Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations
aa r X i v : . [ n li n . S I] S e p Coalescence, Deformation and B¨acklund Symmetries ofPainlev´e IV and II 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 60607-7059September 14, 2020
Abstract
We extend Painlev´e IV model by adding quadratic terms to its Hamiltonian obtainingtwo classes of models (coalescence and deformation) that interpolate between Painlev´e IVand II equations for special limits of the underlying parameters. We derive the underly-ing B¨acklund transformations, symmetry structure and requirements to satisfy Painlev´eproperty.
The Painlev´e equations are second-order differential equations whose solutions have nomovable singular points except poles. This feature (pure poles are the only movablesingularities) of some second order differential equations is known as Painlev´e property.The Painlev´e equations naturally emerge as special scaling limits of integrable models [9, 5,6, 7, 8] and a fundamental conjecture [1] establishes connection between Painlev´e propertyand solvability by inverse scattering. Another basic aspect of Painlev´e equations andtheir Hamiltonian structures is invariance under extended affine Weyl symmetry groups[17, 18]. For example the fourth Painlev´e equation, to which we will refer as P IV , exhibitssymmetry under B¨acklund transformations that form the affine Weyl group of type A (1)2 and the second Painlev´e equation, to which we will refer as P II , is invariant under B¨acklundtransformations from the affine Weyl group A (1)1 . B¨acklund transformations have also beenextensively studied in connection with the Schlesinger transformations, see for instancereferences [10, 15, 23] for the case of Painleve II and IV equations.Hybrid Painlev´e equations have been a focus of several papers, e.g. [14, 20]. Morerecently, in reference [3] we introduced the hybrid P III − V model that was obtained asreduction of a class of integrable models known as multi-boson systems [6, 7] that gen-eralize the AKNS hierarchy [5]. The P III − V model reduces to P III , P V and I , I andI equations from Ince’s list [12, 2] for special limits of its parameters while for remain-ing finite values of its parameters preserves enough symmetry under remaining B¨acklundtransformations of the extended affine Weyl symmetry group to satisfy Painlev´e property[3]. We will conduct here a similar investigation for the hybrid of P II and P IV models andpoint out how the presence of remaining B¨acklund transformations symmetries influences he outcome of the Painlev´e test. Starting from the symmetric Painlev´e IV equations, insection 2, we enlarge its parameter space to allow for extension of symmetry structure byadditional automorphisms π i , ρ i , i = 0 , ,
2. We derive algebraic relations between theseautomorphisms and A (1)2 B¨acklund transformations.We present two different limiting procedures leading to Painlev´e II equation.One way, described in section 3, is to formulate coalescence/degeneracy in a frameworkof symmetric Painlev´e IV equations augmented by a non-zero integration constant. Thisgeneralization of P IV equation remains invariant under the additional automorphism ρ .The underlying Weyl group symmetry reduces from A (1)2 down to A (1)1 in the appropriatelimit and we are able to obtain close expressions for the B¨acklund transformations of P II from their P IV counterparts. In the P II limit the automorphism ρ toggles between twocopies of P II equations each with its own A (1)1 symmetry.In another scheme, presented in section 5, the A (1)2 symmetry group of symmetricPainlev´e IV equation is explicitly broken by addition of a deformation parameter beforethe limit resulting in Painlev´e II equation is taken. The deformed model is formulatedin such a way that it is invariant under additional automorphisms π , ρ . We point outa connection between existence of residual symmetry of the deformed model (invarianceunder one of the original three B¨acklund transformations of A (1)2 ) and passing of theKovalevskaya-Painlev´e test by this model. Such deformed model provides another exampleof hybrid Painlev´e equations with properties that they pass Painlev´e test, retain invarianceunder residual B¨acklund transformations and reduce down to underlying Painlev´e or Inceequations for special values of their parameters.In section 4 we will introduce and study a generalization of P IV Hamiltonian structureof the form : H = H + 1 ǫ ( f + f ) (cid:18) k σz − k f + f ) (cid:19) , (1.1)where ǫ, σ, k , k are complex parameters and H = − f f f + − α + α f + − α − α f + 2 α + α f , (1.2)is the well-known Okamoto’s P IV Hamiltonian [19]. The two basic conditions that guideour construction of such generalization are : (1) that the original cubic Hamiltonian isaugmented only by terms of dimensions lower than three and (2) the Hamilton equationsremain finite and do not violate the Painlev´e property. These conditions restrict theallowed generalization of P IV Hamiltonian structure to be of the form given in equation(1.1). As we will see below the combination f + f appearing in the above expressionensures invariance under a pair of B¨acklund transformations s , ρ , if we used f + f or f + f we would encounter invariance under s , ρ or s , ρ with all these transformationsbeing defined in the forthcomming sections.We show that this natural generalization (1.1) represents either coalescence/degeneracyor A (1)2 deformation of P IV and we present arguments that those two approaches are theonly ones leading from P IV model to P II model under the above conditions.We summarize the novel features of our formalism and reiterate rationale for expandingthe parameter space of Painlev´e IV model by additional parameters in Section 6. IV model, definition and sym-metries This section is devoted to a summary of relevant results on P IV equations, B¨acklundtransformations and coalescence between P IV and P II available in the literature (e.g.[11, 18]). e also generalize the conventional symmetric Painlev´e IV model by adding the newparameter σ in a way that makes the generalized model invariant under additional auto-morphisms π i , ρ i , i = 0 , , IV symmetric equations The starting point of subsection is the Okamoto Hamiltonian (1.2) for P IV equation. Inthe literature the parameters α i , i = 0 , , α + α + α = 1. Herewe find that our discussion of symmetries and coalescence limits will profit from workinginstead with conditions : α + α + α = σ, f = σz − f − f . (2.1)Here we introduced σ as an additional parameter for the P IV model that enables us toextend symmetry group of the model. The advantages of introducing the σ parameterwill be summarized in the concluding Section 6.The corresponding Hamilton’s equations can be cast in a form of the so-called sym-metric P IV system described by e.g. [18]: f ′ = f ( f − f ) + α ,f ′ = f ( f − f ) + α ,f ′ = f ( f − f ) + α , (2.2)where f i = f i ( z ) and ′ = d/dz .Eliminating f = σz − f − f from (2.2) we obtain: f ′ ( z ) = f ( − σz + f + 2 f ) + α ,f ′ ( z ) = f ( σz − f − f ) + α , (2.3)while the third equation in (2.2) can be obtained by summing the above two equations.By further eliminating f or f from (2.3) we get for the remaining component: f ′′ i ( z ) = f ′ i f i − α i f i + (cid:18) σ z + ( − i (2 α + 2 α − α i − σ ) (cid:19) f i − σzf i + 32 f i , i = 0 , . (2.4)Both equations are equivalent to the standard P IV equation [11, 5]: w xx = w x w + 3 w xw + 2 (cid:0) x − A (cid:1) w + Bw (2.5)by setting σ → f ( z ) = w ( x ) √− , z = x √− α = 12 (1 + A − α ) , α = r − B f with the appropriate changes.Equations (2.4) will be referred to as P IV equations throughout this document whileequations (2.2) will be referred to as symmetric P IV equations. Equations (2.2) are manifestly invariant under B¨acklund transformations s i ( i = 0 , , π defined as follows (see e.g. [18]): α α α f f f s − α α + α α + α f f + α f f − α f s α + α − α α + α f − α f f f + α f s α + α α + α − α f + α f f − α f f π α α α f f f (2.7) hese transformations satisfy s i = 1 , ( s i s i +1 ) = 1 , π = 1 , πs i = s i +1 π, i = 0 , , ✄ ,, (2.8)and thus h s , s , s , π i form the extended affine Weyl group A (1)2 [18].Due to the presence of parameter σ introduced in equation (2.1) in the setting ofsymmetric P IV equation (2.2) we have additional automorphisms π i and ρ i , i = 0 , , α α α f f f σπ − α − α − α − f − f − f − σπ − α − α − α − f − f − f − σπ − α − α − α − f − f − f − σ (2.9)and α α α f f f σ zρ − α − α − α f f f − σ − zρ − α − α − α f f f − σ − zρ − α − α − α f f f − σ − z (2.10)that keep equations (2.2) invariant. The automorphisms π i and ρ i square to one π i = 1 , ρ i = 1 , i = 0 , , , (2.11)and satisfy the so-called braid relations π i π j π i = π j π i π j , ρ i ρ j ρ i = ρ j ρ i ρ j , i = j . (2.12)The automorphisms π i and ρ i are related to automorphism π from (2.7) via π = π π = π π = π π = ρ ρ = ρ ρ = ρ ρ (2.13)and satisfy the following commutation relations with the B¨acklund transformations s j : π i s i = s i π i , π i s j = s k π i , ρ i s i = s i ρ i , ρ i s j = s k ρ i , i = j, k = j, i = k . (2.14)We will now describe the B¨acklund transformations for the second order P IV equa-tions (2.4). The procedure will be illustrated by considering the s transformation only.Generalizations to other generators follow easily.First, we consider s ( α i ) , s ( f i ) from (2.9) and eliminate α = σ − α − α and f = σz − f − f to obtain: s ( α ) = σ − α , s ( α ) = σ − α , (2.15) s ( f ) = f + σ − α − α σz − f − f (2.16) s ( f ) = f − σ − α − α σz − f − f . (2.17)Equation (2.3) allows us to write down the following relations between f and f : f = − α + σzf + f ′ − f f (2.18) f = α + σzf − f ′ − f f (2.19)used below to realize s as (1) B¨acklund and (2) auto-B¨acklund transformations, respec-tively as shown below :
1) Eliminating f from the rhs of equation (2.16) and f from the rhs of equation (2.17)yields: s ( f ) = 2 f ( σ − α − α ) − α + σzf + f ′ − f + α + σzf − f ′ − f f , (2.20) s ( f ) = − f ( σ − α − α ) α + σzf − f ′ − f − α − σzf − f ′ + f f . (2.21)(2) Inversely, eliminating f from the rhs of equations (2.17) and f from the rhs ofequations and (2.16) yields (note that in this case we denote s by ˜ s ):˜ s ( f ) = f + 2 ( α f + α f − σf ) − α − σzf + f ′ + f , (2.22)˜ s ( f ) = f − − α f − α f + σf ) − α + σzf + f ′ − f . (2.23)Acting with ρ connects relations (2.20) and (2.21) as well as relations (2.22) and (2.23): ρ ( s ( f )) = s ( f ) , ρ ( ˜ s ( f )) = ˜ s ( f ) . (2.24)As we saw above in items (1) and (2), s ( f i ) , i = 0 , f i , i = 0 , f j , i = j by simple substitutions (2.18) or (2.19). The transformation s that maps f → f and f → f is referred by us as B¨acklund transformation of thesystem of second order P IV equations (2.4) and maps equation (2.4) with i = 0 to thatwith i = 1 and vice versa.The corresponding transformation that maps f → f and f → f is denoted by as˜ s and is referred to as an auto-B¨acklund transformation of the second order P IV equation (2.4) with either i = 0 or i = 1. IV equations In this section we look at coalescence in the setting of symmetric P IV equations. Suchframework makes it easier to see what happens with the B¨acklund symmetries in the ǫ → IV equations (2.2)through the following transformations : f i ( z ) → f i ( z ) + 1 ǫ , z → z + 2 σǫ ,α → ǫα − ǫ , α → ǫα + 1 ǫ , α → ǫα . (3.1)Applying the above transformation to the first order equations (2.2) yields: f ′ ( z ) = f ( f − f ) + f − f ǫ + ǫα − ǫ f ′ ( z ) = f ( f − f ) + f − f ǫ + ǫα + 1 ǫ (3.2) f ′ ( z ) = f ( f − f ) + f − f ǫ + ǫα Now we proceed by the same steps as in the preceding sections. Summing the equationsabove we get: ǫα + ǫα + ǫα = ǫσ, f ′ + f ′ + f ′ = ǫσ. ntegrating equation P i f ′ i = ǫσ yields P i f i = ǫσz + C , where C is an arbitrary constantof integration. Initially C is set to zero but after applying transformation (3.1) on f i and z we obtain : f + f + f + 3 ǫ = ǫσz + 2 ǫ −→ f + f + f = ǫσz − ǫ . with C = − /ǫ . Note that the presence of the non-zero integration constant does not affectthe symmetry of the symmetric P IV equations since we can always work with symmetrytransformations acting on redefined f i ’s as will be done below.Eliminating f and α from (3.2), we get: f ′ ( z ) = ǫ ( α − σzf ) + 2 f + 2 f ǫ + f + 2 f f − σz ,f ′ ( z ) = ǫ ( α + σzf ) + − f − f ǫ − f − f f + σz . (3.3)Substituting α = a /ǫ, α = a /ǫ, σ = σ /ǫ with finite a , a , σ and taking ǫ → ∞ limitwe recover P IV equations (2.3).By eliminating f from (3.3) we obtain: f ′′ ( z ) = 1 ǫf + 1 (cid:18) σ − α − α − σzf + 2 f + ǫ (cid:18) σ z f − α (cid:19) + ǫ (cid:0) − α f − α f + σ z f − σzf + σf (cid:1) + ǫ (cid:18) − α f − α f − σzf + 2 σf + 12 f ′ + 32 f + σ z (cid:19)(cid:19) . (3.4)Taking instead the limit ǫ → f results in two copies of P II equations, namely : f ′′ i ( z ) = ( − i ( − σ + 2 α + 2 α ) − σzf i + 2 f i , i = 0 , . (3.5)The above P II equations transform into each other under the automorphism ρ from (2.10).Since transformations (3.1) are nothing but M¨obius transformations on the variables f i and z , they naturally preserve the Painlev´e property.As a digression we note that equation (3.4) for σ → ǫ becomes for w = f − /ǫ : w ′′ ( z ) = w ′ w + 3 w − w ǫ − w (cid:0) α ǫ + α ǫ − (cid:1) ǫ − (cid:0) α ǫ + 1 (cid:1) wǫ (3.6)in which we recognize the equation XXX (I ) of the Gambier’s classification, that is listedin the classical book of Ince [12] (see also [2] for connection between Painlev´e equationswith additional parameters and equations in [12]) as: I : w ′′ ( z ) = w ′ w + 3 w aw + 2 bw + cw . (3.7)Also, if we make transformation z → z + σǫ − ξ/σ in equation (3.1) (equivalent to adifferent choice of integration constant C in P i f i = σz + C ) with some new parameter ξ and take the limit ǫ → f we obtain f ′′ ( z ) = 2 f − σz − ξ ) f − σ + 2 α + 2 α . (3.8)By taking σ = 0 we arrive at Ince’s I equation: I : w ′′ = 2 w + aw + b . (3.9) .1 The B¨acklund Transformations in the coalescence limit In this subsection we will show how A (1)2 symmetry group reduces to A (1)1 symmetry inthe appropriate limit. A (1)2 symmetry is maintained in equations (3.2) Equations (3.2) are invariant under: α α α f f f s ǫ − α α + α − ǫ α + α − ǫ f α ǫ − ǫ f + ǫ + f f − α ǫ − ǫ f + ǫ s α + α + ǫ − α − ǫ α + α + ǫ f − α ǫ + ǫ f + ǫ f α ǫ + ǫ f + ǫ + f s α + α α + α − α α ǫf + ǫ + f f − α ǫf + ǫ f π α + ǫ α − ǫ α − ǫ f f f (3.10)and the automorphism ρ from (2.10).After we eliminate f and α , we still have invariance under s , s , but no longer under π and the s transformation is modified to: s ( f ) = f − ǫ ( − α − α + σ ) f + f − σzǫ , s ( f ) = f − ǫ ( − α − α + σ ) − f − f + σzǫ ,s ( α ) = σ − α , s ( α ) = σ − α . (3.11) A (1)1 symmetry in the ǫ → limit It is now easy to see from equation (3.10) that the transformations s , s and π divergein the limit ǫ →
0. Also s becomes trivial in this limit. The way around this problem isto form the composition s s s that will be shown not to diverge in the limit ǫ → A (1) l to A (1) l − k appeared in [16].The main conclusion of this subsection is that for the P IV system of equations (3.2)for f , f (obtained after elimination of f ) the ǫ → s s s (or identically s s s ) and s as the two B¨acklund transformations that maintainP II invariant.Explicitly, the action of s s s on all variables is: s s s ( f ) = f − ( α + α ) ǫ ( ǫf + 1) α ǫ + ǫf f + f + f ,s s s ( f ) = ( α + α ) ǫ ǫf + 1 + α ǫ + α α ǫ + α ǫ + α ǫ ( ǫf + 1) ( − α ǫ + ǫf f + f + f ) + f ,s s s ( α ) = − α , s s s ( α ) = − α . (3.12)Now just looking at transformations of the parameters α , α and using notation β := α + α (since they are always together from now on), we see that they have a A (1)1 groupstructure due to: β α s s s − β β + α s α + β − α (3.13)and s = 1 , ( s s s ) = 1 , ( s s s ) s = s ( s s s ) . From now on we will use for brevity the following notation: S = s s s , S = s . (3.14) he relations (2.18), (2.19) obtained in section (2.2) generalize to the following relations f = α ǫ + σzǫ f − ǫf ′ − ǫf − f + σzǫ ǫf + 1) ,f = − α ǫ + σzǫ f + ǫf ′ − ǫf − f + σzǫ ǫf + 1) , (3.15)obtained from (3.3). Using relations (3.15) in an exactly the same way as we did belowequations (2.18), (2.19) we obtain two expressions for auto-B¨acklund transformations ˜ S and B¨acklund transformations S from those given in equation (3.11)˜ S ( f ) = f + 2 ( − α − α + σ ) ( ǫf + 1) α ǫ + σzǫf − f ′ − f + σz , ˜ S ( f ) = f − − α − α + σ ) ( ǫf + 1) − α ǫ + σzǫf + f ′ − f + σz ,S ( f ) = α ǫ + f (cid:0) σzǫ − (cid:1) + ǫ ( σz − f ′ ) − ǫf ǫf + 2 + 2 ( − α − α + σ ) ( ǫf + 1) − α ǫ + σzǫf + f ′ − f + σz ,S ( f ) = − α ǫ + f (cid:0) σzǫ − (cid:1) + ǫ ( f ′ + σz ) + ǫ (cid:0) − f (cid:1) ǫf + 2 − − α − α + σ ) ( ǫf + 1) α ǫ + σzǫf − f ′ − f + σz . As in relations (2.24), these two B¨acklund transformations S and ˜ S are related by theautomorphism ρ . Repeating the same steps for S we obtain : S ( f ) = f − α + α + α ǫf + α ǫf ) α ǫ + σzǫf + f ′ − f + σz , ˜ S ( f ) = − (cid:0) α ǫ + α α ǫ − α − α (cid:1) ( ǫf + 1) ( − α ǫ − α ǫ − σzǫf + f ′ + f − σz ) + − α ǫ − α ǫ + σzǫ f − ǫf − f + σzǫ ǫf + 1) − ǫf ′ ǫf + 1) . The B¨acklund transformations obtained in this way have non trivial limits for ǫ → S ( f ) = 2 ( − α − α + σ ) − f ′ − f + σz + f , ˜ S ( β ) =2 σ − β , (3.16) S ( f ) = 2 ( − α − α + σ ) f ′ − f + σz − f , S ( β ) =2 σ − β , (3.17)˜ S ( f ) = − α + α ) f ′ − f + σz + f , ˜ S ( β ) = − β , (3.18) S ( f ) = − − α − α ) f ′ + f − σz − f , S ( β ) = − β . (3.19)These expressions agree with B¨acklund transformations for P II equation and they obeythe A (1)1 group structure described in the literature [11][13] although the whole A (1)1 groupstructure requires presence of an additional automorphism to be introduced below. Π automorphism for P II model In this subsection we will construct automorphisms Π , e Π of P II equation that satisfy A (1)1 -type relations : Π( f ) = f , Π( f ) = f , Π( β ) = σ − β, (3.20)Π S i = S j Π , i, j = 0 , , Π = 1 nd e Π( f ) = − f , e Π( f ) = − f , e Π( β ) = σ − β (3.21) e Π ˜ S i = ˜ S j e Π , i, j = 0 , , e Π = 1 , with A (1)1 transformations S i , ˜ S i , i = 0 , A (1)2 transformations to be defined below. Note that f i , i = 0 , II equations (3.5).We now return to P IV model where we define P and P − : P := πs = s π, P − := s π = π s , (3.22)with π and s i defined by relations (2.7) from P IV model. The actions of P and P − onB¨acklund transformations S i (3.14) satisfy the following relations : S P = P S : { α → α , α → α + σ } , P S = S P : { α → α − σ, α → α } ,S P − = P − S : { α → α + σ, α → α } , P − S = S P − : { α → α , α → α − σ } . (3.23)Accordingly P and P − satisfy the product rules with S i identical to those given inrelations (3.20) and (3.21) although valid in the context of P IV model.Further one finds using the table (3.10) and relations (3.15) to calculate the actions P and P − on f i , i = 0 , e Π in the ǫ → P on f i , i = 0 , P ( f ) = π ( f ) = f , (3.24) P ( f ) = π (cid:18) f + α ǫ − /ǫ f + 1 /ǫ (cid:19) = f + α ǫ + 1 /ǫ f + 1 /ǫ , (3.25)where as we recall f = σǫz − f − f − /ǫ . The relations (3.15) can now be used tosubstitute f by f on the right hand side of equation (3.24) and f by f on the righthand side of equation (3.25) giving in the limit ǫ → f and substitute it by f on the right hand side of equation (3.25)gives in the limit ǫ → II − IV equations and its Hamiltonian We will now consider the following class of generalizations of P IV equations (2.3) by addingnontrival terms parametrized by constants k , k : f ′ = α − σzf + f + 2 f f + 1 ǫ ( − k σz + k ( f + f )) , (4.1) f ′ = α + σzf − f − f f + 1 ǫ ( k σz − k ( f + f )) . (4.2)We will determine values of constants k , k for which the above equations reproduce P II equation in the ǫ → IV Hamiltonian (1.2) due to addition ofquadratic terms with constants k , k .First let us comment on how general are such extensions of P IV model. Replace theterm k ( f + f ) on the right hand sides of equations (4.1) and (4.2) with a more general ombination k f + k f such that k = k . In such case the resulting second orderequation for f and f would be divergent in the limit ǫ →
0. For example, f ′′ wouldcontain the term ( k − k ) f / (2 f ǫ + k ǫ ) that would go to infinity for ǫ → k = k . Thus, we have to set k = k as we did in equations (4.1) and (4.2). Theaddition of terms proportional to zf i is also forbidden for the same reason.For α i = a i ǫ, i = 0 , , , σ = σ ǫ the second order equation for f in the ǫ → f ′′ = 2 f + 2( k − k ) σ zf + k a + k a − k σ . (4.3)Thus as long as k = k , (4.4)the system (4.1)-(4.2) will have P II equation as a limit.We now discuss the conditions for the system (4.1)-(4.2) to remain invariant under the A (1)2 symmetry.Insert f = σz − f − f back into equations (4.1), (4.2) and rewrite them as : f ′ = f ( f − f ) + α + 1 ǫ ( k ( f + f ) − k σz ) ,f ′ = f ( f − f ) + α − ǫ ( k ( f + f ) − k σz ) ,f ′ = f ( f − f ) + α , (4.5)with α = σ − α − α .Following Appendix A we now introduce¯ f = f + dǫ , ¯ f = f + dǫ , ¯ f = f (4.6)in an effort to remove through this shift of f i ’s the extra terms with k , k constants fromthe generalized P IV equations (4.1)-(4.2). In this way we obtain¯ f ′ = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + α + dǫ − dǫ (cid:0) ¯ f + ¯ f − ¯ f (cid:1) + 1 ǫ ( k ( ¯ f + ¯ f − dǫ ) − k σz ) , ¯ f ′ = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + α − dǫ + dǫ (cid:0) ¯ f + ¯ f − ¯ f (cid:1) − ǫ ( k ( ¯ f + ¯ f − dǫ ) − k σz ) , ¯ f ′ = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + α . (4.7)In the first equation in (4.7) the terms with ( ¯ f + ¯ f ) and the terms with σz will appearas − (2 d − k ) ǫ ( ¯ f + ¯ f ) + σzǫ ( d − k ) , (4.8)after eliminating f from this equation. The same terms but with the opposite sign willappear in the second equation in (4.7).With condition (4.4) satisfied we now describe two possible cases, the first case co-incides with the P IV coalescence model discussed in section 3 and the second definesdeformation of P IV model to be discussed in section 5.Case 1. Both terms in equation (4.8) vanish. This can only occur for2 d = k , d = k , which requires k = 2 k . (4.9)Condition (4.9) allows to restore the full A (1)2 symmetry in the generalized P IV equations (4.1)-(4.2). Recall that such mechanism took place in the P IV coalescencemodel. For example, for k = 1 , k = 2 , σ = ǫσ we recognize the coalescence caseof (3.3). ase 2. Only one term in equation (4.8) vanishes. Accordingly, we consider k = 2 k and k = k (preserving (4.4)). Setting the variable d to eliminate one of the two termsin (4.8), say d = k , results in 2 d − k = 2 k − k = 0. Consequently the only non-zero extra term in thefirst equation in (4.7) is − (2 k − k ) ǫ ( ¯ f + ¯ f ) . (4.10)Such system will be referred to as a deformed P IV model and will be discussed inthe subsequent section. One easily verifies that choosing d = k / IV model As we have seen in section 4, P II equation can also be obtained from deformation of P IV that changes its symmetry structure even before the limit is taken.Following derivation presented in section 4 we now propose the following P IV model :¯ H = − f f f + − α + α f + − α − α f + 2 α + α f + X i,j,k η i ( f j + f k ) , (5.1)as a generalization of the structure in (1.2). The summation in (5.1) is over all threeindices i, j, k being distinct. The parameters η i , i = 0 , , f ,z = f ( f − f ) + α − η ( f + f ) + η ( f + f ) ,f ,z = f ( f − f ) + α + η ( f + f ) − η ( f + f ) ,f ,z = f ( f − f ) + α − η ( f + f ) + η ( f + f ) . (5.2)Equations (5.2) are invariant under automorphisms (2.9), (2.10) augmented by π i ( η i ) = − η i , π i ( η j ) = − η k i, j, k distinctand ρ i ( η i ) = η i , ρ i ( η j ) = η k i, j, k distinct . Introduce ¯ f i = f i + ξ i , i = 0 , , , (5.3)with ξ i = 12 ( η j + η k ) , i, j, k distinct . (5.4)Note, that X i ¯ f i = X f i + X ξ i = σz + η + η + η . (5.5)The equations (5.2) can then be recast back into the original form of P IV symmetricequations: ¯ f ,z = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + ¯ α , ¯ f ,z = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + ¯ α , ¯ f ,z = ¯ f (cid:0) ¯ f − ¯ f (cid:1) + ¯ α . (5.6) ut with the z -dependent coefficients:¯ α = α + 14 ( η − η ) + 12 ( η − η ) σz , ¯ α = α + 14 ( η − η ) + 12 ( η − η ) σz , ¯ α = α + 14 ( η − η ) + 12 ( η − η ) σz , (5.7)that still satisfy P ¯ α i = P α i = σ .For η i = η j , i = j the z -dependence will disappear from ¯ α k = α k , k = i, k = j andthe system will become invariant under one specific B¨acklund transformation ¯ s k definedas one of the following transformations:¯ α ¯ α ¯ α ¯ f ¯ f ¯ f ¯ s − ¯ α ¯ α + ¯ α ¯ α + ¯ α ¯ f ¯ f + ¯ α ¯ f ¯ f − ¯ α ¯ f ¯ s ¯ α + ¯ α − ¯ α ¯ α + ¯ α ¯ f − ¯ α ¯ f ¯ f ¯ f + ¯ α ¯ f ¯ s ¯ α + ¯ α ¯ α + ¯ α − ¯ α ¯ f + ¯ α ¯ f ¯ f − ¯ α ¯ f ¯ f . (5.8)Now set η = η = 0 and η = 2 /ǫ in (5.2). We see that in such case (5.2) becomes(4.5) with k = 0 and k = 2 and since k = k we know from equation (4.3) that thelimit will still be P II .The condition η i = η j for i = j and corresponding invariance under s k transformationturns out to be a condition for the model to pass Kovalevskaya-Painlev´e test as we willnow explain. (5.1) Assume that solutions of the extended P IV (5.2) equations have the form f i = a i z + b i + c i z + d i z + e i z + · · · , i = 0 , , . (5.9)Substituting into (5.2) yields0 = a i ( a i +1 − a i − ) + a i , (5.10)0 = a i ( b i +1 − b i − ) + b i ( a i +1 − a i − ) − η i +1 ( a i + a i − ) + η i − ( a i + a i +1 ) , (5.11) c i = a i ( c i +1 − c i − ) + α i + c i ( a i +1 − a i − ) + b i ( b i +1 − b i − )+ η i − ( b i + b i +1 ) − η i +1 ( b i + b i − ) , (5.12)2 d i = b i ( c i +1 − c i − ) + c i ( b i +1 − b i − ) + a i ( d i +1 − d i − )+ d i ( a i +1 − a i − ) + η i − ( c i + c i +1 ) − η i +1 ( c i + c i − ) , (5.13)and etc for i = 0 , ,
2. Since P i a i = 0 there are three (up to a sign and an overallconstant) possible nontrivial solutions of the top equation in (5.10)( a , a , a ) = (0 , , − , (5.14)( a , a , a ) = ( − , , , (5.15)( a , a , a ) = (1 , − , , (5.16)which correspond to a i = 0 for i = 0 or i = 1 or i = 2. The automorphism π j will takethe configuration with a i = 0 into the one with a k = 0 for the three distinct indices i, j, k .We will show that for a given i such that a i = 0 the solution (5.9) will pass theKovalevskaya-Painlev´e test [25] as long as η j = η k . e will illustrate the argument for a = 0 as in (5.14). Plugging the sequence from(5.14) into (5.11) we find that b = −
12 ( η + η ) , b = b + 12 ( η − η ) . (5.17)Thus, in the case of (5.14) all the parameters b i are determined with exception of one,either b or b . For (5.15) the determined coefficient in term of η -coefficients will be b with one of b or b coefficients being undetermined. For (5.16) the determined coefficientwill be b while one of the two other coefficients remaining undetermined. This is a generalfeature which is present independently of whether the η deformation terms are present ornot.From (5.12) we find that all the coefficients c i multiplying z are determined in termsof the lower coefficients: c = − α + ( η + η ) / − η + b ( η − η ) , (5.18) c = 13 (3 α + 2 α + α ) + 13 b ( η − η − η − b ) + 16 η ( η − η )+ ( η − η ) / − η η , (5.19) c = 13 (3 α + 2 α + α ) + 13 b (2 η + 2 η − η + b )+ 16 ( η η − η η − η η ) + η / . (5.20)By summing the above coefficients one confirms that they satisfy the condition c + c + c = α + α + α = σ , (5.21)as expected from their definition in (5.9).Let us rewrite equation (5.13) as2 d i − d i ( a i +1 − a i − ) − a i ( d i +1 − d i − ) = b i ( c i +1 − c i − ) + c i ( b i +1 − b i − )+ η i − ( c i + c i +1 ) − η i +1 ( c i + c i − ) , (5.22)where we have grouped the terms with d i on the left hand side of the equation. In allthree (5.14), (5.15) and (5.14) cases summing the left hand side of (5.22) over i = 0 , , d + d + d ) while the sum of the right hand side of (5.22) over i = 0 , , P i d i = 0 as expected from thedefinition in (5.9).For the choice (5.14) the left hand side of equation (5.22) vanishes for i = 0 while theright hand side is equal to12 ( c + c + c )( η − η ) = 12 σ ( η − η ) . Thus consistency requires in the case of (5.14) that η = η . Similarly for the case (5.15)we find the left hand side of equation (5.22) vanishes for i = 1 while the right hand side isequal to σ ( η − η ) / i = 2 while the right hand side is equal to σ ( η − η ) / η j = η k for the case of a i = 0 with d i being the only undetermined coefficient among d , d , d . Generalizing the equation (5.22)to coefficient f ( k ) i of z k gives an equation with a left hand side: kf ( k ) i − f ( k ) i ( a i +1 − a i − ) − a i ( f ( k ) i +1 − f ( k ) i − ). This relation can be cast in terms of the 3 × k ( k − k +1). Correspondingly, the undetermined coefficients only appear for k = 0 and k = 2 as one of b i and d i coefficients consistent with what we have seen above. Togetherwith a position of the pole this leaves exactly three parameters as arbitrary with all the emaining coefficients fully determined. This demonstrates existence of a solutions withsimple pole structure and dependence on 3 arbitrary constants that are consistent whentwo of the deformations parameters are equal to each other.Thus we have connected the integrability property associated with the fact of passingthe Kovalevskaya-Painlev´e test to presence of the B¨acklund symmetry under s i emergingfrom the consistency condition η j = η k . II limit of the deformed symmetric P IV equation The starting point here are equations f ,z = f ( f − f ) + α + η ( f + f ) ,f ,z = f ( f − f ) + α − η ( f + f ) ,f ,z = f ( f − f ) + α . (5.23)of the deformed P IV obtained from (5.2) by setting η = η, η = η = 0. The parameter η is equal to the constant − (2 k − k ) /ǫ in equation (4.10) and as we have learned in section4 equations (5.23) will have P II limit which we elaborate in this section in greater detailsincluding application of the Painlev´e test.We recall that for η = η, η = η = 0 equation (5.23) is invariant under s B¨acklundsymmetry, π automorphism from the table (2.9) with π ( η ) = − η and ρ from the table(2.10) with ρ ( η ) = η .Using association f = − q with f + f + f = σz we get from (5.23) the followingequation for q : q zz = q z q − η + 12 q − η (cid:0) q + 2 q (2 σz − η ) + q (2 α + 4 α − σ − ησz + σ z )+ q (3 ση − α η − ησ z + 2 η σz − α η ) − ση − α + α η + ηα σz (cid:1) . (5.24)For η → IV equation : q zz = q z q + 3 q q + 2 q σz + q ( α + 2 α − σ + 12 σ z ) − α q , (5.25)that agrees with equation (2.4) for f = − q .For σ → Q = q − η/ Q zz = Q z Q + 32 Q + 2 Q η + Q (cid:18) α + 2 α + 34 η (cid:19) − Q (cid:18) α + 14 η (cid:19) , (5.26)which is I for η = 0.For σ = σ /η, α = a /η, α = a /η and in the limit η → ∞ we get : q zz = 2 q − qσ z − a + σ = 2 q − qσ z + ( a + a ) . (5.27)More generally for f and f from equation (5.23) we obtain in the limit η → ∞ ; f ′′ i ( z ) = ( − i ( α + α ) − σ zf i + 2 f i , i = 0 , . (5.28)in which we recognize two P II equations for i = 0 and i = 1 that again are transformedinto each other under the automorphism ρ from (2.10) but differ from P II equations in(3.5) by the values of the constant coefficients on the right hand sides.Because of the presence of deformation parameter η in the denominator in relation(5.24) it appears that the three cases η ≪ , η ≫ η -finite need to be considered eparately. For the first two cases we are in P IV and P II regimes, respectively but forfinite η it makes sense to make a change of variables q → Q = q − η/ Q zz = Q z Q + 32 Q + 2 Q ( σz + η ) + Q (cid:18) σ z + 12 ησz + α + 2 α − σ + 34 η (cid:19) + 12 ση + 12 Q (cid:18) ησzα − α − η α − η σ z + 14 η σz − η (cid:19) . (5.29)In Appendix B we provide details of the Painlev´e test applied on equation (5.29). Thatequation (5.29) passes the direct Painlev´e test agrees with the result of the Kovalevskaya-Painlev´e test that established the consistency of the extended P IV (5.2) as long as twoout three η i parameters are equal (which is the case here). II equations as a limit of the deformedmodel Here we will show how starting from equations (5.23) to obtain the first order system ofequations underlying the P II equations (5.28) and their A (1)1 B¨acklund transformations ina limit η → ∞ . We set σ = σ /η, α = a /η, α = a /η with constants σ , a , a andrepresent f , f as f = − q, f = − η p, f = σ η z + 2 η p + q . (5.30)Plugging these substitutions into (5.23) we obtain − η p z = − η p (cid:18) σ η z + 2 η p + 2 q (cid:19) + a η , (5.31) − q z = − q (cid:18) − σ η z − q − η p (cid:19) + a η − η (cid:18) σ η z + 2 η p (cid:19) . (5.32)Considering large η and neglecting the terms of order O (1 /η ) in the first equation andthe terms of order O (1 /η ) one obtains in such limit equations p z = 2 pq − a , (5.33) q z = − q + σ z + 2 p . (5.34)Taking the derivative with respect to z on both sides of (5.34) gives P II equation (5.27)(or (5.28) with f = − q ).Let us now repeat the above analysis to obtain the P II equation (5.28) with a differentsign of the constant term. We consider f = − y, f = − η h, f = σ η z + 2 η h + y . (5.35)that follows from identificiation (5.30) via acting with ρ automorphism and replacing q, p with y, h to emphasize that we are working with a different P II equation. Pluggingsubstitutions (5.35) into (5.23) like in (5.32) and considering large η we arrive at thesystem of first order equations : h z = − hy − a , (5.36) y z = y − σ z − h , (5.37)that lead to the second P II equation namely (5.28) with f = − y . rom equation (5.34) we derive p = 12 ( q z + q − σ z ) . (5.38)In order to conveniently introduce all the A (1)1 symmetry generators in the setting ofequations (5.33), (5.34) let us define an auxiliary quantity v obtained from p given in(5.34) by transformation q → − q : v = 12 ( − q z + q − σ z ) = − p + q − σ z . (5.39)Taking a derivative on both sides of equation (5.39) we obtain the counterparts of equa-tions (5.33), (5.34) valid for v, q : v z = − vq + 12 ( − σ − a − a ) , (5.40) q z = q − σ z − v . (5.41)Note that the transformation :Π : q → − q, p → v, σ → σ , a + a → − a − a , (5.42)takes equations (5.33), (5.34) into equations (5.40), (5.41) and does not change P II equa-tion (5.27). Also note that the transformation :¯ ρ : z → − z, σ → − σ , p → v, q → q , (5.43)will have the same effect.Alternatively, equations (5.40), (5.41) can be obtained directly from symmetric de-formed P IV equations (5.23) through the following substitution of f , f : f = q, f = 2 η v, f = σ η z − η v − q, a = − ( σ + a + a ) , (5.44)for large η values.Recall that for the deformed P IV equations (5.2) with η = η = 0 and η = η = 0 thesurviving symmetry generator is s : f s −→ f − α f , f s −→ f , α s −→ − α , (5.45)or in terms of variables p, q used above : q s −→ q − a p , p s −→ p, a s −→ − a , (5.46)after cancellation of η . One easily checks that indeed equations (5.33), (5.34) are invariantunder s transformation as shown in (5.46). Note that s : a + a → − ( a + a ) + 2 σ .Similarly inserting representation (5.44) into expression for the s -transformation (5.45)produces after cancellation of ηq ˜ s −→ q + σ + a + a v , v ˜ s −→ v, a + a s −→ − ( a + a ) − σ , (5.47)where we denoted s by ˜ s when it acts on q, v system to distinguish it from s as definedin relations (5.46). Both transformations s and ˜ s defined in (5.46) and (5.47) keep theP II equation (5.27) invariant and square to one. It is interesting to compare action of ˜ s tothat of the automorphism ˜ S = s s s from equation (3.18). Introducing γ = σ + a + a we can rewrite the nontrivial part of transformation (5.47) as q ˜ s −→ q + γ − q z + q − σ z , γ ˜ s −→ − γ , (5.48) here we inserted the definition of v from equation (5.39). Comparing with expression(3.18) we see that the action of ˜ s almost agree with the limit of s s s and the differenceis only due to the difference between constant terms of P II equations given in (5.28) versus(3.5).Using relation (5.39) between v and p one also derives formulas for actions of ˜ s on p and s on v : v s −→ v − qp a + a p , (5.49) p ˜ s −→ p + qv ( σ + a + a ) + ( σ + a + a ) v . (5.50)These completes all the information on the B¨acklund transformations of the A (1)1 symmetrygroup consisted of s , ˜ s , ¯Π of P II equation obtained as a limit of the deformed P IV model.Note that equations (5.33), (5.34) and equations (5.40), (5.41) can be compactly sum-marized as a system of equations q z = p − v ,p z = 2 pq − a ,v z = − vq + 12 ( − σ − a − a ) . (5.51)manifestly invariant under ¯Π and ¯ ρ .Similarly, equations (5.33), (5.34) would enter into a system of equations y z = u − h ,h z = − hy − a ,u z = 2 uy + 12 ( − σ − a − a ) . (5.52)that lead to the other copy of P II equations in (5.28) with its own A (1)1 symmetry. We would like to make few comments on special novel features of our formalism.By enlarging a parameter space of P IV model we extended the A (1)2 symmetry structureby additional automorphisms π i , ρ i , i = 0 , ,
2. In particular, the presence of the auto-morphism ρ facilitated the reduction process from A (1)2 to A (1)1 . The authomorphism ρ together with the B¨acklund transfomation s remain a symmetry for P II − IV and survivethe P II limit while they also commute with each other. A crucial feature of P II limitof P IV generalized models is that it consists of two P II equations (see (3.5), (5.33),(5.34)and (5.36),(5.37)) connected via authomorphism ρ . Each of the two P II equations isinvariant under A (1)1 symmetry. Thus the presence of ρ is critically important for the fullunderstanding of all features of the formalism. Note that in order to define the action ofauthomorphisms π i , ρ i , i = 0 , , σ that transforms nontrivially under these authomorphisms, see tables(2.9), (2.10). The presence of σ affords us also an opportunity to include in the formalismthe solvable Painlev´e equations (classified by Gambier) that appear on Ince’s list [12] (seealso [2]). In particular, equations I , I given in equations (3.7),(3.9) were obtained herein the σ → σ remains non-zero there exists a transformation α i = ˜ α i σ, f i ( z ) = √ σ ˜ f i (˜ z ) , z = ˜ z/ √ σ iven in terms of quantities entering equations (2.2) that allows for absorbing σ in theformalism and thus effectively setting it to 1. Note however that the possibility of redefin-ing σ = σ /η is essential for ability of taking P II limit for η → ∞ in the deformed P IV model. Likewise, setting σ = σ /ǫ and taking ǫ → ∞ limit was crucial for recovering P IV equations from equation (3.3). Since we are interested in models that interpolate betweenP II and P IV and in automorphisms π i , ρ i , i = 0 , , σ .As pointed out below the definition of the Hamiltonian H in (1.1) the choice of ad-ditional terms in H ensured invariance under ρ . As explained below (1.1), equivalenttheories with invariance under ρ or ρ could be introduced via simple redefinitions of theadditional terms. We have provided arguments that there is one unique (up to simpleredefinition of such additional terms) generalization of P IV model allowing addition ofquadratic terms to the hamiltonian and requiring finite limits and Painlev´e property.To summarize, in this work we focused on the symmetry properties for the two gener-alizations, namely coalescence and deformation, of P IV model contained in P II − IV Hamil-tonian of (1.1). We derived B¨acklund transformations of the P II − IV model and uncovereda connection between presence of symmetry and passing of Painlev´e property test. Thiswork raises an interesting question whether other Painlev´e/Ince equations can be unifiedwithin some mixed model similar to the one presented in this paper. A About introducing two integration constantsinto the P IV system Above, we have studied the transformation (4.6) with P i ¯ f i = P i f i + 2 d/η . We wouldtherefore now investigate the P IV systems that generally allow for P i f i = σz + C .Given is the P IV system f ′ i = f i ( f i +1 − f i +2 ) + α i , i = 0 , , s i , i = 0 , , π . The two constraints of the P IV system X i α i = σ, X i f i = σz + C (A.2)define two possible integration constants σ, C of the P IV system. Customarily, people set C = 0 and σ = 1. Recall that setting σ = 0 reduces P IV to Ince’s XXX equation (see also[2]).The integration constant C can be absorbed by redefining f i ’s : f i → g i so that P i g i = σz and the system is obviously still invariant under B¨acklund symmetries s i , i = 0 , , π .There appear (at least) three ways of changing variables to eliminate C from theconstraint P i f i . In each of these cases, the constant C will appear explicitly in theresulting differential equations.1. g i = f i + η, i = 0 , , , η = − C (A.3)with the shifted P IV system : g ′ i = g i ( g i +1 − g i +2 ) + α i − η ( g i +1 − g i +2 ) , i = 0 , , h i = f i + η, i = 0 , , h = f , η = − C (A.5) ith the shifted P IV system : h ,z = h ( h − f ) + α + η − η ( h + h − f ) ,h ,z = h ( f − h ) + α − η − η ( f − h − h ) f ,z = f ( h − h ) + α (A.6)with f = σz − h − h − η = σz − h − h + C .3. d = f + η, η = − C (A.7)with the shifted P IV system : d ,z = d ( f − f ) + α − η ( f − f ) ,f ,z = f ( f − d ) + α + ηf f ,z = f ( d − f ) + α − ηf . (A.8)We refer the reader to section 4 where the above scheme 2. was employed to transformaccordingly generalized P IV equations. B Painlev´e test of equation (5.29)
In this appendix we will apply the Painlev´e test to equation (5.29). Following the standardprocedure of this test we first insert Q ( z ) = a ( z − z ) µ and focus on the dominant behavior near singularity on both sides of equation (5.29) toobtain µ ( µ − a ( z − z ) µ − = a µ ( z − z ) µ − a ( z − z ) µ + 32 a ( z − z ) µ with contributions on the right hand side originating from the first and the second termof the right hand side of equation (5.29). This way we obtain: a = 1 , µ = − µ is a negative integer for a movable polewith no branching. Next, to check the resonance condition we plug Q ( z ) = a ( z − z ) − + η ( z − z ) − R into equation (5.29) and keep only the terms linear in η to obtain the resonance equationfor R : ( R + 1)( R −
3) = 0This resonance structure suggests that a Laurent expansion q ( z ) = ∞ X j =0 a j ( z − z ) j − = a ( z − z ) − + a + a ( z − z )+ h ( z − z ) + a ( z − z ) + · · · (B.1)expresses expansion around an arbitrary pole at z where we identified a = h as the singlearbitrary coefficient. Inserting expression (B.1) into (5.29) and looking on coefficients of ower of η = z − z we get:0 = − a + a a a + a η − a a + a σz a a σz + 12 a a − a a + a ησz + 3 a η / a σ + 4 a α − a σ + 18 a a + 2 a α + a σ z + 12 a a η σηa + 12 a a σz + 8 a a α + 12 a a σz + 12 a a + 12 a a σ + 12 a a + 36 a a a + 2 a a z ησ + 3 a a η + 4 a a α + 12 a a η − a a σ + 2 a σ z − a a + a ησ + 2 a a σ z + 12 a a η (B.2)The top equation gives two possible non-zero solutions a = ± a = −
12 ( η + σz ) (B.3)for both values a = 1 and a = − a : a = ησz σ z − α − σ − α a = 1 and a as given in (B.3) and a = − ησz − σ z
12 + 2 α − σ + α a = − a as given in (B.3).Consider now the fourth equation. The coefficient a drops from this equation forboth values of a = ± R = 3). The solutions of the the fourthequation for a are a = ησz σ z − α − σ − α a = 1 and a as given in (B.3) and a = − η + σz ) (5 σ z η + σ z + 16 ση + 12 σ z + 4 σz η − α η − α σz − α η − α σz ) (B.7)for a = − a as given in (B.3). We see that a given in (B.4) and (B.6) are equal sothe solution to the recursive problem is in this case consistent. In addition a and highercoefficients will depend on a but a is not fixed by the scheme and can be taken to anyvalue including zero.However a given in (B.5) and (B.7) differ by ση/ (3( σz + η )). Hence the secondsolution for a = − η = 0 , σ = 0 or σ = 0 , η = 0.It seems therefore that as long as a = 1 there is a solution to the recurrence relationsthat does not fail the Painlev´e test. 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. eferences [1] Ablowitz M J, Ramani A and Segur H 1978 Nuovo Cimento J.Math. Phys. J. Phys.: Conf. Ser.
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