Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlev\'e equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlev\'e equation. It is shown that each of these equations describes B\"acklund transformations of Veselov-Shabat periodic dressing lattices with odd period known also as Noumi-Yamada systems of type $A_{2(n-1)}^{(1)}$.


Introduction
The main goal of the present paper is to exhibit an approach, which aims to construct a broad community of one-dimensional discrete systems over finite number of fields sharing the property of having Lax pair. It is achieved by similarity reduction of equations of the so-called extended discrete KP (edKP) hierarchy [18,19,20], which itself is proved to be a basis for construction of many classes of integrable differential-difference systems (see, for example, [20] and references therein). By analogy with the situation when Painlevé equations and its hierarchies arise as a result of similarity reduction of integrable partial differential evolution equations [2,6], discrete equations (like dP I ) and its corresponding hierarchies also can be obtained by application of group-theoretic methods.
In this connection it is necessary to mention various approaches aiming to select integrable purely discrete one-dimensional equations, for instance, by applying the confinement singularity test [9], from Bäcklund transformations for the continuous Painlevé equations [8,14], by imposing integrable boundary conditions for two-dimensional discrete equations and reductions to onedimensional ones [12,11,10,15], deriving discrete systems from proper chosen representations of some affine Weyl group [16].
In the article we propose a scheme for constructing purely discrete equations by similarity reductions of equations of edKP hierarchy and then investigate more specific cases. Namely we analyze one-component generalizations of dP I -discrete equations which turn out to serve as discrete symmetry transformation for Veselov-Shabat periodic dressing lattices with odd period. The latter also known in the literature as Noumi-Yamada systems of type A (1) l due to series of works by Noumi with collaborators (see, for example, [17]) where they selected the systems of ordinary differential equations admitting a number of discrete symmetries which realized as automorphisms on field of rational functions of corresponding variables and constitute some representation of extended affine Weyl groupW (A (1) l ). The paper is organized as follows. In Section 2, we give necessary information on Darboux-KP chain, its invariant manifolds and the edKP hierarchy. In Section 3, we show that self-similar ansatzes yield a large class of purely discrete systems supplemented by equations governing deformations with respect to parameters entering these systems. In Section 4, we show one-component discrete equations naturally generalizing dP I . We prove that these discrete equations together with deformation equations are equivalent to Veselov-Shabat periodic dressing chains. Finally, in this section we show that each of these discrete systems or more exactly its autonomous version has continuous limit to P I .

Darboux-KP chain and its invariant submanifolds
In this section we give a sketch of the edKP hierarchy on the basis of approach using the notion of Darboux-KP (DKP) chain introduced in [13]. Equations of DKP chain are defined in terms of two bi-infinite sets of formal Laurent series. The first set consists of generating functions for Hamiltonian densities of KP hierarchy, and second one is formed by Laurent series a(i) each of which relates two nearest neighbors h(i) and h(i + 1) by Darboux map h(i) → h(i + 1) = h(i) + a x (i)/a(i). The DKP chain can be interpreted as a result of successive iterations of Darboux map applying to any fixed solution of KP hierarchy, say h(0), in forward and backward directions. It is given by two equations first of which defines evolution equations of KP hierarchy in the form of local conservation laws, and the second one serves as compatibility condition of KP flows with Darboux map. The Laurent series (1) is the current of corresponding conservation law constructed as special linear combination over Faà di Bruno polynomials h (k) = (∂ + h) k (1) by requiring to be projection of z k on H + = 1, h, h (2) , . . . [4].
In [19,20] we have exhibited two-parameter class of invariant submanifolds of DKP chain S n l each of which is specified by condition Here The invariant submanifolds S 1 l were presented in [13]. It was shown there that the restriction of DKP chain on S 1 0 is equivalent to discrete KP hierarchy [21]. Restriction of DKP chain on S 1 0 , S 2 0 , S 3 0 and so on leads to the notion of edKP hierarchy which is an infinite collection of discrete KP like hierarchies attached to multi-times t (n) ≡ (t (n) 1 , t (n) 2 , . . .) with n ≥ 1. All these hierarchies "live" on the same phase-space M which can be associated with hyperplane whose points are parametrized by infinite number of functions of discrete variable: {a k = a k (i) : k ≥ 1}. One can treat these functions as analytic ones whose the domain of definition is restricted to Z. We have shown in [19] that the flows on S n 0 are given in the form of local conservation laws as Here the coefficients q l ≡ Λ s (a l )} (for definition, see below (5)). In what follows, we refer to (3) as nth subhierarchy of edKP hierarchy. In the following subsection we show Lax pair for edKP hierarchy and write down evolution equations on the functions q (n,r) k generated by (3) in its explicit form.

Lax representation for edKP hierarchy
First, let us recall the relationship between Faà di Bruno iterations and formal Baker-Akhiezer function of KP hierarchy [4,13] These relations allow representing the DKP chain constrained by relation (2) as compatibility condition of auxiliary linear systems. As was shown in [19,20], the restriction of DKP chain on S n 0 leads to linear discrete systems (see also [18]) with the pair of discrete operators Here Λ is a usual shift operator acting on arbitrary function f = f (i) of the discrete variable as (Λf )(i) = f (i + 1). The coefficients q We assign to a l ] = l. One says that any polynomial Q k in a l is a homogeneous one with degree k if Q k → ǫ k Q k when a l → ǫ l a l . 1 As was mentioned above, in fact, these are polynomials in a The consistency condition for the pair of equations (4) reads as Lax equation and can be rewritten in its explicit form as It is important also to take into account algebraic relations coded in permutability operator relation Q r 1 +r 2 . It is quite obvious that equations (6) and (7) admit reductions with the help of simple conditions q (n,r) s ≡ 0 (∀ s ≥ l + 1) for some fixed l ≥ 1. As was shown in [19,20], these reductions can be properly described in geometric setting as double intersections of invariant manifolds of DKP chain: S n,r,l = S n 0 ∩ S ln−r l−1 . The formula (6) when restricting to S n,r,l is proved to be a container for many integrable differential-difference systems (lattices) which can be found in the literature (for reference see e.g. [20]).

Conservation laws for edKP hierarchy
The conserved densities for edKP hierarchy can be constructed in standard way as residues Corresponding currents are easily derived from (6). One has with J (n) We observe that by virtue of relations (7) with r 1 = ln, r 2 = sn and k = l + s, the relation Q (n,ln) l,s = Q (n,sn) s,l with arbitrary l, s ∈ N is identity and therefore we can write down exactness property relation Looking at (9), it is natural to suppose that there exist some homogeneous polynomials ξ If so, then one must recognize that {ξ (n) s : s ≥ 1} is an infinite collection of conserved densities of nth subhierarchy of edKP hierarchy, i.e.
Then the formula (11) says that each ξ (n) s is equivalent to h (n) s /n modulo adding trivial densities. Substituting (11) into (7) with r 1 = 1 and r 2 = kn leaves us with the following equation 2 : s−j (i + 1 − jn).

One can check that solution of this equation is given by
For example, we can write down the following: q l (i + (j − 1)n) δξ (n) s δq l (i + (j − 1)n). 2 Here and in what follows q k (i) ≡ q (n,1) k (i). It is important to note that the mapping {a k } → {q k } is invertible and one can use {q k } as coordinates of M.

Self-similar ansatzes for edKP hierarchy and integrable discrete equations
In what follows, we consider nth edKP subhierarchy restricted to the first p time variables {t (n) 1 , . . . , t (n) p } with p ≥ 2. This system is obviously invariant under the group of scaling transformations k are homogeneous polynomials in q k of corresponding degrees, the transformation g yields Let us consider similarity reductions of nth edKP subhierarchy requiring Then any homogeneous polynomial in q l satisfies corresponding self-similarity condition. In particular, one has More explicitly, one can rewrite (14) as j . We observe that α (n,n) do not depend on evolution parameters p } provided that (13) is valid. Indeed, taking into account exactness property (10) one has For Baker-Akhiezer function, one has the corresponding self-similarity condition in the form One can rewrite the above formulas in terms of self-similarity ansatzes At this point we can conclude that similarity reduction of nth edKP subhierarchy yields the system of purely discrete equations supplemented by deformation equations s,k , k = 1, . . . , p − 1.
These equations appear as a consistency condition of the following linear isomonodromy problem The pair of equations (15) and (16) when restricting dynamics on S n,r,l becomes a system over finite number of fields. If one requires x (n,r) k (i) ≡ 0 (∀ k ≥ l + 1) then l-th equation in (15) is specified as Since it is supposed that x (n,r) l ≡ 0 then the constants α Remark that this equation makes no sense in the case r = ln. In the following section we consider restriction on S n,1,1 corresponding to Bogoyavlenskii lattice.
4 Examples of one-field discrete equations 4.1 One-field discrete system generalizing dP I and its hierarchies Let us consider reductions corresponding to S n,1,1 with arbitrary n ≥ 2. For p = 2 one obtains one-field system 3 together with deformation equation which is nothing else but Bogoyavlenskii lattice. The pair of equations (18) is a specification of (16) and (17). In the case n = 2, (18) becomes dP I while (19) turns into Volterra lattice. Hierarchy of dP I appears to be related with a matrix model of two-dimensional gravity (see, for example, [7] and references therein). In [5] Joshi and Cresswell found out representation of dP I hierarchy with the help of recursion operator. We are in position to exhibit hierarchy of more general equation (18) by considering the cases p = 3, 4 and so on. From (12) one has Substituting this into (16) leads to where α (n) i 's are constants which are supposed to solve the same algebraic equations as in (18). As an example, for p = 3, one has Equation (20) should be complemented by deformation ones which are higher members in Bogoyavlenskii lattice hierarchy.

Continuous limit of stationary version of (18)
Let us show in this subsection that stationary version of the equation (18) T + n−1 s=1−n for any n, is integrable discretization of P I : w ′′ = 6w 2 + t.
One divides the real axis into segments of equal length ε. One considers discrete set of values {t = iε : i ∈ Z}. Values of the function w, respectively, are taken for all such values of the variable t. Therefore one can denote w(t) = w i . Let Substituting (24) in the equation (23) and taking into account the relations of the form and turning then ε to zero we obtain, in continuous limit, the equation which, by suitable rescaling, can be deduced to canonical form of P I . This situation goes in parallel with that when Bogoyavlenskii lattice (19), for all values of n ≥ 2, has continuous limit to Korteweg-de Vries equation [3].

Conclusion
We have considered in the paper edKP hierarchy restricted to p evolution parameters {t (n) 1 , . . ., t (n) p }. It was shown that self-similarity constraint imposed on this system is equivalent to purely discrete equations (15) and (16) supplemented by (p − 1) deformation equations which in fact are evolution equations governing (p − 1) flows of nth edKP subhierarchy. What is crucial in our approach is that we selected quantities α (n,n) which enter these discrete equations and turn out to be independent on evolution parameters due to exactness property for conserved densities.
Discrete systems over finite number of fields arise when one restricts edKP hierarchy on invariant submanifold S n,r,l . In the present paper we considered only one-field discrete equations corresponding to S n,1,1 with n ≥ 2. It is our observation that in this case discrete equation under consideration supplemented by deformation one is equivalent to Veselov-Shabat periodic dressing lattice with odd period. It is known due to Noumi and Yamada that this finite-dimensional system of ordinary differential equations admits finitely generated group of Bäcklund transformations which realizes representation of extended affine Weyl groupW (A (1) 2(n−1) ). It is natural to expect that restriction of edKP hierarchy to other its invariant submanifolds also can yield finitedimensional systems invariant under some finitely generated groups of discrete transformations. We are going to present relevant results on this subject in subsequent publications.