Integrable Origins of Higher Order Painleve Equations
aa r X i v : . [ n li n . S I] O c t October, 2010
Integrable Origins of Higher Order Painlev´e Equations
H. AratynDepartment of PhysicsUniversity of Illinois at Chicago845 W. Taylor St.Chicago, Illinois 60607-7059J.F. Gomes and A.H. ZimermanInstituto de F´ısica Te´orica-UNESPRua Dr Bento Teobaldo Ferraz 271, Bloco II,01140-070 S˜ao Paulo, Brazil
ABSTRACT
Higher order Painlev´e equations invariant under extended affine Weyl groups A (1) n are obtained through self-similarity limit of a class of pseudo-differential Laxhierarchies with symmetry inherited from the underlying generalized Volterra latticestructure. 1 Introduction
This paper investigates a self-similarity limit of a special class of pseudo-differentialLax hierarchies of the constrained KP hierarchy with symmetry structure definedby B¨acklund transformations induced by a discrete structure of Volterra type lat-tice. The underlying integrable hierarchy is realized in terms of 2 M Lax coeffi-cients e i , c i , i = 1 , . . ., M forming M “Darboux-Poisson”canonical pairs with respectto the second Gelfand-Dickey bracket of the underlying constrained Kadomtsev-Petviashvili (KP) hierarchy.It is shown that in the self-similarity limit the second t -flow equations of thathierarchy transform to the higher order Painlev´e equations : f ′ i = f i ( f i +1 − f i +2 + f i +3 − f i +4 + · · · − f i − ) + α i , i = 0 , , . . . , M (1.1)under a change of variables from e i , c i , i = 1 , . . ., M to f i , i = 1 , . . ., M , whichis described in subsection 4.1. Equation (1.1) introduced f = − P Mi =1 f i − x andconstants α i satisfying P Mi =0 α i = −
2. The system satisfies periodicity conditions f M + i = f i − , α M + i = α i − , i = 0 , , , . . ., M . These equations are invariantunder B¨acklund transformations forming the extended affine Weyl group A (1)2 M . Theextended affine Weyl group A (1) n is generated by n + 1 transformations s , s , . . ., s n in addition to cyclic permutation π which together satisfy relations s i s j s i = s j s i s j ( j = i ± , s i s j = s j s i ( j = i ± πs i = s i +1 π, π n +1 = 1 , s i = 1 . The symmetric Painlev´e equations with their extended affine Weyl symmetry group A (1) n first appeared in Adler’s paper [1] in the setting of periodic dressing chains andlater were discussed in great details by Noumi and Yamada [8, 9] (see also [10]).Imposing second-class constraints on the second KP Poisson bracket structurevia Dirac scheme reduces a number of 2 M Lax coefficients to 2 M − M − k ) coefficients and via self-similarity reduction reproduces Painlev´e equationswith the extended affine Weyl symmetry A (1)2 M − . Here we just present results ofthe special case of the extended affine Weyl symmetry A (1)3 and the correspondingPainlev´e V equation.In section 2 we present the underlying integrable Lax hierarchy focusing onthe second flow equations and B¨acklund transformation keeping the Lax equationsinvariant. In Section 3 the self-similarity limit is taken and Hamiltonians governingthe t flow equations in this limit are derived. Next, in Section 4 the Hamiltoniansfound in Section 3 are shown to reproduce Hamiltonian structure of the higherPainlev´e equations invariant under the extended affine Weyl symmetry A (1)2 M whenexpressed in terms of canonical variables. This is illustrated for special cases of M = 1 , , A (1)2 M symmetry down to a model characterized by A (1)2 M − symmetry.This is illustrated in Section 5 for M = 2 when the reduced model is nothing butthe Painlev´e V system. Concluding comments are given in Section 6.2 The Integrable Hierarchy, its Second Flow andGeneralized Volterra Symmetry Structure
It is well-known that symmetry of many continuum KP-type hierarchies are governedby discrete lattice-like structures. A standard example is provided by the AKNShierarchy and the Toda lattice structure of its B¨acklund transformations leadingto Hirota type equations for the Toda chain of tau-functions [2]. There also existsthe so-called two-boson formulation of the AKNS hierarchy which is invariant undersymmetry transformations on a “half-integer” lattice which generalizes Toda lattice[2]. We now present a general “half-integer” lattice (or the generalized Volterralattice) following closely reference [3]. The foundation of this formalism rests on twospectral equations: λ / e Ψ n + = Ψ n +1 + A (0) n +1 Ψ n + M X p =1 A ( p ) n − p +1 Ψ n − p (2.1) λ / Ψ n = e Ψ n + + B (0) n e Ψ n − (2.2)and “time” evolution equations: e Ψ n + = (cid:0) ∂ − B (0) n − A (0) n (cid:1) e Ψ n − ; Ψ n +1 = (cid:16) ∂ − B (0) n − A (0) n +1 (cid:17) Ψ n (2.3)which both involve objects labeled by integers and half-integers. After removing theterm P Mp =1 A ( p ) n − p +1 Ψ n − p from equation (2.1) the above system yields the Volterrachain equations. For that reason we will refer to equations (2.1)–(2.3) as a general-ized Volterra system. As shown in [3], upon eliminating the half-integer modes, thegeneralized Volterra system (2.1)–(2.3) reduces to the Toda lattice equations. From(2.1)–(2.3) we find: λ / e Ψ n + = (cid:18) ∂ − B (0) n + M X p =1 A ( p ) n − p +1 ( ∂ − B (0) n − p − A (0) n − p +1 ) − · · ·· · · ( ∂ − B (0) n − − A (0) n ) − (cid:19) Ψ n (2.4)and λ / Ψ n = (cid:0) ∂ − A (0) n (cid:1) e Ψ n − (2.5)Eliminating half-integer modes from the last two relations yields a spectral equationof a form λ Ψ n = L ( M +1) n Ψ n (2.6)with Lax operator L ( M +1) n given by recurrence relation: L ( M +1) n = e R B (0) n − (cid:16) ∂ − A (0) n + B (0) n − (cid:17) L ( M ) n (cid:0) ∂ − A (0) n (cid:1) − e − R B (0) n − (2.7)3here L ( M ) n = ∂ + M X p =1 A ( p ) n − p ( ∂ + B (0) n − −B (0) n − p − −A (0) n − p ) − · · · ( ∂ + B (0) n − −B (0) n − −A (0) n − ) − (2.8)Using equation (2.5) it is easy to shift the spectral equation (2.6) to the half-integerlattice: λ e Ψ n − = e L ( M +1) n e Ψ n − , e L ( M +1) n = (cid:0) ∂ − A (0) n (cid:1) − L ( M +1) n (cid:0) ∂ − A (0) n (cid:1) (2.9)The similarity transformation responsible for transformation from integer to half-integer lattice will be shown below to play a central role as a B¨acklund transforma-tion of the higher order Painlev´e equations. M -bose constrained KP hierarchy The recurrence relation (2.7) is realized by the 2 M -bose constrained KP hierarchywith Lax operators L M , M = 1 , , . . . : L M = ( ∂ − e M ) Y k = M − ∂ − e k − M X l = k +1 c l ! ∂ − M X l =1 c l ! × M Y k =1 ∂ − e k − M X l = k c l ! − (2.10)given here in terms of the “Darboux-Poisson”canonical pairs ( c k , e k ) Mk =1 . Recallthat the KP hierarchy is endowed with bi-Hamiltonian Poisson bracket structuresresulting from the two compatible Hamiltonian structures on the algebra of pseudo-differential operators. Remarkably, for the above Lax hierarchy the second bracket ofhierarchy is realized in terms of ( c k , e k ) Mk =1 as a Heisenberg Poisson bracket algebra: { e i ( x ) , c j ( y ) } = − δ ij δ x ( x − y ) , i, j = 1 , , . . . , M (2.11)The Lax operator (2.10) realizes the recursive relation (2.7) rewritten in this contextas follows: L M = e R c M ( ∂ + c M − e M ) L M − ( ∂ − e M ) − e − R c M , L ≡ ∂ (2.12)for M = 1 , , . . . . The corresponding second flow equations can be obtained fromthe second bracket structure as follows ∂ f∂t = { f, H } , (2.13)where the Hamiltonian H is an integral of the coefficient u ( M ) appearing in frontof ∂ − in the Lax operator (2.12) when cast in a conventional KP form : L M = ∂ + u ( M ) ∂ − + u ( M ) ∂ − + . . .
4s a consequence of equation (2.12) we obtain the recursive relations for the coeffi-cients : u ( M ) = u ( M −
1) + ( e ′ M + e M c M ) u ( M ) = u ( M −
1) + u ′ ( M −
1) + 2 u ( M − c M + ( e ′ M + e M c M ) ( e M + c M ) . with solutions u ( M ) = M X i =1 ( e ′ i + e i c i ) u ( M ) = M − X i =1 ( M − i ) ( e ′ i + e i c i ) ′ + 2 M − X i =1 u ( i ) c i +1 + M X i =1 ( e ′ i + e i c i ) ( e i + c i )The Darboux-B¨acklund transformation of the Lax operator L M defined in equation(2.10) takes a form L M → ( ∂ − e M ) − L M ( ∂ − e M )and since e M ∼ A (0) n we see from equation (2.5) that it represents transformation onthe Volterra lattice from integer modes to half-integer modes. In terms of coefficientsthis results for coefficients with highest indices in : g ( e M ) = e M − + c M , g ( c M ) = − e M − + e M − c ′ M c M g ( e M − ) = e M − + e M − − e M + c M + c M − + c ′ M c M g ( c M − ) = − e M − + e M − − (cid:16) − e M − + e M − c M − c M − − c ′ M c M (cid:17) ′ (cid:16) − e M − + e M − c M − c M − − c ′ M c M (cid:17) (2.14)in addition to g e k + M X l = k +1 c l ! = e k − + M X l = k c l , ≤ k ≤ M, g e + M X l =2 c l ! = M X l =1 c l . (2.15)For a special example of the so-called two Bose system with M = 1 (which below willbe shown to correspond to the symmetric Painlev´e IV equations) the Lax operatoris: L = ( ∂ − e )( ∂ − c )( ∂ − e − c ) − Such a Lax operator possesses a Darboux-B¨acklund symmetry: L → ( ∂ − e ) − L ( ∂ − e ) = ( ∂ − c )( ∂ − e + c x /c )( ∂ − e − c + c x /c ) − which keeps its form unchanged and transforms e , c as follows : g ( e ) = c , g ( c ) = e − c x /c , (2.16)5 Hamiltonians and t Flow Equations in the Self-similarity Limit of the M -bose constrained KPhierarchy Second flow equation (2.13) results in the following expressions for the Lax coeffi-cients: ∂ c j ∂t = dd x c ′ j − c j − e j c j + 2 M X i = j +1 c ′ i − M − X i = j c i c i +1 ! , j = 1 , . . ., M∂ e j ∂t = dd x − e ′ j − e j − e j c j − u ( j − − e j M X i = j +1 c i ! , j = 1 , . . ., M (3.1)Effectively, the action of the self-similarity reduction replaces ∂f /∂t with − ( xf ) x / e ′ j + 2 j − X i =1 e ′ i = 2 xe j − e j − e j M − X i = j c i +1 ! − j X i =1 e i c i + ¯ κ j c ′ j + 2 M X i = j +1 c ′ i = − xc j + c j + 2 c j M − X i = j c i +1 + 2 c j e j + κ j (3.2)for j = 1 , . . ., M and with integration constants κ j , ¯ κ j . The above equations areHamiltonian in a sense that : e ′ j + 2 j − X i =1 e ′ i = ∂ H M ∂c j , c ′ j + 2 M X i = j +1 c ′ i = − ∂ H M ∂e j (3.3)with H M = − M X j =1 e j c j ( e j + c j − x ) − X ≤ j i, j ≤ i . The equations of motion (3.6)-(3.7) are reproduced through e ′ j = { e j , H M } , c ′ j = { c j , H M } . (3.12)7 Connection to Higher Order Painlev´e Equations
Let q i , p i , i = 1 , . . ., M be canonical coordinates satisfying the canonical brackets { q i , p j } = − δ ij , { q i , q j } = 0 = { p i , p j } , i = 1 , . . ., M . Relations q i = f i , p i = i X k =1 f k − , i = 1 , . . ., M (4.1)define new variables f k , k = 1 , . . ., M and map the canonical brackets into thefollowing Poisson brackets : { f i , f i +1 } = 1 , { f i , f i − } = − , i = 1 , . . ., M .
We now propose a conversion table mapping e i , c i i = 1 , . . ., M into a special setof canonical coordinates as well as Painlev´e variables f k , k = 1 , . . ., M that willsatisfy the higher order Painlev´e equations (1.1).First, we list the result for e i i = 1 , . . ., M : e M = q M + p M + 2 x + k M p M − p M − = M X i =1 f i − + f M + 2 x + k M f M − e M − = − p M − = − M − X i =1 f i − e M − = − q − · · · − q M − = − f − · · · − f M − e M − = − p M − + p = − f − f − · · · − f M − · · · = · · · e = − f M − − f M = ( − q M/ − q M/ − M even − p ( M +1) / + p ( M − / M odd e = − f M − = ( − p M/ + p M/ − M even − q ( M − / M odd (4.2)8nd next for c i i = 1 , . . ., M : c M = − p M + p M − = − f M − ,c M − = p M − p M − − q M − q M − = f M − − f M − − f M c M − = p M − + q + q + · · · + q M − + q M − + 2 q M − + 2 q M + 2 xc M − = − p M − − p − q − q − · · · − q M − − q M − − q M − − q M − − q M − x,c M − = p + p M − + q + · · · + q M − + q M − ++ 2 q M − + 2 q M − + 2 q M + 2 x · · · = · · · c M − k = p k − + p M − k − + q k + · · · + q M − k − + 2 q M − k + 2 q M − k +1 + · · ·· · · + 2 q M − + 2 q M + 2 x, k = 1 , , , . . .c M − (2 k − = − ( p k − + p M − k + q k − + · · · + q M − k − + 2 q M − k + · · · +2 q M − + 2 q M + 2 x ) , k = 2 , , . . . (4.3)Thus from eq. (4.3) it follows that c = − (cid:0) p M/ − + p M/ + q M/ − + 2 q M/ + . . . + +2 q M + 2 x (cid:1) M even p ( M − / + p ( M − / + q ( M − / +2 q ( M +1) / + . . . + 2 q M + 2 x M odd (4.4)The Hamiltonian H M defined in (3.4) reads in terms of q i , p i , i = 1 , . . ., M definedin equations (4.2)-(4.3) as follows H M = M X j =1 p j q j ( p j + q j + 2 x ) + 2 X ≤ ji q j + 2 x ! + α i are equivalent to the higher Painlev´e equations as given in equation (1.1) withidentification of variables provided by relation (4.1).In the following subsections of this section we will illustrate the above generalresult for M = 1 , ,
3. 9 .2 The case of M = 1 and Painlev´e IV Equations For M = 1 the equations (3.6)-(3.7) take the form of the Levi system : e ′ = 2 xe − ( e + 2 c ) e + ¯ k c ′ = − xc + ( c + 2 e ) c + k , (4.6)which is kept invariant under transformations (2.16) when accompanied by trans-formations g ( k ) = 2 − ¯ k , g (cid:0) ¯ k (cid:1) = − k of integration constants. Note that eqs.(4.6) are Hamiltonian in the following sense e ′ = ∂ H ∂c = { e , H } , c ′ = − ∂ H ∂e = { c , H } , where as follows from the previous subsection : H = 2 xe c − e c − e c + ¯ k c − k e , { e , c } = 1 . (4.7)Also we derive from ∂ H ∂x = 2 e c that H xx = 2 e ′ c + 2 e c ′ = 2 e c − c e + 2¯ k c + 2 k e and 2 ( H − x H x ) = 2 (cid:0) − e c − c e + ¯ k c − k e (cid:1) Accordingly, H xx + 2 ( H − x H x ) = 4 c (cid:0) e c − ¯ k (cid:1) = 2 c (cid:0) H x − k (cid:1) and H xx − H − x H x ) = 4 e ( e c + k ) = 2 e ( H x + 2 k ) . Thus, H satisfies the Jimbo-Miwa equation [7] of Painlev´e IV system:( H xx + 2 ( H − x H x )) ( H xx − H − x H x )) = 2 H x (cid:0) H x − k (cid:1) ( H x + 2 k )Connection of M = 1 example (4.6) to A (1)2 symmetric Painlev´e IV set of equa-tions f x = f ( f − f ) + α f x = f ( f − f ) + α f x = f ( f − f ) + α with α + α + α = − f i = − c , f i +1 = − e + c x c f i +2 = e + c − c x c − x, α i = k , α i +2 = − k − ¯ k for i = 0 , , g defined in (2.16)and accordingly mapping f i to f i +1 , α i → − α i +1 and α i +2 → α i + α i +1 . This isconsistent with realization of g as g = πs i for i = 0 , ,
2, where generators s i of theaffine Weyl group A (1)2 act as s i ( α i +2 ) = α i + α i +2 . The above solutions together withthe idea of introducing permutation symmetry of the extended affine Weyl group A (1)2 by associating f i ’s to any of the solutions of the Levi system was discussed in[4]. 10 .3 The Four-Bose system and A (1)4 Painlev´e Equations
We now consider a four-boson case with M = 2 and ( c k , e k ) k =1 subject to equations e ′ = 2 xe − ( e + 2 c + 2 c ) e + ¯ k e ′ = 2 xe − xe − ( e + 2 c ) e + (2 c + 2 e + 4 c ) e + ¯ k c ′ = − xc + 4 xc + ( c + 2 e ) c + (2 c − c − e ) c + k c ′ = − xc + ( c + 2 e ) c + k (4.8)as follows from equations (3.6)-(3.6). The corresponding Hamiltonian is H = − e c ( e + c + 2 c ) − e c ( e + c ) + 2 x X i =1 e i c i − k e + ¯ k c − ( k + 2 k ) e + (¯ k + 2¯ k ) c . (4.9)We find that in the case of M = 2 the vector fields E j , C j , i = 1 , E = ∂∂c , E = ∂∂c − ∂∂c , C = 2 ∂∂e − ∂∂e , C = − ∂∂e and according to (3.10) they lead to the Poisson brackets : { e , c } = 1 , { e , c } = 0 , { e , c } = − , { e , c } = 1 (4.10)consistent with equations of motion (4.8) through the Poisson brackets : e ′ j = { e j , H } , c ′ j = { c j , H } , j = 1 , . (4.11)The symmetry transformations (2.14)-(2.15) read here : g ( e ) = e + c , g ( e ) = e − e + c + c + c ′ c (4.12) g ( c ) = − e + e − c ′ c , g ( c ) = e − (cid:16) − e + e − c − c − c ′ c (cid:17) ′ (cid:16) − e + e − c − c − c ′ c (cid:17) , which keep equations (4.8) invariant for : g ( k ) = − k + 2¯ k , g (¯ k ) = 2 − ¯ k − ¯ k − k − k ,g ( k ) = 2 − ¯ k − ¯ k , g (¯ k ) = − k + 2¯ k + 2 k + 5 k . (4.13)In order to see the meaning of this transformation from the group theoretic pointof view we cast equations (4.8) into the symmetric A (1)4 Painlev´e equations: f ′ i = f i ( f i +1 − f i +2 + f i +3 − f i +4 ) + α i , i = 0 , . . . , f i = f i +5 and P i =0 α i = −
2. We propose the following identification f = − e f = g ( f ) = − (cid:18) e − e + c + c + c ′ c (cid:19) = − e − e − c − c + 2 x − k c f = − c f = g ( f ) = − (cid:18) − e + e − c ′ c (cid:19) = e + e + c − x + k c f = − f − f − f − f − x = e + c + 2 c − x and α = − ¯ k , α = 2 − ¯ k − ¯ k − k − k α = k , α = − k + ¯ k α = − X i =1 α i = − k + k + 2 k . Alternatively we can write relations between e i , c i , i = 1 , f i , i = 0 , , . . ., e = − f , e = − f − f + f ′ /f = − f − f + α f c = f − f − f , c = − f (4.15)in agreement with equations (4.2) and (4.3). Accordingly one can rewrite the g -transformation from (4.12) as g ( f ) = f , g ( f ) = f g ( f ) = − f + f + f − f ′ f + f ′ f = f − α f + α f g ( f ) = f + f − f + f ′ f = f + α f . (4.16)Comparing with definitions of transformations s i , i = 1 , , , g from (4.12)-(4.13) agrees with g = πs + πs − π as applied on both f ’s and α ’s.More generally, associating − e and − c to f i and f i +2 , respectively, with i takingall the values i = 1 , , , g = πs i + πs i +2 − π ≡ g i for each realization, where s i are generators of the affine Weyl A (1)4 . This relates g with s i + s i +2 and by varying i over all its values makes possible to recover all theaffine Weyl A (1)4 generators s i from one Darboux-B¨acklund transformation g . Forinstance s = ( − I + π ( g + g + g − g − g )) / .4 M = 3 Bose System and the symmetric A (1)6 Painlev´eEquations
For M = 3, equations (3.6)-(3.7) become e ′ = 2 xe − ( e + 2 c + 2 c + 2 c ) e + ¯ k e ′ = 2 xe − xe − ( e + 2 c + 2 c ) e − ( − c − e − c − c ) e + ¯ k e ′ = 2 xe − xe + 4 xe − ( e + 2 c ) e − ( − c − e − c ) e − (2 c + 2 e + 4 c + 4 c ) e + ¯ k c ′ = − xc + 4 xc − xc + ( c + 2 e ) c + (2 c − c − e ) c + (2 c − c + 2 c + 4 e ) c + k c ′ = − xc + 4 xc + ( c + 2 e ) c + (2 c − c − e ) c + k c ′ = − xc + ( c + 2 e ) c + k . (4.17)These equations are Hamiltonian as in eqs.(3.3) with H = 2 x e c + 2 x e c + 2 x e c − e c ( e + c + 2 c + 2 c ) − e c ( e + c + 2 c ) − e c ( e + c ) + ¯ k c + (¯ k + 2 ¯ k ) c + (¯ k + 2 ¯ k + 2 ¯ k ) c − ( k + 2 k + 2 k ) e − ( k + 2 k ) e − k e . (4.18)The symmetric A (1)6 Painlev´e equations f ′ i = f i ( f i +1 − f i +2 + f i +3 − f i +4 + f i +5 − f i +6 ) + α i , i = 0 , , , . . . , , (4.19)with condition f i = f i +7 and with f = − X i =1 f i − x are satisfied by f = e + e + c + 2 c − xf = e + c + 2 c + 2 c − xf = − e f = − e − e − c − c − c + 2 xf = − e + e − c − c − c ′ c = − e − e − c − c + 2 x − k c f = − c f = − (cid:18) e − e − c ′ c (cid:19) = e + e + c − x + k c (4.20)13urthermore α = − k + 2 k + ¯ k + ¯ k = − κ + ¯ κ − ¯ κ α = − k + 2 k + 2 k + ¯ k = − κ + ¯ κ α = − ¯ k = − ¯ κ α = 2 − k − k − k − ¯ k − ¯ k = 2 − κ − ¯ κ + ¯ κ α = 2 − ¯ k − ¯ k − k − k = 2 − ¯ κ + ¯ κ − κ − κ α = k = κ α = − k + ¯ k = − κ − ¯ κ in terms of objects e i , c i , k i , ¯ k i , i = 1 , , M = 3 equations (4.17).The logic of deriving the above association is as follows. We initially set f = f M − = − c M = − c and f = f M = g ( f ). This is suggested by equation of motionfor c M which has the unique 2 xc M term on the right hand side making it a goodcandidate for f i , since Painlev´e equations have this structure after elimination of f . But equations of motion for f and f after elimination of f only involve sums f + f and f + f , respectively. Therefore these quantities are derived from theseequations to be f + f = − e = − e M − and f + f = g ( f + f ) = − g ( e ). Next,we turn into equation for f in (4.19), which we rewrite as f ′ = f ( − e − c − c + 2 x + f ) + α after we substituted f = − c , f + f = − e and f = − ( f + f ) − e − k /c andthe values for f and f + f determined previously by g transformations. We findthat the solution to these equations is given by f = e + c + 2 c + 2 c − xα = 2 + k + 2 k + 2¯ k + ¯ k From that result we derive f as − e − f and from eq. for f we derived f and f = − e .Applying similarity transformation (2.2) L → ( ∂ − e ) − L ( ∂ − e )on the Lax operator obtained from (2.10) by setting M = 3. g ( e ) = e + c , g ( c ) = − e + e − c ′ c g ( e ) = e + e − e + c + c + c ′ c g ( c ) = − e + e − (cid:16) − e + e − c − c − c ′ c (cid:17) ′ (cid:16) − e + e − c − c − c ′ c (cid:17) (4.21)in addition to g ( e + c + c ) = X l =1 c l . f i : g ( f ) = f − α f , g ( f + f ) = f + f g ( f ) = f , g ( f ) = f + α f − α f g ( f ) = f , g ( f ) = f + α f , which agrees with the action of g = πs + πs − π (4.22)as applied on f i and α i .Now the B¨acklund transformation of c is equal to g ( c ) = e − ( g ( c ) + g ( c ) − c − c − c ) ′ g ( c ) + g ( c ) − c − c − c (4.23)in terms of g ( c ) and g ( c ) from eqs. (4.21).In terms of f i and α i ’s it takes a form g ( c ) = − f − (cid:16) f − α f (cid:17) ′ f − α f = − f − f − f + f + f + α f − α + α f − α f This gives rise to the following transformation of f : g ( f ) = f − α + α f − α f Thus representation of the B¨acklund transformation g in terms of generators of theextended affine Weyl group A (1)6 from (4.22) needs to be augmented as follows: g = πs + πs − π + s ( πs − π ) = πs + πs s − π as applied on f i and α i .By generalizing eq. (4.20) by substituting i = 5 with arbitrary i between 0 and M one obtains g i = πs i + πs i − s i − − π, i = 0 , , . . . , f i = − c M providing a scheme to reproduce all generators s , s , . . . , s of the affine Weyl groupof A (1)6 by one B¨acklund transformation g .15 Reduction of M = 2 case. Painlev´e V We will follow reference [3] and perform a Dirac reduction of M = 2 case (seesubsection 4.3), by redefining variables as follows :( e , c , e , c ) → (˜ e = e , ˜ c = c , ˜ e = ( e − c ) / , ˜ c = ( c − e ) / , which is equivalent to setting a second-class constraint c = c = − e with the Dirac bracket: { c, c } = 12 δ x ( x − y ) . The self-similarity reduction applied on the resulting t evolution equations (3.1)yields − xc = c ′ + 2 c ′ − c − e c − c c + k (5.1) − xe = − e ′ − e − e c − e c + ¯ k (5.2) − xc = e ′ + e c + k . (5.3)Eliminating c and c from equations (5.1)- (5.3) yields the following expression for y = e / x : y zz = − z y z + (cid:18) y + 12( y − (cid:19) y z − αyz ( y − − β ( y − z y − γz y ( y − − δy ( y − y −
1) (5.4)with constants α = 18 ( k + 1)( k + ¯ k + 1) + ¯ k
32 = 18 (cid:18) k + 1 + ¯ k (cid:19) β = − ¯ k
32 = − (cid:18) ¯ k (cid:19) , γ = k + k + 12 σ , δ = − σ (5.5)after a change of coordinate x → z such that z = σx .The above equation takes on a conventional form of the Painlev´e V equation for w = y/ ( y −
1) and δ = − / L → ( ∂ + c ) − L ( ∂ + c )on the Lax operator for the reduced 4-boson system [3]. This induces the followingtransformations for variables of the reduced subspace: g ( e ) = e + c + 2 cg ( c ) = e − ( c + e + 2 c ) ′ c + e + 2 cg ( c ) = − e − c
16t follows that g transforms the constants k, k , ¯ k as g ( k ) = − k − ¯ k g (¯ k ) = k + ¯ k + 2 kg ( k ) = ¯ k − w = y/ ( y −
1) of a conventionalPainlev´e V equation yields g ( w ) = 1 − zσwFF = + zw z − w (cid:18) k + 1 + ¯ k (cid:19) + w (cid:18)
12 ( k + 1) + zσ (cid:19) + 14 ¯ k (5.7)In terms of quantities c g = 12 (cid:18) k + 1 + ¯ k (cid:19) , a g = 14 ¯ k with properties c g = 2 α, a g = − β the function F from relation (5.7) can be rewritten as F = + zw z − w c g + w ( c g − a g + zσ ) + a g in complete agreement with [5, 6]. We have here derived the higher order Painlev´e equations by taking self-similaritylimit of the special class of integrable models and shown how the extended affineWeyl groups A (1) n symmetries are induced by B¨acklund transformations generatedby translations on the underlying “half-integer” Volterra lattice. The Hamiltonianof the integrable model reduced by the self-similarity procedure has been explicitlyshown to transform under change of variables into the Hamiltonian for the higherPainlev´e equations. In the forthcoming publication we plan to provide explicitproof for formulas governing such change of variables and include in the formalismthe Painlev´e equations with the extended affine Weyl groups A (1)2 n − symmetries. Wewill also employ a link between on the one hand integrable hierarchies and on theother hand higher order Painlev´e equations to derive the corresponding higher orderPainlev´e hierarchies. Acknowledgments
JFG and AHZ thank CNPq and FAPESP for partial financial support. Work of HAwas partially supported by FAPESP. HA thanks Nick Spizzirri for discussions. Theauthors thank Danilo Virges Ruy for discussions.17 eferences [1] V. E. Adler, Recuttings of Polygons, Functional Analysis and Its Applications, :2, 141–143 (1993)[2] H. Aratyn, L. A. Ferreira, J. F. Gomes and A. H. Zimerman, Toda and Volterralattice equations from discrete symmetries of KP hierarchies, Phys. Lett. B ,85 (1993), [arXiv:hep-th/9307147][3] H. Aratyn, E. Nissimov, S. Pacheva and A. H. Zimerman, Reduction Of TodaLattice Hierarchy To Generalized KdV Hierarchies And Two Matrix Model,Int. J. Mod. Phys. A , 2537 (1995) [arXiv:hep-th/9407112].[4] H. Aratyn, J.F. Gomes and A. H. Zimerman, On the symmetric formulationof the Painleve IV equation, [arXiv:0909.3532][5] V. I. Gromak, Solutions of Painlev´e’s fifth equation, Differential Equations ,519–521 (1976).[6] V.I. Gromak, I. Laine and S. Shimomura, “Painlev´e Differential Equations inthe Complex Plane”, de Gruyter Studies in Mathematics, Volume 28, 2002.[7] M. Jimbo and T. Miwa, Monodromy Preserving Deformation of Linear Ordi-nary Differential Equations with rational coefficients II, Physica D, 407–448(1981)[8] M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems andPainlev´e equations, Comm. Math. Phys. , 281–295 (1998)[9] M. Noumi and Y. Yamada, Higher order Painlev´e equations of type A (1) l ,Funkcial. Ekvac. , 483–503 (1998).[10] M. Noumi, Painlev´e equations through symmetry, in: Translations of Mathe-matical Monographs, vol. 223, American Mathematical Society Providence, RI,2004.[11] Y. Sasano and Y. Yamada, Symmetry and holomorphy of Painlev´e type sys-tems, RIMS. Kokyuroku B2