity limit from (5.6) yields v z = α + 20 v + 42 vv z − v v z + 14 v z v − v v z + 56 v z v z v + 70 v z v z − zv , (5.12)after performing an integration. One of the two B¨acklund transformations we find in thiscase is : v → v + b, b = 32 − α denom , α → − α , (5.13)where denom = − v − v z − v z + 70 v z − v v z + 140 v v z ++ 14 v z v z + 14 v (20 v z v z − v z ) + 70 v (cid:0) v z + v z (cid:1) + 32 z . Common for all these higher-order Painlev´e equations derived from higher mKdV equa-tions is invariance under the extended affine Weyl group obtained from the B¨acklundgauge symmetries in the self-similarity limit.
In the recent publications [3, 4, 5] we advanced a notion of symmetry being a relevantmeasure of integrability by proposing models that only kept a part of the affine extendedWeyl group symmetry while maintaining (not violating) the Painlev´e property. Along the ame line of the investigation as presented in [3] we can here propose a deformation of thedressing chain (3.6) given for N = 3 by( j + j ) z = − ( j − j ) ( j + j ) + α + η ( j + j )( j + j ) z = − ( j − j ) ( j + j ) + α − η ( j + j )( j + j ) z = − ( j − j ) ( j + j ) + α (6.1)This model is still manifestly invariant under the transformation generated by U fromequation (4.26) such that j → j + κj + j , j → j − κj + j , j → j , (6.2)with κ = − α and α i → α i + α , i = 1 , U i , i =1 , π does not keep the equations (6.1) invariantanymore. Instead equations (6.1) are invariant under a new π automorphism [3] π : j → − j , j → − j , j → − j , α → − α , α → − α , α → − α , η → − η , with the transformed j i , α i , i = 1 , , P i =1 j i = − z/ P i =1 α i = − / j i = a i z + b i + c i z + d i z + e i z + · · · , i = 1 , , , (6.3)with the pole chosen at 0. After substituting the above equation back into equations (6.1)we find that coefficients of the first term must satisfy relations a i + a i +1 − a i + a i +1 =0 , i = 1 , ,
3. For the solution a = 1 , a = 0 , a = − η we findthat b and d (or d ) will remain undetermined while all the other coefficients will befixed. This is a general feature which occurs independently of whether the η terms arepresent. Thus the pole solutions of the type shown in (6.3) will depend on three parameters( b , d and position of the pole). This agrees with integrability property derived from theKovalevskaya-Painlev´e test of the N = 3 dressing chain and shows that the η -deformationpreserving the symmetry (6.2) also results in passing the integrability test.The issue of finding an underlying integrable model behind the deformed dressingchain (6.1) remains under investigation. A Basic facts about sl ( N ) algebra Since sl ( N ) Lie algebra is isomorphic to the algebra of trace-less N × N matrices, we findit convenient to work with a matrix representation of sl ( N ) with basic matrices beingunit upper diagonal matrices E α a , E α a + α a +1 , . . ., E α a + ... + α b , a, b = 1 , . . ., N − E α a ) ij = δ i,a δ j,a +1 , (cid:0) E α a + α a +1 (cid:1) ij = δ i,a δ j,a +2 , . . ., ( E α a + ... + α b ) ij = δ i,a δ j,b +1 (A.1)and their corresponding lower diagonal conjugated matrices E − α a = E † α a etc, with i, j =1 , . . ., N . The so-called Cartan subalgebra of G is generated by trace-less combinations ofthe N × N diagonal matrices h α a = h e a − h e a +1 , a = 1 , . . ., N − h e a ) ij = δ i,a δ j,a . It is often convenient to relate the eigenvalues of Cartan subalgebra matrices h α a obtained when acting on E α a matrices to N − sl ( N ). This is facilitated y introducing N vectors e a obtained as follows. Let { ˆ e , . . ., ˆ e N } be an orthonormal basisin R N . Define N vectors e i = ˆ e i − N N X j =1 ˆ e j , that by definition satisfy N X j =1 e j = 0and have inner product h e i , e j i = δ ij − N . (A.2)Define now N − sl ( N ) as α i = e i − e i +1 , i = 1 , . . ., N − N − × ( N −
1) Cartan matrix K ij : K ij = h α i , α j i = i = j − i = j ±
10 otherwise (A.4)for products of simple roots defined in (A.3).For fundamental weights defined asΛ i = i X j =1 e j , i = 1 , . . ., N − h Λ i , α j i = δ ij . The Cartan matrix transforms the basis given in term of fundamental weights to the basisgiven in terms of roots according to: N − X j =1 K ij Λ j = e i − e i +1 = α i The Weyl group of sl(N) is a permutation of N vectors e i .We are interested in generalizing the above structure to the loop algebra b sl ( N ) spannedby λ n E ± ( α a + ... + α b ) , λ n h α a on which we define the principal gradation operator: Q = N λ ddλ + H Λ (A.6)where Λ = N − X i =1 Λ i and H Λ acts by an adjoin action. One proves that:Λ = ( N − e + ( N − e + . . . + e N − This gives for N = 3 , , N = 3 : Λ + Λ = 2 e + e = α + α N = 4 : Λ + Λ + Λ = 3 e + 2 e + e = 3 α + 4 α + 3 α N = 5 : Λ + Λ + Λ + Λ = 4 e + 3 e + 2 e + e = 2 α + 3 α + 3 α + 2 α he above expression allows an alternative simple expression for Q as: Q = N λ ddλ + ( N − h e + ( N − h e + . . . + ( N − i ) h e i + . . . + h e N − , (A.7)with Q N =3 = 3 λ ddλ + h α + h α , Q N =4 = 4 λ ddλ + 3 h α + 4 h α + 3 h α , etc. One fundamental feature that is of great relevance for the formalism we present isthe fact that loop algebra b sl ( N ) decomposes into different grade sectors according to theireigenvalues under Q : b sl ( N ) = ⊕ i G i , [ Q, G i ] = i G i . (A.8)This property provides foundation for the zero-curvature approach of section 2 used todetermine flows of an integrable hierarchy.The operator Q in (A.7) induces the following graded subspaces: G nN = { h ( n )1 , · · · , h ( n ) N − } , G nN +1 = { E ( n ) α , · · · , E ( n ) α N − , E ( n +1) − α ···− α N − } , G nN +2 = { E ( n ) α + α , E ( n ) α + α , · · · , E ( n ) α N − + α N − , E ( n +1) − α ···− α N − , E ( n +1) − α ···− α N − } , ... = ... G nN + N − = { E ( n +1) − α , · · · , E ( n +1) − α N − , E ( n ) α + ··· + α N − } . (A.9)with each subspace G m of grade m ∈ Z containing generators of grade m . B From periodic dressing chain to Painlev´e equa-tions
For completeness and notational consistency we will provide here a number of usefulidentities to prove that the periodic dressing equation (3.6) can be rewritten as higherPainlev´e equations (3.11) and (3.12) for N odd and even, respectively.Under substitution (3.10) the dressing equation (3.6) can be rewritten as f i z = f i (cid:2) f i − + . . . − ( − k f i − k + ( − k j i − k − f i +1 + . . . + ( − k f i + k + ( − k +1 j i + k +1 (cid:3) + α i , (B.1)after replacing j i − = f i − − j i − and j i +2 = f i +2 − j i +3 etc.For N odd using the periodicity condition (3.5) we conclude that j i − k − j i + k +1 = 0 for 2 k + 1 = N , (B.2)and therefore equation (B.1) becomes for 2 k + 1 = N : f i z = − f i [ k X m =1 ( − m f i + m − k X m =1 ( − m f i − m ] + α i . (B.3)Due to periodicity, f i − m = f i − m + N and we can rewrite (B.3) in a standard from : f i z = f i (cid:2) k X r =1 f i +2 r − − k X r =1 f i +2 r ] + α i . (B.4)From condition (3.4) we derive N X i =1 f i = 2 N X i =1 j i = N z . (B.5) hus it follows that the dressing chain equation (3.6) gives A (1)2 k = A (1) N − Painlev´e equa-tions for N odd with P Ni =1 f i = N z/ P Ni =1 α i = N/ α i = β i + .For N even the condition (3.4) can now be cast as n X r =1 f i +2 r = n X r =1 f i +2 r +1 = n z . (B.6)We next rewrite the dressing equation (3.6) as: f i z = f i (2 j i +1 − f i ) + α i , i = 1 , . . ., N . (B.7)To eliminate j i +1 from the above equation we sum the derivatives as follows: n X r =1 f i +2 r z = n n X r =1 f i +2 r j i +2 r +1 − n X r =1 f i +2 r + n X r =1 α i +2 r . (B.8)The squared terms can be eliminated due to relation f i = n X s =1 f i +2 s ± − n − X s =1 f i +2 s leaving : n X r =1 (cid:18) − α i +2 r (cid:19) = 2 n X r =1 f i +2 r j i +2 r +1 − n X r =1 f i +2 r n X s =1 f i +2 r +2 s ± − n − X s =1 f i +2 r +2 s ! . (B.9)To rewrite terms explicitly depending on j i +2 r +1 we use the formula : j i +1+2 r − j i +1 = n − X s = r f i +2 s +1 − n − X s = r f i +2 s +2 , (B.10)that follows from expansion j i +1+2 r = f i +2 r +1 − f i +2 r +2 ... carried until we reach + f i +2 n − − f i +2 n .In this way we obtain : nzj i +1 = f i nz ) − n − X r =1 f i +2 r n − X s = r f i +2 s +1 ! + 2 n − X r =1 f i +2 r n − X s = r f i +2 s +2 ! + n − X r =1 f i +2 r n X s =1 f i +2 r +2 s ± ! − n X r =1 f i +2 r n − X s =1 f i +2 r +2 s ! + n X r =1 (cid:18) − α i +2 r (cid:19) . (B.11)One can show that the second and the fourth summation terms on the right hand side of(B.11) cancel each other while the difference of the first and the third summation termson the right hand side of (B.11) combine to yield : n − X r =1 r X s =1 f i +2 s − f i +2 r − n − X r =1 f i +2 r n − X s = r f i +2 s +1 !! (B.12a)= X ≤ s ≤ r ≤ n − f i +2 s − f i +2 r − X ≤ r ≤ s ≤ n − f i +2 r f i +2 s +1 . (B.12b)To summarize we obtain an expression for j i +1 , i = 1 , . . ., N : j i +1 = 1 nz f i nz ) + X ≤ r ≤ s ≤ n − f i +2 r − f i +2 s − X ≤ r ≤ s ≤ n − f i +2 r f i +2 s +1 + n X r =1 (cid:18) − α i +2 r (cid:19) . (B.13) lugging it back into (B.7) yields: nzf i z = f i X ≤ r ≤ s ≤ n − f i +2 r − f i +2 s − X ≤ r ≤ s ≤ n − f i +2 r f i +2 s +1 + n X r =1 (1 − α i +2 r ) + nzα i , i = 1 , . . ., N , (B.14)reproducing A (1)2 n − = A (1) N − invariant Painlev´e equations for even N = 2 n with conditions(B.6). Acknowledgments
JFG and AHZ thank CNPq and FAPESP for financial support.VCCA thanks S˜ao Paulo Research Foundation (FAPESP) for financial support by grants2016/22122-9 and 2019/03092-0.
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