Higher order Painlevé equations and their symmetries via reductions of a class of integrable models

Higher order Painlevé equations and their symmetry transformations belonging to extended affine Weyl groups A(1)n are obtained through a self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized Volterra lattice structure. In particular, an explicit example of the Painlevé V equation and its Bäcklund symmetry is obtained through a self-similarity limit of a generalized KdV hierarchy from Aratyn et al (1995 Int. J. Mod. Phys. A 10 2537).


Introduction
The aim of this work is to explore integrable origins of higher order Painlevé equations and their extended affine Weyl symmetry groups. With this goal, we investigate a self-similarity limit of a special class of pseudo-differential Lax hierarchies of the constrained Kadomtsev-Petviashvili (KP) hierarchy with symmetry structure defined by Bäcklund transformations induced by a discrete structure of the Volterra type lattice. The underlying integrable hierarchy is parametrized in terms of 2M Lax coefficients e i , c i , i = 1, . . . , M. In terms of these coefficients, the second Gelfand-Dickey bracket of the underlying constrained KP hierarchy [3] simplifies to a Heisenberg Poisson bracket algebra: where δ x (x − y) = ∂ δ(x − y)/∂x. It is shown that in the self-similarity limit the second t 2 -flow equations of that hierarchy reduce to the higher order Painlevé equations:  (1) 2M . The extended affine Weyl group A (1) n is generated by n + 1 transformations s 0 , s 1 , . . . , s n in addition to a cyclic permutation π . Together they satisfy the relations s i s j s i = s j s i s j (j = i ± 1), s i s j = s j s i (j = i ± 1) πs i = s i+1 π, π n+1 = 1, To obtain Painlevé systems invariant under the extended affine Weyl group A (1) n for odd n we impose second-class constraints working with Dirac's modified second KP Poisson bracket structure. This procedure effectively reduces a number of 2M Lax coefficients of the original Lax hierarchy to 2M − 1 (or in general 2M − k) coefficients and via the self-similarity reduction reproduces Painlevé equations with the extended affine Weyl symmetry A (1) 2M−1 . Here, we present results for the special case of the extended affine Weyl symmetry A (1) 3 with the corresponding Painlevé V equation: with the parameter δ = −1/2. The symmetric Painlevé equations with their extended affine Weyl symmetry group A (1) n first appeared in Adler's [1] and Veselov and Shabat's papers [19] in the setting of periodic dressing chains and later were discussed in great detail from the purely affine Weyl group symmetry point of view by Noumi and Yamada [11,12] (see also [14] and [5] for accessible presentations). As shown in [13] (see also [16,17]) the higher order Painlevé equation of type A (1) n can be obtained by self-similarity reduction of the (n + 1)-reduced modified KP hierarchy with (n + 1) × (n + 1) Lax pair with construction based on a standard regular element in the principal Drinfeld-Sokolov hierarchy. Equivalence of both approaches has been explained in [17] (see also [10,18]) within the Hamiltonian framework. Our investigation provides a different origin of the higher order Painlevé systems as a self-similarity limit of a class of the constrained KP hierarchies defined in terms of the pseudo-differential Lax operators naturally connected with unconventional (not principal) gradations [3].
In section 2, we present the underlying integrable Lax hierarchy focusing on the second flow equations and Bäcklund transformation keeping the Lax equations invariant. In section 3, the self-similarity limit is taken and the Hamiltonians governing the t 2 flow equations are derived in the self-similarity limit. Next, in section 4, the Hamiltonians found in section 3 are shown to reproduce the Hamiltonian structure of higher Painlevé equations invariant under the extended affine Weyl symmetry A (1) 2M when expressed in terms of the canonical variables. This is illustrated for the special cases of M = 1, 2 for which the generators of the extended affine Weyl symmetry group are derived from the Bäcklund transformation of section 2. The Dirac reduction scheme when applied on the original integrable constrained KP hierarchy leads to reduction of the model with A (1) 2M symmetry down to a model characterized by A (1) symmetry. This is illustrated in section 5 for M = 2 with the reduced model being nothing but the Painlevé V equation with its Bäcklund symmetry. Concluding comments are given in section 6, in which we also announce future plans for obtaining solutions to the higher order Painlevé equations and generalizing these equations to the higher Painlevé hierarchies by making use of the results presented here.

A 'half-integer' lattice
It is well known that symmetries of many continuum KP-type hierarchies are governed by discrete lattice-like structures. A standard example is provided by the AKNS hierarchy and the Toda lattice structure of its Bäcklund transformations leading to Hirota-type equations for the Toda chain of tau-functions [2]. There also exists the so-called two-boson formulation of the AKNS hierarchy which is invariant under symmetry transformations on a 'half-integer' lattice which generalizes Toda lattice [2]. We now present a general 'half-integer' lattice (or the generalized Volterra lattice) following closely [3]. The foundation of this formalism rests on two-spectral equations: and 'time' evolution equations: which both involve objects labeled by integers and half-integers and ∂ = ∂/∂x. If we remove the term M p=1 A (p) n−p+1 n−p from equation (2.1), the remaining system yields the Volterra chain equations. For that reason we will refer to equations (2.1)-(2.3) as a generalized Volterra system. As shown in [3], upon eliminating the half-integer modes, the generalized Volterra system (2.1)-(2.3) reduces to the Toda lattice equations. From (2.1)-(2.3) we find where (2.8) Using equation (2.5) it is easy to shift the spectral equation (2.6) to the half-integer lattice: (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 Bäcklund transformation of the higher order Painlevé equations.

Basic facts about the 2M-Bose-constrained KP hierarchy
The recurrence relation (2.7) is realized by the Lax operators: Recall that the KP hierarchy is endowed with the bi-Hamiltonian Poisson bracket structure resulting from two compatible Hamiltonian structures on the algebra of pseudo-differential operators. Remarkably, for the above Lax hierarchy the second bracket of the hierarchy is realized in terms of (c k , e k ) M k=1 as the Heisenberg Poisson bracket algebra (1.1). The Lax operator (2.10) realizes the recursive relation (2.7) rewritten in this context as follows: for M = 1, 2, . . .. The corresponding second flow equations can be obtained from the second bracket structure through where the Hamiltonian H 2 is an integral of the coefficient u 1 (M) appearing in front of ∂ −2 in the Lax operator (2.11) when cast in a conventional KP form: As a consequence of equation (2.11) we obtain The Darboux-Bäcklund transformation of the Lax operator L M defined in equation (2.10) takes a form and since e M ∼ A (0) n we see from equation (2.5) that it represents transformation on the Volterra lattice from integer modes to half-integer modes. For coefficients with highest indices this transformation results in For a special example of the so-called two-Bose system with M = 1 (which below will be shown to correspond to the symmetric Painlevé IV equation) the Lax operator is The Lax operator L 1 possesses a Darboux-Bäcklund symmetry: which keeps its form unchanged while transforming e 1 , c 1 as follows:

Hamiltonians and the t 2 -flows in the self-similarity limit of the 2M -Bose-constrained KP hierarchy
The second flow equation (2.12) results in the following expressions for the Lax coefficients: Effectively, the action of the self-similarity reduction replaces ∂f/∂t 2 with −(xf ) x /2. Integrating all equations obtained by taking the self-similarity limit we find where κ j ,κ j are integration constants. It follows that equations (3.2) can be rewritten as for j = 1, . . . , M with appropriately redefined constants k j ,k j : Equations (3.4)-(3.5) can be reproduced through with a Poisson bracket structure: where E j,i is an element of a strictly lower triangular matrix equal to

General construction
Let q i , p i , i = 1, . . . , M, be canonical coordinates satisfying the canonical brackets Relations define new variables f k , k = 1, . . . , 2M, and map the canonical brackets into the following Poisson brackets: We now propose a conversion table between e i , c i , i = 1, . . . , M, and a special set of canonical coordinates and equivalently Painlevé variables f k , k = 1, . . . , 2M, that will satisfy the higher order Painlevé equations (1.2). First, we list the result for e i , i = 1, . . . , M: in agreement with [15]. The corresponding Hamilton equations are equivalent to the higher Painlevé equations as given in equation (1.2) with identification of variables provided by relation (4.1).
In the following subsections of this section we will illustrate the above general result for M = 1 and M = 2.

The case of M = 1 and the Painlevé IV equation
For M = 1 equations (3.4)-(3.5) simplify to and agree with the self-similar limit of the so-called two-boson formulation of the AKNS hierarchy [2,4]. H 1x = 2e 1 c 1 , H 1xx = 2e 1x c 1 + 2e 1 c 1x = 2e 2 1 c 1 − 2c 2 1 e 1 + 2k 1 c 1 + 2k 1 e 1 +k 1 c 1 − k 1 e 1 which leads to the Jimbo-Miwa equation [9] of the Painlevé IV system: for H 1 . Connection of M = 1 example (4.5) to the A (1) 2 manifestly symmetric Painlevé IV set of equations with α 0 + α 1 + α 2 = −2 can be made explicit by setting for i = 0, 1, 2. For each of the values 0, 1, 2 of the index i we denote by g i the Darboux-Bäcklund transformation derived from g from relation (2.15) by replacing variables e i , c i by f i according to relation (4.7). It is easy to check that each such g i maps f i → f i+1 , α i → −α i+1 and α i+2 → α i + α i+1 and is realized as πs i in terms of the extended affine Weyl operators from (1.3). Note that the generators s i of the affine Weyl group A (1) 2 act as s i (α i+2 ) = α i + α i+2 . The above construction together with the idea of introducing permutation symmetry of the extended affine Weyl group A (1) 2 by associating f i 's with any of the solutions of the self-similar limit of the so-called two-boson formulation of the AKNS hierarchy was discussed in [4].

The four-Bose system and A (1)
4 Painlevé equations We now consider a four-boson case with M = 2 and (c k , e k ) 2 k=1 subject to equations e 1 x = 2xe 1 − (e 1 + 2c 1 + 2c 2 )e 1 +k 1 e 2 x = 2xe 2 − 4xe 1 − (e 2 + 2c 2 )e 2 + (2c 1 + 2e 1 + 4c 2 )e 1 +k 2 The symmetry transformations (2.13)-(2.14) read here and keep equations (4.8) invariant for g(k 2 ) = −4 + 3k 1 + 2k 2 + 2k 1 + 5k 2 . (4.11) In order to see the meaning of this transformation from the group theoretic point of view we cast equations (4.8) into the symmetric A (1) 4 Painlevé equations: 4 ( 4 .12) with conditions f i = f i+5 and 4 i=0 α i = −2. We propose the following identification: Alternatively, we can write relations between e i , c i , i = 1, 2, and f i , i = 0, 1, . . . , 4, as in agreement with equations (4.2) and (4.3). Accordingly, one can rewrite the g-transformation from (4.10) as Comparing with definitions of transformations s i , i = 1, 2, 3, 4 (see for instance [14]), we see that the expression for the transformation g from (4.10)-(4.11) agrees with g = πs 1 + πs 3 − π (4.16) as applied on both f 's and α's. Generalizing relation (4.13), we next associate −e 1 and −c 2 with f i and f i+2 , respectively, for each index i value of 1, 2, 3, 4. In that way we obtain the following realizations of the Darboux-Bäcklund transformation g defined in relation (4.10): by replacing e i , c i in relation (4.10) with f i . In the above expression s i are generators of the affine Weyl group A (1) 4 . Note that obviously g 1 agrees with g as given in equation (4.16) for f 1 = −e 1 , f 3 = −c 2 . Thus, in a close analogy to a reasoning presented for the Painlevé IV system at the end of subsection 4.2 as well as discussion in [4], by employing different associations of the symmetric Painlevé system variables f i and f i+2 with i = 1, 2, 3, 4 with variables −e 1 and −c 2 of the underlying integrable model we are able to recover all of the affine Weyl A (1) 4 generators s i from a Darboux-Bäcklund transformation g defined in relation (4.10). For instance, s 1 = (−I + π(g 1 + g 2 + g 3 − g 0 − g 4 ))/2.

Reduction of the M = 2 case: Painlevé V equation and its symmetries
We will now follow [3] and perform a Dirac reduction of the M = 2 case (see subsection 4.3), by redefining variables as follows: of a so-called SL(3, 1) KdV hierarchy from [3]. The self-similarity reduction of the above equation yields Eliminating c from equation (5.4) and plugging it into equation (5.3) yields the following expression for c 1 : in terms of e 1 and its derivative. Next, we plug the above expression into equation (5.2) together with the expression for c in terms of e 1 and c 1 to obtain the second-order equation for y = e 1 /2x: .
A change of coordinate from x to z such that z = σ x 2 yields The above equation takes on a conventional form of the Painlevé V equation (1.4) for w = y/(y − 1) and δ = −1/2 or σ 2 = 1 (see [20]).
To study the Darboux-Bäcklund transformation of the Painlevé V system we follow the method of equation (2.9) and perform the similarity transformation on the Lax operator for the reduced four-boson system [3]: with B 1 = e 1 + c 1 + c. Note that L can be rewritten in a generalized KdV form L = u 1 (∂ − u 2 ) −1 + u 3 + ∂ 2 of unconventional gradation for appropriate coefficients u i , i = 1, 2, 3 [3].
Substituting c by (2xy x + 2y + 2xyc 1 + k) and c 1 , as follows from formula (5.5), by x y x y − 1 2k 1 y + k +k 1 + 1 (5.11) or g(y(z)) = with properties c 2 g = 2α, a 2 g = −2β the function F from relation (5.13) can be rewritten as F = +zw z − w 2 c g + w c g − a g + zσ + a g in complete agreement with the expression for the Bäcklund transformation obtained in [7,8]. Thus, this construction establishes the Painlevé V equation with its Bäcklund symmetry structure as a limit of an integrable pseudo-differential Lax hierarchy. To the best of our knowledge, this is the first time such explicit derivation of a general Painlevé V equation with three independent Painlevé coefficients was carried out by taking a self-similarity limit of an integrable model defined by the pseudo-differential Lax operator L = u 1 (∂ − u 2 ) −1 + u 3 + ∂ 2 .

Outlook
We have derived here the higher order Painlevé equations by taking a self-similarity limit of the special class of integrable models and showed how the extended affine Weyl groups A (1) n symmetries are induced by the Bäcklund transformations generated by translations on the underlying 'half-integer' Volterra lattice. The Hamiltonians of the integrable model reduced by the self-similarity procedure have been explicitly shown to transform under the change of variables into the Hamiltonians for the higher Painlevé equations. In a forthcoming publication, we plan to provide explicit proof for formulas governing such a change of variables and include in the formalism the Painlevé equations with the extended affine Weyl groups A (1) 2n−1 symmetries for n > 2. We will also employ on the one hand a link between integrable hierarchies and on the other hand the higher order Painlevé equations to derive the corresponding higher order Painlevé hierarchies generalizing the construction of the Painlevé IV hierarchy in [6] to higher orders. Another future goal is to use our approach to provide solutions of the higher order Painlevé equations by taking appropriate limits of soliton solutions to the original integrable models similarly to what was accomplished in [4] for the Painlevé IV equation.