Hierarchies of q-discrete Painlevé equations

Huda Alrashdi; Nalini Joshi; Dinh Thi Tran

doi:10.1080/14029251.2020.1757235

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Volume 27, Issue 3, May 2020, Pages 453 - 477

Hierarchies of q-discrete Painlevé equations

Authors

Huda Alrashdi

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia,halr6311@uni.sydney.edu.au*

Nalini Joshi

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia,nalini.joshi@sydney.edu.au

Dinh Thi Tran

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia,dinhthi.tran@sydney.edu.au

^*

current email: halrashidi@ksu.edu.sa

Received 10 May 2019, Accepted 11 November 2019, Available Online 4 May 2020.

DOI: 10.1080/14029251.2020.1757235 How to use a DOI?
Keywords: q-discrete Painlevé equations; Lax pair; Hierarchies; Bäcklund transformations
Abstract: In this paper, we construct a new hierarchy based on the third q-discrete Painlevé equation (qP_III) and also study the hierarchy of the second q-discrete Painlevé equation (qP_II). Both hierarchies are derived by using reductions of the general lattice modified Korteweg-de Vries equation. Our results include Lax pairs for both hierarchies and these turn out to be higher degree expansions of the non-resonant ones found by Joshi and Nakazono [29] for the second-order cases. We also obtain Bäcklund transformations for these hierarchies. Special q-rational solutions of the hierarchies are constructed and corresponding q-gamma functions that solve the associated linear problems are derived.
Copyright: © 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Following widespread interest in the Painlevé equations, it is natural to ask whether their integrable discrete versions share their fundamental properties. In this paper, we consider and answer the following question about multiplicative or q-discrete Painlevé equations, namely the construction of new associated infinite sequences of discrete equations called hierarchies. The term ‘hierarchy’ here refers to a sequence of q-difference equations sharing a linear problem.

In the literature, discrete hierarchies are known for some additive discrete Painlevé [10,16], and q-discrete Painlevé equations [21,32,45] but not for most of the known discrete Painlevé equations. Our paper extends the class of known hierarchies of q-discrete Painlevé equations by providing a new hierarchy, associated with the q-discrete third Painlevé equation.

Our approach starts with an integrable partial difference equation, also known as a lattice equation, and considers higher-order reductions than those that have been constructed before. (See for example [19, 25, 34, 37–39].) Our starting point is equation (3.1), which is a slightly more general (multi-parameter) form of a standard lattice equation denoted by H3δ=0 in the ABS classification [3, 4]. Further background information about lattice equations is given for the interested reader in §1.1.

In particular, we obtain two q-discrete hierarchies, whose starting points are the q-discrete second and third discrete Painlevé equations, denoted by qP_II and qP_III respectively below. We will refer to the n^thmember of the respective hierarchies by qPII(n), qPIII(n). In fact, we also obtain additional hierarchies, but we focus on qP_II and qP_III here to provide a self-contained exposition. The term hierarchy is used because each sequence shares a linear problem, which is one of a pair of linear problems known as Lax pairs. The corresponding Lax pairs have the form

Φ(qx,t)=A(x,t)Φ(x,t),(1.1a)

𝒯(Φ(x,t))=B(x,t)Φ(x,t),(1.1b)

where 𝒯 is a time-deformation operator, whose action iterates the Painlevé variable t in the resulting hierarchy. The variables x and t are often called spectral and Painlevé variables respectively. The matrix A is polynomial in x and given by equations (1.3), while B is given by equation (1.6) or (1.9). (See our main result Theorem 1.1 in §1.2.)

For Equations (1.1a) and (1.1b) to be compatible, we must have

𝒯(A(x,t))B(x,t)=B(qx,t)A(x,t).(1.2)

For each n ∈ ℕ corresponding to the degree of A in x, we show that equation (1.2) gives rise to either qPII(n) or qPIII(n) (corresponding to the choice of B). Equations (1.8) and (1.11) in Section 2 give the complete form of these hierarchies. In the same way that qP_II is a symmetric case (or projective reduction) of qP_III, we can show that the n^thmember of the qP_II hierarchy can be obtained from the n^thmember of the qP_III hierarchy.

Previous approaches [10, 21] for constructing hierarchies have started by extending the degree of the matrix A in x and using the resulting compatibility conditions to derive new equations. The key idea here relies on fixing one matrix while we change the degree of the other one. However, the calculations are not straightforward, and become more technical in the case of q-discrete equations. Instead, we pursue a simpler approach given by higher-order reductions of systems of lattice equations, as explained in Section 3. In finding the appropriate reductions, we were motivated by the Lax pairs and the actions of the time-deformation operator 𝒯 on parameters given in [29], which allow us to hypothesize the lattice parameters that we use for the reductions.

Given an integer d, a well-known procedure for finding reductions of lattice equations governing a function x(l,m) is to assume a periodic (d,1)-reduction, i.e., impose x_l,m₊₁ = x_l_+d,m or x_l,m₊₁ = 1/x_l_+d,m. Such periodic reductions of the lattice mKdV (modified Korteweg-de Vries) equation, also known as H₃, were obtained for some integer d in [21]. We note that this led to a qP_II hierarchy, which we also find here. However, the calculations are intricate and further assumptions were required to find Lax pairs. In contrast, our approach immediately provides hierarchies as well as their Lax pairs, without intermediate assumptions.

The 2 × 2 Lax pairs we find have certain advantages for analysis. The most important properties are that each matrix A corresponding to a member of the hierarchy is non-singular at x = 0 and ∞ and explicitly factorisable into linear factors in x. The former property enables Birkhoff and Carmichael’s classical theory of non-resonant q-linear difference equations [6, 9] to be applied immediately.

1.1. Background

In this section, we provide some background material about Painlevé and discrete Painlevé equations, and integrable lattice equations.

The Painlevé equations appear widely in physical applications (see for example [46]). They were discovered originally by Painlevé [40], Gambier [14] and Fuchs [13] in the search for new transcendental functions that satisfy ordinary differential equations (ODEs) and their general solutions are known to be higher transcendental functions, called the Painlevé transcendents. Later, they were found as reductions of completely integrable partial differential equations (PDEs), such as the Korteweg-de Vries equation [1, 2]. More recently, integrable difference equations with properties that are very similar to those of the classical Painlevé equations have been identified. They are now known as discrete Painlevé equations and there are three types. We focus here on discrete Painlevé equations of q-discrete or multiplicative type; see Sakai [43].

The search for discrete versions of integrable PDEs has also been a very active area of recent research. Many partial difference equations with properties that are very similar to those of integrable PDEs are known to share a geometric property called consistency around a cube (CAC) or multi-dimensional consistency [35,36]. Classifications of scalar equations with this property [3,4,7] led to a list of scalar partial difference equations. By convention, they are denoted as A-, D-, H- or Q-type. In this paper, we focus on an example of H-type, denoted H₃, which is also known as a lattice mKdV equation.

Many reductions of lattice equations to the q-discrete Painlevé equations are already known [8, 12, 19, 22, 23, 26, 34, 37–39]. For example, the qP_II and qP_III equations were derived from a so-called geometric reduction of H₃ and a special case of an equation labelled D₄ in [7]; see [30].

Motivated by the existence of hierarchies (infinite sequences) of PDEs associated with each integrable PDE, hierarchies of Painlevé equations have also been found [15, 28]. Correspondingly, hierarchies for a restricted set of discrete Painlevé equations have also been obtained [10, 21, 33]. However, to the best of our knowledge, very few q-discrete Painlevé hierarchies have been constructed.

Although generic solutions of discrete Painlevé equations are higher transcendental functions, there also exist special function and rational solutions for special cases of parameters. Each such special-parameter case can be mapped to another through transformations called Bäcklund transformations [24]. Explicit solutions of qP_II were studied and corresponding solutions of its Lax pair found in [31].

Starting with a seed solution corresponding to an initial parameter, one can generate an infinite series of solutions of the same equation with different parameters, provided that the Bäcklund transformation we use does not terminate. Confusingly, the word hierarchy may also be used in this context to refer to an infinite sequence of solutions generated by a Bäcklund transformation when successively applied to a seed solution.

1.2. Main Result

Our main result shows that each member of the hierarchy of qP_II and qP_III shares one spectral linear problem equation (1.1a), with the coefficient matrix given by

A(x,t)=∏j=0nAn−j,n⩾2,(1.3a)

where

Al:=(−iblλhl1−1−iblhlλx).(1.3b)

Here, b_l, l = 0,1,...,n, λand q are the non-zero complex parameters and h_l, l = 0,1,...,n are dependent variables. We find that the product of all h_l turns out to be constant in the case of the qP_II and qP_III hierarchies. We take this constant to be λ ²without loss of generality, to match with the second-order case given in [29], that is,

∏j=0nhj=λ2.(1.4)

We also take

bn=q(1.5)

to match with the base cases given in the same paper [29]. In fact, one can take b_n to be any constant from the construction in Section 3.

We use the following notation. Given 2 ⩽ n ∈ ℕ, the n^thmember of the hierarchy will be described by using a deformation operator referred to as T˜ and T^ for the case associated with qP_II and qP_III respectively. These operators act on a set of parameters and independent variables denoted by b₀, b₁, ... , b_n, λ, q, x and on the dependent variables, which we will denote by h₀, ..., h_n.

The following theorem collects our main results.

Theorem 1.1

Let n ∈ ℕ, n ≥ 2. The compatibility condition (1.2) is satisfied on solutions of the qP_II and qP_III hierarchy respectively when {𝒯, B} is replaced by one of the following choices:

(1) { T˜, B_II} and (2) {T^, B_III}. The qP_II and qP_III hierarchies are given as follows.

(1)

Take
BII=(−ib0λh0x1−1−ib0h0λx),(1.6)
and define T˜ by the action
T˜:(b0,b1,…,bj,…,bn−1,bn,λ,q,x)→(b1,b1,…,bj+1,…,qb0,bn,λ,q,x).(1.7)

Then the resulting hierarchy of equations is given by
{T˜(hr)=hr+1,0⩽r⩽n−2,T˜(hn−1)=λ2(1+b0∏j=1n−1hj)∏j=0n−1hj(b0+∏j=1n−1hj).(1.8)
(2)

Take
BIII=(−ib1λh1x1−1−ib1h1λx)(−ib0λh0x1−1−ib0h0λx),(1.9)
and define T^ by the action
T^:(b0,b1,…,bj,…,bn−2,bn−1,bn,λ,q,x)→(b2,b3,…,bj+2,…,b0q,b1q,bn,λ,q,x),(1.10)

Then the resulting hierarchy of equations is given by
{T^(hr)=hr+2,0⩽r⩽n−3,T^(hn−2)=λ2(1+b0∏j=1n−1hj)∏j=0n−1hj(b0+∏j=1n−1hj),T^(hn−1)=λ2(1+b1(∏j=2n−1hj)T^(hn−2))(∏j=1n−1hj)T^(hn−2)(b1+(∏j=2n−1hj)T^(hn−2)).(1.11)

Remark 1.1

Variables h_l (l = 0,1,...,n) are functions of t on which these operators T˜ and T^ act respectively as follows

{t→pt=q1nt, for equation(1.8) by taking b0=t, b1=q1nt,t→pt=q2nt, for equation(1.11) by taking b0=t, b1=bt,b2=q2nt,b3=q2nt for even n,

where b is a constant for the latter case. For the former case, one can easily see that b1=pt=T˜(t)=T˜(b0). Similarly, for the latter case we have b2=T^(b0), b3=T^(b1). Using these operators, we find that

{bj=qjnt for 0⩽j⩽n−1, for equation (1.8),b2j=q2jnt,b2j+1=bq2jnt, for even n and 0⩽j⩽n/2, for equation (1.11).

Remark 1.2

The results of Theorem 1.1 can be separated into two hierarchies in each case. These are distinguished by whether n is odd or even. The cases corresponding to odd n reduce to one of lower order in each case. At the base level, we get degenerate limits of qP_II and qP_III. See further details for the case n = 3 in Remark 2.1.

Remark 1.3

It is intriguing to note that the Lax pair A(x,t) given in (1.3) has a form that is analogous to the one given by Kajiwara et al [32]. This is because these matrices both come from periodic reductions of lattice equations.^a

Because of the observations in Remarks 1.2 and 2.1, we will assume that n is even in the remainder of the paper. (Nevertheless, all the results are satisfied also for the case when n is odd.)

1.3. Outline of the paper

This paper is organized as follows. In Section 2, we provide examples of the first and second members of each hierarchy. In Section 3, we use periodic reductions of the general mKdV to derive the hierarchies of q-discrete second and third Painlevé equations. Their associated Lax pairs and Bäcklund transformations are obtained automatically from this method, in Sections 3 and 4. In Section 5 we describe the rational solutions of the hierarchies and deduce the corresponding solutions of their associated Lax pairs.

2. Second- and Fourth-order members of the hierarchies

In this section, we consider the cases n = 2, 3, 4 in Theorem 1.1 explicitly. The case n = 2 corresponds to the well-known qP_II and qP_III equations. The case n = 4 is the next member of the hierarchy corresponding to each of these equations. The odd case where n = 3 is deduced here for illustrative purposes, to show that the result is a second-order equation that is a degenerate version of qP_II and qP_III. Thereafter, we will refer to the iteration of the variable h_l (l = 0,...,n) under the deformation operators T˜ and T^ by h˜l and h^l, respectively.

2.1. The case n= 2

In this subsection, we consider the case n = 2 in Theorem 1.1. Recall that equations (1.5) and (1.4) give the constraints

h0h1h2=λ2 and b2=q.

Note that the deformation operators are given as follows

T˜:(b0,b1,b2,λ,q)→(b1,qb0,b2,λ,q),T^:(b0,b1,b2,λ,q)→(qb0,qb1,b2,λ,q).

So the first members of the qP_II and qP_III hierarchies are given by following equations

qPII(2):{h˜0=h1,h˜1=λ2(1+b0h1)h0h1(b0+h1),(2.1)

qPIII(2):{h^0=λ2(1+b0h1)h0h1(b0+h1),h^1=λ2(1+b1h^0)h^0h1(b1+h^0).(2.2)

Equations (2.1) and (2.2) are the second order of the q-discrete second and third Painlevé equations given in [29].

2.2. The case n = 3

Remark 2.1

For the case where n is odd, we can always reduce the order of the associated equations by unity, and that leads to a degenerate version of the hierarchy.

We consider the case n = 3 to illustrate the argument. Recall that equations (1.4) and (1.5) give the constraints

h0h1h2h3=λ2 and b3=q.

Note that the deformation operators are given as follows

T˜:(b0,b1,b2,b3,λ,q)→(b1,b2,qb0,b3,λ,q),T^:(b0,b1,b2,b3,λ,q)→(b2,qb0,qb1,b3,λ,q).

We consider each case listed in Theorem 1.1 separately below.

(1)

We obtain
h˜0=h1,h˜=h2,h˜2=λ2(1+b0h1h2)h0h1h2(b0+h1h2),
which is equivalent to
h˜˜˜2=λ2(1+b0h˜0h˜˜0)h0h˜0h˜˜0(b0+h˜0h˜˜0).(2.3)

Defining h˜0h˜˜0=f, equation (2.3) becomes
f˜f˜=λ2(1+b0f)(b0+f).(2.4)

The resulting equation is a degenerate version of the qP_II equation that was first obtained in [42]; for details see [17, 18, 42].
(2)

We obtain
h^0=h2, h^1=λ2(1+b0h1h2)h0h1h2(b0+h1h2) and h^2=λ2(1+b1h2h^1)h1h2h^1(b1+h2h^1),
which is equivalent to
h^1h1=λ2(1+b0h1h^0)h0h^0(b0+h1h^0) and h^^0h^0=λ2(1+b1h^1h^0)h1h^1(b1+h^1h^0).(2.5)

Defining f = h₀h₁ and g=h^0h1, equation (2.5) becomes
f^f=λ2(1+b0g)(b0+g), g^g=λ2(1+b1f^)(b1+f^).(2.6)

The cases listed above cover all the possibilities for n = 3. These illustrate the assertion made in Remark 2.1. The general case of odd n will be consider in a separate paper.

2.3. The case n = 4

Here we consider the second member (fourth-order) of the qP_II and qP_III hierarchies. Recall that equations (1.4) and (1.5) give the constraints

h0h1h2h3h4=λ2 ang b4=q.

Noting that the deformation operators are given by

T˜:(b0,b1,b2,b3,b4,λ,q)→(b1,b2,b3,qb0,b4,λ,q),T^:(b0,b1,b2,b3,b4,λ,q)→(b2,b3,qb0,qb1,b4,λ,q),

we obtain the second member of the qP_II and qP_III hierarchies as follows

qPII(4):{h˜0=h1,h˜1=h2,h˜2=h3,h˜3=λ2(1+b0h1h2h3)h0h1h2h3(b0+h1h2h3),(2.7)

or, equivalently

h˜˜2h˜˜2=λ2(1+b0h˜2h2h˜2)h˜2h2h˜2(b0+h˜2h2h˜2).(2.8)

Moreover, we have

qPIII(4):{h^0=h2,h^1=h3,h^2=λ2(1+b0h1h2h3)h0h1h2h3(b0+h1h2h3),h^3=λ2(1+b1h2h3h^2)h1h2h3h^2(b1+h2h3h^2),(2.9)

or, equivalently

{h^2h^2=λ2(1+b0h^3h2h3)h^3h2h3(b0+h^3h2h3),h^3h^3=λ2(1+b1h2h3h^2)h2h3h^2(b1+h2h3h^2).(2.10)

3. Proof of Theorem 1.1

In this section we prove Theorem 1.1 by applying a periodic reduction, often called a staircase method, to a lattice equation, which is a multi-parametric version of the discrete modified mKdV equation, given by

Q(w0,w1,w2,w12;αi,βi)=α1w0w2−α2w1w12−β1w0w1+β2w2w12=0,(3.1)

where the arguments w₀,w₁,w₂,w₁₂ are associated with vertices of a quadrilateral. Interpreting these as points on a lattice with directions l and m, we assume w₀ = w_l,m, w₁ = w_l_+1,m, w₂ = w_l,m₊₁, and w₁₂ = w_l_+1,m+1, as shown in Figure 3.1.

Note that α_i and β_i, i = 1,2 are parameters, which may also depend on l and m. We assume that the parameters (α₁,α₂) and (β₁,β₂) correspond to the l and m directions respectively on the lattice and embed this equation in a cube with a third direction associated with parameters (γ₁,γ₂), see Figure 3.1 cf. [47].

It is easy to check that if we put the equation with respective associated parameters on each face of the cube and initial values are given at vertices w₀,w₁,w₂,w₃, we obtain the same value of w₁₂₃ from the three possible ways of computing it [3, 24]. This property is called consistency around a cube, or CAC.

The Lax pair of the lattice mKdV equation is well known [24]. In Appendix A, we provide a derivation of the following Lax pair whose compatibility condition, namely M(w₁,w₁₂,β₁,β₂,x)L(w₀,w₁,α₁,α₂,x) = L(w₂,w₁₂,α₁,α₂,x)M(w₀,w₂,β₁,β₂,x) is the multi-parametric equation (3.1):

{L(w0,w1,α1,α2,x)=(−α1w0w1xl1l2−α2w1w0x),M(w0,w2,β1,β2,x)=(−β1w0w2xl1l2−β2w2w0x),(3.2)

where we consider l₁ and l₂ as constants and x as a spectral variable.

3.1. qP_II hierarchy

In this section, suppose n ⩾ 2 is a given, fixed integer. We will derive the qP_II hierarchy by using the (n,1)-reduction of equation (3.1) i.e., by imposing w_l,m = w_l_+n,m+1 and taking u_i = w_l,m where i = l − nm [37, 41].

Figure 3.2 represents this reduction with associated parameters, which are different on each edge. Consider the last edge on the horizontal line in this figure, which joins u_n to u_n₊₁. We assume that the corresponding parameters are given by (qα_0,1,qα_0,2). This can be seen as a non-autonomous reduction of the general mKdV.

On the quadrilateral on the right of Figure 3.2, we have the equation Q(u_n,u_n₊₁, u₀,u₁; qα_0,1,qα_0,2,β₁,β₂) = 0, which is given by

qα0,1unu0−qα0,2un+1u1−β1unun+1+β2u0u1=0.

We can solve for u_n₊₁ from this equation to find the shift map

S:(u0,u1,…,un−1,un;α0,i,…,αn−1,i) ↦(u1,u2,…,un,un+1;α1,i,…,αn−1,i,qα0,i)(3.3)

where i=1,2 and

un+1=u0(qα0,1un+β2u1)qα0,2u1+β1un.

Let h_j = u_j₊₁/u_j for j = 0,1,2 ,...,n− 1, then we obtain the map

T˜:(h0,h1,…,hn−1,α0,i…,αn−1,i) ↦(h1,h2,…,hn−1,T˜(hn−1),α1,i,…,αn−1,i,qα0,i),(3.4)

where

T˜(hn−1):=h˜n−1=qα0,1h1h2…hn−1+β2h0h1…hn−1(qα0,2+β1h1h2…hn−1).(3.5)

We note that the map (3.4) with (3.5) is a general version of the qP_II hierarchy given in Theorem 1.1.

Now we will construct the Lax pair for the map (3.4) from the reduction described in Figure 3.2. We start by constructing a monodromy matrix associated with the (n,1)-reduction (see [41]). Walking along the vertices labelled by u₀ in Figure 3.2, we have steps taken along horizontal edges, which are represented by L, and one step up in the vertical, which is represented by M, leading to the composition:

ℒ=(M(un,u0,β1,β2,x))L(un−1,un,αn−1,1,αn−1,2,x)L(un−2,un−1,αn−2,1,αn−2,2,x) …L(u1,u2,α1,1,α1,2,x)L(u0,u1,α0,1,α0,2,x).(3.6)

This matrix ℒ is called a monodromy matrix.

Applying the deformation T˜ on ℒ we have

T˜(ℒ)=(M(un+1,u1,β1,β2,x))L(un,un+1,qα0,1,qα0,2x)L(un−1,un,αn−1,1,αn−1,2,x) …L(u2,u3,α2,1,α2,2,x)L(u1,u2,α1,1,α1,2,x).(3.7)

The Lax pair for the quad-equation Q(u_n,u_n₊₁,u₀,u₁,qα_0,1,qα_0,2,β₁,β₂) = 0 is given by

M(un+1,u1,β1,β2,x)L(un,un+1,qα0,1,qα0,2,x) =L(u0,u1,qα0,1,qα0,2,x)M(un,u0,β1,β2,x).

Substitute this in equation (3.7), we get

T˜(ℒ)=(L(u0,u1,qα0,1,qα0,2,x))M(un,u0,β1,β2x)L(un−1,un,αn−1,1,αn−1,2,x) …L(u2,u3,α2,1,α2,2,x)L(u1,u2,α1,1,α1,2,x), =L(u0,u1,qα0,1,qα0,2,x)ℒ(L(un,u0,β1,β2,x))−1, =L(u0,u1,α0,1,α0,2,qx)ℒ(L(u0,u1,α0,1,α0,2,x))−1,

where we have used L(α_i,qx) = L(qα_i,x).

Thus, we get

T˜(ℒ)L(u0,u1,α0,1,α0,2,x)=L(u0,u1,α0,1,α0,2,qx)ℒ.

Letting A = ℒ and B = L(u₀,u₁,α_0,1,α_0,2,x), we obtain a Lax pair of qP_II hierarchy.

Taking the parameters as the following

β1=ibn/λ, β2=ibnλ, αl,1=iblλ, and αl,2=ibl/λ,(3.8)

where l = 0,1 ,...,n − 1, and b_n = q, we obtain the hierarchy of qP_II given by (1.8).

3.2. qP_III hierarchy

First of all, we notice that the deformation operator of qPIIIT^ is a two-fold composition of the deformation operator of qPII(T˜), (i.e. T^=T˜2). Hence, we use the same methods with one extra iteration of the reduction and the corresponding Lax matrices L and M.

We consider the hierarchy of qP_III starting with (n,1)-reduction, and it can be described by the following Figure:

On the quadrilateral on the right of Figure 3.3, we have the following equations

Q(un,un+1,u0,u1;qα0,i,βi)=0,Q(un+1,un+2,u1,u2;qα1,i,βi)=0,

which are given by

qα0,1unu0−qα0,2un+1u1−β1unun+1+β2u0u1=0,(3.9)

qα1,1un+1u1−qα1,2un+2u2−β1un+1un+2+β2u1u2=0.(3.10)

We can solve for u_n₊₁ and u_n₊₂ from these two equations to find the map

(u0,u1,…,un−1,un;α0,i,α1,i,…,αn−1,i)↦(u2,u3,…,un+1,un+2;α2,i,α3,i,…,αn−1,i,qα0,i,qα1,i).

Similar to the hierarchy of qP_II, we define hj=uj+1uj where j = 0,1,2 ,...,n − 1. Taking parameters as (3.8), where b_n = q, then we obtain the map

T^:(h0,h1,…,hn−2,hn−1;b0,b1,…,bn−2,bn−1,bn)↦(h2,h3,…,T^(hn−2),T^(hn−1);b2,b3,…,qb0,qb1,bn),(3.11)

where

T^(hn−2):=h^n−2=λ2(b0∏i=1n−1hi+1)∏i=0n−1hi(b0+∏i=1n−1h1),T^(hn−1):=h^n−1=λ2(b1h^n−2∏i=2n−1h1+1)h^n−2∏i=1n−1hi(b1+h^n−2∏i=2n−1hi).(3.12)

We note that the map (3.11) with (3.12) is the qP_III hierarchy given in Theorem 1.1.

Now as in the qP_II hierarchy, we will construct the Lax pair for the map (3.11) from the reduction described in Figure 3.3. Similarly, the monodromy matrix associated with the (n,1)-reduction is given by

ℒ=M(un,u0,β1,β2,x)L(un−1,un,αn−1,1,αn−1,2,x)L(un−2,un−1,αn−2,1,αn−2,2,x) …L(u1,u2,α1,1,α1,2,x)L(u0,u1,α0,1,α0,2,x).(3.13)

Applying the deformation T^ on ℒ we have

T^(ℒ)=M(un+2,u2,β1,β2,x)L(un+1,un+2,qα1,1,qα1,2x)L(un,un+1,qα0,1,qα0,2,x) …L(u3,u4,α3,1,α3,2,x)L(u2,u3,α2,1,α2,2,x).(3.14)

The Lax pairs for the quad-equations

Q(un,un+1,u0,u1;qα0,1,qα0,2,β1,β2)=0,

and

Q(un+1,un+2,u1,u2;qα1,1,qα1,2,β1,β2)=0,

satisfy

M(un+1,u1,β1,β2,x)L(un,un+1,qα0,1,qα0,2,x) =L(u0,u1,qα0,1,qα0,2,x)M(un,u0,β1,β2,x),(3.15)

and

M(un+2,u2,β1,β2,x)L(un+1,un+2,qα1,1,qα1,2,x) =L(u1,u2,qα1,1,qα1,2,x)M(un+1,u1,β1,β2,x).(3.16)

Using (3.16) and (3.15) in equation (3.14), we get

T˜(ℒ)=L(u1,u2,qα1,1,qα1,2,x)M(un+1,u1,β1,β2x)L(un,un+1,qα0,1,qα0,2,x) …L(u3,u4,α3,1,α3,2,x)L(u2,u3,α2,1,α2,2,x), =L(u1,u2,qα1,1,qα1,2,x)L(u0,u1,qα0,1,qα0,2,x)M(un,u0,β1,β2,x) =L(un−1,u2,qα1,1,qα1,2,x)……L(u2,u3,α2,1,α2,2,x) =L(u1,u2,qα1,1,qα1,2,x)L(u0,u1,qα0,1,qα0,2,x)ℒ(L(u0,u1,α0,1,α0,2,x))−1 (L(u1,u2,α1,1,α1,2,x))−1,

where we have used L(α_i,qx) = L(qα_i,x). Thus, we obtain

T^(ℒ)L(u1,u2,α1,1,α1,2,x)L(u0,u1,α0,1,α0,2,x)=L(u1,u2,α1,1,α1,2,qx)L(u0,u1,α0,1,α0,2,qx)ℒ.

If we take A = ℒ, B = L(u₁,u₂,α_1,1,α_1,2,x)L(u₀,u₁,α_0,1,α_0,1,x), we have the Lax pair of qP_III hierarchy.

Remark 3.1

The entries (1,2) and (2,1) of matrix A_j, which we choose to be 1 and − 1 in the matrices, can be replaced with any constants and the compatibility condition still holds.

4. Bäcklund Transformations of the hierarchies

One of the interpretations of CAC property is a connection with Bäcklund transformations. The CAC property can be regarded as a Bäcklund transformation between a top and a bottom equation in a cube [5]. It can be described as follows.

We consider the following quad-equation which is CAC

Q(u0,u1,u2,u12;α,β)=0.(4.1)

We then embed this equation (4.1) in the third direction associated with variables v and lattice parameters γ, this parameter will be a Bäcklund transformation parameter (see Figure 3.1 where w₃ is replaced with v). A Bäcklund transformation between two equations which are depicted by the top and bottom faces in the cube is given by

Q(u0,u1,v0,v1;α,γ)=0,Q(u0,u2,v0,v2;β,γ)=0.

We note that this is an auto-Bäcklund transformation as the top and bottom equations are the same.

In this section, we use the Bäcklund transformation of the lattice mKdV to derive the Bäcklund transformation for the qP_II and qP_III hierarchies. Moreover, we give a few examples of some rational solutions for second and fourth order.

4.1. Bäcklund transformation of the qP_II hierarchy

To find a Bäcklund transformation for the qP_II hierarchy, we embed the (n,1) periodic reduction in three dimensions with a slight modification. A third direction in the 𝕑 × 𝕑 × 𝕑 lattice is associated with parameters γ₁,γ₂ and variables v_i’s. Instead of imposing the (n,1) periodic reduction for the v’s variables, we use a twist reduction v_l_+n,m+1 = v_l,m/d cf. [8]. This is because we want to create a Bäcklund transformation between two equations with different parameters. Parameters along the staircase of the v variables are as the same as the ones we use for the u variables. For example the (2,1)- reduction in three dimension corresponding to the first member in the qP_II hierarchy can be described in Figure 4.1.

We now illustrate a method of finding a Bäcklund transformation for the qP_II hierarchy by using the base case qP_II which is associated with n = 2.

The twisted reduction for variables v’s gives the top shaded equation which is given by

Q(v2,v3v0/d,v1/d;qα0,1qα0,2,β1,β2)=qα0,1v2v0/d−qα0,2v3v1/d−β1v2v3+β2v0v1/d2=0.(4.2)

This gives us a shift map

Sv:(v0,v1,v2,α0,i,α1,i,βi)↦(v1,v2,v3,α1,i,qα0,i,βi), with i=1,2,(4.3)

where v₃ can be solved from (4.2).

As we discussed above, a Bäcklund transformation can be inherited from CAC. Thus, a Bäcklund transformation between the shaded equations in Figure (4.1) is the following system

Q(u0,u1,v0,v1;α0,1,α0,2,γ1,γ2)=0,Q(u1,u2,v1,v2;α1,1,α1,2,γ1,γ2)=0,Q(u2,u0,v2,v0/d;β1,β2,γ1,γ2)=0.(4.4)

We consider this system as the system of variables v₀,v₁,v₂. By writing v₂ and v₁ in terms of v₀, we obtain a quadratic equation in v₀ which leads to non-rational solutions in general. However, if we take γ₂ = 0 then system (4.4) becomes a linear system and it can be solved uniquely with the following solution

v0=γ1d(α0,2α1,2u0u2+α0,2β1u1u2+α1,1β1u0u1)(dα0,1α1,1β1−α0,2α1,2β2)u0,v1=γ1(dα0,1α1,2u0u2+α0,1β1u1u2+α1,2β1u0u1)(dα0,1α1,1β1−α0,2α1,2β2)u1,v2=γ1(dα0,1α1,1u0u2+α0,2β2u1u2+α1,1β2u0u1)u2(dα0,1α1,1β1−α0,2α1,2β2).

This defines the Bäcklund transformation BT : (u₀,u₁,u₂) ↦ (v₀,v₁,v₂). We note that the Bäcklund transformation should be compatible with the shift map S_v and S, i.e. we have

Sv∘BT=BT∘S.

This implies d = q. Similar to the qP_II equation, we introduce gi=vi+1vi for i = 0,1. Using the parameters given in (3.8), we obtain the equation

g˜1g˜1=λ2(b0qg1+1)qg1(b0+qg1), where b0=t.(4.5)

This suggests that we introduce H_i = qg_i for i = 0, 1, 2, in which case equation (4.5) can be written as

H˜1H˜1=q2λ2(tH1+1)H1(t+H1).

Therefore, H₁ satisfies qP_II, with parameter λ ²q²instead of λ ². Hence, the transformation from h_i to H_i, where h_i = u_i₊₁/u_i, and i = 0,1,2, defines the Bäcklund transformation for qP_II. Thus, we can write the Bäcklund transformation of qPII(2) as

H1=qg1=qv2v1=q1/2(q1/2tλ2h1+q1/2λ2+h1h˜1)h1(q1/2h1h˜1+th1+1), by taking b1=q12t,(4.6)

where H₁ = H(t),h₁ = h₁(t) and h˜1=h1(q1/2t). This is equivalent to the known Bäcklund transformation for qP_II [27].

We can see that this transformation produces a solution, H of qP_II with parameter λ²q²from a solution, h corresponding to λ².

Proposition 4.1

The simplest rational solution (seed solution) of the qP_II and qP_III hierarchies, equations (1.8) and (1.11), respectively is

hl=±1, l=0,1,…,n, and λ=±1.(4.7)

This can be checked by substituting the above values into (1.8) and (1.11), respectively. Applying the Bäcklund transformation (equation (4.6)) repeatedly on the seed solution will give us an infinite number of solutions of equation (2.1) for λ²↦q²λ². In the next few examples we will use the notation h(t;λ²) to refer to a solution of equation corresponding to λ².

Example 4.1

Let us look at few examples of rational solutions of qPII(2). Starting with λ ²= 1, we have

h1(t;1)=1,h1(t;p4)=pP0(pt)P0(t),h1(t;p8)=p4P0(t)P1(t)P0(pt)P2(t),h1(t;p12)=p5P0(pt)P2(t)(P0(p2t)P2(t)P2(pt)+p5tP0(pt)P1(pt)P2(t)+p9P0(t)P1(t)P1(pt))P0(t)P1(t)(p9P0(p2t)P2(t)P2(pt)+p4tP0(pt)P1(pt)P2(t)+p8P0(t)P1(t)P1(pt)),

where p²= q and

P0(t)=1+p+t,P1(t)=1+p3P0(p2t)P0(t)+p2tP0(p2t)P0(pt),P2(t)=p5+p2P0(p2t)P0(t)+ptP0(p2t)P0(pt).

We note that we can start with another seed solution h₁ = − 1 so, we have another infinite solution for qPII(2) which was already generated in [44].

Following the same method above, a Bäcklund transformation for the n^thmember of the qP_II hierarchy (Figure 3.2) is given by the system below

Q(u0,u1,v0,v1;α0,1,α0,2,γ1,γ2)=0, ⋮Q(un−1,un,vn−1,vn;αn−1,1,αn−1,2,γ1,γ2)=0,Q(un,u0,vn,v0/d,β1,β2,γ1,γ2)=0.

Taking γ₂ = 0, we obtain a linear system of v₀,v₁,...,v_n. We also want that this Bäcklund transformation is compatible with the shift map; thus d = q.

Let g_i = v_i₊₁/v_i, and let the parameters be as given in (3.8). Then, we obtain a Bäcklund transformation of equation (1.8), which is

gihi=N/D,(4.8)

where

N=1qλ2n−2−1∑j=0n−1hj∏s=0j−1hs2bjλ2j−2i−1+∑j=i+1n−1hj∏s=0j−1hs2bjλ2j−2i−1+λ2i+1∏s=0n−1hsqλ2n−2−1,D=1qλ2n−2−1∑j=0n−1hj∏s=0j−1hs2bjλ2j−2i−1+∑j=in−1hj∏s=0j−1hs2bjλ2j−2i+1+λ2i−1∏s=0n−1hsqλ2n−2−1,

and the following equation

gn=λ2(qb0g1g2…gn−1+1)qg0g1…gn−1(qg1g2…gn−1+b0).(4.9)

To obtain the form of qPII(n), we take Hi=q1n−1gi to cancel out each q present in the above equation. Thus, we obtain a Bäcklund transformation h_i ↦ H_i of (1.8) where λ2↦q2n−1λ2.

Proposition 4.2

Let h_i be a solution of qPII(n) (equation (1.8)) with parameter λ². Then

Hi=q1n−1gi, where gi is given by equation (4.8), (i=0,1,2,…,n)

is also a solution of qPII(n), with parameter q q2n−1λ2.

Corollary 4.1

The Bäcklund transformation of qPII(4) takes h_i ↦ H_i = q^1/3g_i, where i = 0,1,2,3,4, λ²↦ λ²q^2/3and

H2=q1/2(q1/2λ4+q1/4λ2h˜˜2h˜2+h2h˜22h˜˜2+q1/2λ4th˜2h2h˜2+q3/4λ4h˜˜2h˜22h22h˜2)h2(q1/4λ2+h˜˜2h˜2+q3/4λ4h˜˜2h˜22h2+q1/4λ2th˜2h2h˜2+q1/2λ2h˜˜2h˜22h22h˜2),(4.10)

where H₂ = H₂(t),h₂ = h₂(t) and h˜2=h2(q1/4t).

Example 4.2

We generate the first three rational solutions of the qPII(4). Starting with λ²= 1, we have

h2(t;1)=1,h2(t;p83)=p1/3P(p2t)P(pt),h2(t;p163)=p2/3P(pt)(p3Q(t)+Q(pt)+p9Q(p2t)+p5tP(p4t)P(t)+p6Q(p3t))P(p2t)(Q(t)+p9Q(pt)+p6Q(p2t)+p2tP(p4t)P(t)+p3Q(p3t)),

where p⁴= q and

P(t)=1+p+p2+p3+t,Q(t)=P(t)P(pt).

4.2. Bäcklund transformation of the qP_III hierarchy

We have the qP_III hierarchy given by equation (1.11). We can apply the method given in Subsection 4.1 to the qP_III hierarchy because Figure 3.3 is the same as Figure 3.2 with the square on the right extended one step. Hence, we can deduce the Bäcklund transformation of the qP_III hierarchy.

Proposition 4.3

Let h_i be a solution of qPII(n) (equation (1.11)) with parameter λ². Then

Hi=q1n−1gi, (i=0,1,2,…,n),

is also a solution of qPIII(n), with parameter q2n−1λ2 where

gi={1h0N0D0, i=0,1h1N1D1, i=1,…,n−1,1hnN2D2, i=n,(4.11)

and

N0=q(qλ2n−2−1)(∑j=1n−1hj∏s=0j−1hs2bjλ2j−2i+1+h0∏s=0nhsqλ+hn∏s=0n−1hs2qb0λ),D0=q(qλ2n−2−1)(∑j=1n−1hj∏s=0j−1hs2bjλ2j−2n+3+h0∏s=0nhsqλ3+hn∏s=0n−1hs2qb0λ3)+h0b0λ,N1=1qλ2n−2−1∑j=1n−1hj∏s=0j−1hs2bjλ2j−2i−1+∑j=i+1n−1hj∏s=0j−1hs2bjλ2j−2i−1+hn∏s=0n−1hs2b0λ1−2i+h0∏s=0n−1hsλ1−2i,D1=1qλ2n−2−1∑j=1n−1hj∏s=0j−1hs2bjλ2j−2i+1+∑j=in−1hj∏s=0j−1hs2bjλ2j−2i+1+hn∏s=0n−1hs2b0λ3−2i+h0∏s=0n−1hsλ3−2i,N2=1(qλ2n−2−1)(∑j=1n−1hj∏s=0j−1hs2bjλ2j−2i−1+λ2n−1h0∏s=0nhs+λhn∏s=0n−1hs2qb0),D2=1qλ2n−2−1∑j=1n−1hj∏s=0j−1hs2bjλ2j−2i+1+hn∏s=0n−1hs2b0λ3−2n+h0∏s=0n−1hsλ3−2n.

This proposition provides us with the following Bäcklund transformation for qPIII(n)

h1↦Hi, and λ2↦q2n−1λ2.(4.12)

Corollary 4.2

The Bäcklund transformation of qPIII(2) is given by

H0=λ2p(bth1+b+q+h0h1)h0(bth1+λ2bq+qh0h1),H1=q(λ2bth1+λ2b+h0h1)h1(bth1+b+qh0h1),(4.13)

where H₀ = H₀(t),H₁ = H₁(t),h₀ = h₀(t) and h₁ = h₁(t). This system of two equations provide us a solution of qP_III with λ² ↦ q²λ ².

Example 4.3

We consider solutions of qPIII(2) which are given when λ² ↦ q²λ ². Starting with λ²= 1, we have

h0(t;1) and h1(t;1)=1,h0(t;q2)=qR0(t)R1(t) and h1(t;q2)=q(1+b+bt)R0(t),h0(t;q4)=q2R1(t)(bt+R2(t)+qR3(t))R0(t)(bq3t+R2(t)+qR3(t)) and h1(t;q4)=R0(t)(bq2t+q2R2(t)+R3(t))(1+b+bt)(bt+R2(t)+qR3(t)),

where

R0(t)=bt+b+q and R1(t)=bq+bt+q,R2(t)=bqt2(1+b+bt)R0(t) and R3(t)q2t(1+b+bt)R1(t).

In the next Corollary we give the explicit form of Bäcklund transformation for the fourth-order of the qP_III equation.

Corollary 4.3

The Bäcklund transformation of qPIII(4) is given by

H2=λ4bq1/2th2h3h^3+λ4bq1/2+λ4qh22h3h^2h^32+λ2q1/2h^2h^3+bh2h^2h^32q1/6h2(λ4bq1/2h2h^2h^32+λ2bth2h3h^3+λ2b+λ2q1/2h22h3h^2h^32+h^2h^3),H3=q1/3(λ6bq1/2th2h3h^3+λ6bq1/2+λ4q1/2h^2h^3+λ2bh2h^2h^32+h22h3h^2h^32)h3(λ4bq1/2th2h3h^3+λ2bq1/2+λ4qh22h3h^2h^32+λ2q1/2h^2h^3+hb2h^2h^32),

where H₂ = H₂(t),H₃ = H₃(t),h₂ = h₂(t),h₃ = h₃(t) and h^2=h2(t/p), h^3=h3(t/p) with p²= q. This system of two equations give us solutions H₂ and H₃ from solutions h₂ and h₃ corresponding to λ²↦p^4/3λ².

Example 4.4

Let us give a few solutions of qPIII(4).For l = 0,1,2, we have λ ²= 1, p^4/3, p^8/3, and

h2(t;1)=h3(t;1)=1,h2(t;p43)=F0(pt)p13F1(t) and h3(t;p43)=p23F1(pt)F0(pt),h2(t;p83)=p13F1(t)(p2Q0(t)+bp4tF1(pt)F1(t/p)+Q1(t))F0(pt)(Q0(t)+bp2tF1(pt)F1(t/p)+p3Q0(pt)), andh3(t;p83)=p13F0(pt)(p3Q0(t)+bp5tF1(pt)F1(t/p)+Q0(pt))F1(pt)(p2Q0(t)+bp4tF1(pt)F1(t/p)+Q1(t)),

where

F0(t)=bp+bt+b+p2+p,F1(t)=1+b+p+bp+bt,Q0(t)=F0(t)(bpF1(t/p)+F1(t)),Q1(t)=F0(t)(bF1(t)+p5F1(pt)).

5. Exact solution of the Lax pairs for q-discrete second and third Painlevé equations

Because the general solutions of discrete Painlevé equations are new transcendental functions, the corresponding solutions of their associated linear problems are highly nontrivial. In this section, we consider special q-rational solutions of the qP_II and qP_III equations that exist for special values of parameters and deduce the solutions of their respective linear problems. The results we obtain are similar to those in [20, 31], where rational solutions of qP_II and qP_III and their associated linear problem were studied. In the latter paper, the monodromy matrix A was shown to be a product of diagonal matrices and we show that this is also true for the qP_II hierarchy as well as the hierarchies of qP_III. The main results of this section are stated in Propositions 5.1 and 5.2.

We use the notation Γ_q(1 − z) to denote the product

Γq(1−z)=1(z;q)∞=∏k=0∞1(1−qkz),(5.1)

to be consistent with the terminology used in [31]. Note that this convention differs from the definition of Γ_q(z) in [11, Chapter 5].

The qP_II hierarchy equation (1.8) is solved by (4.7), and this allows its linear system to be diagonalized by a constant matrix. Hence, we can solve it in terms of q-Gamma functions.

Proposition 5.1

Suppose h_i are solutions of the qP_II hierarchy given by Proposition 4.1. When B and 𝒯 are as given in (1.6) and (1.7), respectively, there exists a solution Φ(x,t) of Lax pair (1.1a,1.1b) given by

Φ(x,t)=(i−i11)(c0(−i)μ(n+1)+μ¯∏j=0nΓq(1−bjx)c1iμ(n+1)+μ¯∏j=0nΓq(1−bjx)),

where μ=lnxlnq, μ¯=nlntlnq,b₀ = t, bl=qlnt(l=1,…,n−1), b_n = q and c₀ and c₁ are constants.

Proof

Substituting the special solution (4.7) (take λ = − 1) into the Lax pair (1.1a) and (1.1b), where A is given in (1.3), and B is given in (1.6) gives

Φ(qx,t)=AΦ(x,t),(5.2)

𝒯(Φ(x,t)=BΦ(x,t),(5.3)

where

A=∏j=0n(ibn−jx1−1ibn−jx), B=(ib0x1−1ib0x).

We denote the solution matrix of (5.2) and (5.3) as Φ(x,t)=(ϕ1ϕ2).

We now diagonalise both A and B with constant matrix P=(i−i11). By taking Φ(x,t) =PΨ(x,t) we obtain

Ψ(qx,t)=P−1APΨ(x,t).(5.4)

where

P−1AP=((−i)n+1∏l=0n(1−blx)00in+1∏l=0n(1+blx)).

Let (u(x,t)v(x,t)) be the matrix solution of equation (5.4), i.e (ϕ1ϕ2)=P(C0(t)u(x,t)C1(t)v(x,t)) where C₀(t) and C₁(t) are constants, and

u(qx,t)=(−i)n+1∏l=0n(1−blx)u(x,t),(5.5a)

v(qx,t)=in+1∏l=0n(1+blx)v(x,t).(5.5b)

Equation (5.5a) is solved by writing

u(x,t)=(−i)μ(n+1)∏l=0nu1(x,t).(5.6)

where μ=nlnxlnq and

ul(qx,t)=(1−blx)ul(x,t), for l=0,1,…,n.(5.7)

These equations (5.7) can be solved in terms of q-Gamma function; therefore we get

u(x,t)=(−i)μ(n+1)∏l=0nΓq(1−blx).(5.8)

Similarly, equation (5.5b) has a solution

v(x,t)=(i)μ(n+1)∏l=0nΓq(1+blx).(5.9)

Therefore

(ϕ1ϕ2)=P(C0(t)(−i)μ(n+1)∏j=0nΓq(1−bjx)C1(t)iμ(n+1)∏j=0nΓq(1+bjx)).

To find C₀(t) and C₁(t), we use the deformation problem

𝒯(Ψ(x,t)=P−1BPΨ(x,t),(5.10)

where

P−1BP=(−i(1−b0x)00i(1+b0x)).

This implies

T˜(u(x,t))=−i(1−b0x)u(x,t),T˜(v(x,t))=i(1+b0x)v(x,t).

Using (5.8) and (5.9), we obtain C0(t)=c0(−i)nlntlnq and C1(t)=c1inlntlnq, where c₀ and c₁ are constants.

Therefore, we get

(ϕ1ϕ2)=P(c0(−i)μ(n+1)+μ¯∏j=0nΓq(1−bjx)c1iμ(n+1)+μ¯∏j=0nΓq(1+bjx)),

where μ=lnxlnq and μ¯=nlnxlnq.

Remark 5.1

The simplest solution of the qP_II hierarchy given by Proposition 4.1 also happens to be a solution of the qP_III hierarchy under the condition b = 1. Since the qP_II and qP_III hierarchies share the same spectral linear problem, it follows that the solution of the linear problem given in Proposition 5.1 is also a solution of the corresponding linear problem for the qP_III hierarchy. The difference is only on the values of parameters b_i (i = 0,1,...,n), in the case of qP_III the values take: b₀ = b₁ = t, bl+1=q1nt(l=2,3,6,…,n−2), and b_n = q.

Proposition 5.2

When B and 𝒯 were given in (1.9) and (1.10), respectively, and the qP_III hierarchy equation (1.11) is solved by (4.7), there exists a solution Φ(x,t) of Lax pair (1.1a, 1.1b) given by

Φ(x,t)=(i−i11)(c0(−1)μ¯(−i)μ(n+1)Γq(1−bnx)∏j=0n(Γq(1−bjx))2c1(−1)μ¯iμ(n+1)Γq(1−bnx)∏j=0n(Γq(1+bjx))2)

where μ=lnxlnq, μ¯=nlnx2lnq and c₀ and c₁ are constants.

6. Conclusion

In this paper, we have presented two hierarchies of Painlevé equations, one of them is a qP_II hierarchy which is found in [21] and another is a qP_III hierarchy which is new. Each of these hierarchies was obtained by reduction of the multi-parametric lattice mKdV equation, by using the staircase method. The explicit forms of these hierarchies are given in equations (1.8) and (1.11), with second members of each hierarchy provided by equations (2.8) and (2.10).

In addition to explicit construction of these hierarchies, we provided some properties which are deduced for the first time. One of these is a method to construct Bäcklund transformations for every member of the qP_II and qP_III hierarchies. We found so-called seed solutions for each member of these hierarchies. We then used these transformations on seed solutions to find rational solutions of second-order and fourth-order members of the qP_II and qP_III hierarchies. From the seed solution for the hierarchies, we also deduced the corresponding solutions of their Lax pair; see Propositions 5.1 and 5.2.

It is noteworthy that the spectral problem in each Lax pair in each hierarchy involves a 2 × 2 coefficient matrix, A from equation (1.1a), which satisfies the conditions of non-resonance in Birkhoff’s theory of linear q-difference equations. To our knowledge, this is the first time such linear problems have been constructed for q-Painlevé hierarchies.

There still remain open questions. In the PDE setting, members of a hierarchy are related by recursion operators. However, such operators are not known in the difference equation setting. We have also not touched upon continuum limits, although there is reason to believe that the hierarchies we have provided have well known Painlevé hierarchies as continuum limits.

Finally, we note that the construction methods in this paper also lead to other hierarchies. These will be the subject of future publications.

A. Derivation of the Lax pair (3.2)

Here we provide how we derive the Lax pair (3.2). Using the CAC property, we obtain a Lax pair of equation (3.1)

L1(w0,w1,α1,α2)=(−α1w0w1k1w0k2w1−α2) and M1(w0,w2,β1,β2)=(−β1w0w2k1w0k2w1−β2),(A.1)

where k₁ and k₂ are spectral parameters. Using the gauge matrix

𝒢(z)=(1001z),

we have a new-looking matrix

L(w0,w1,α1,α2)=𝒢−1(w1)L(w0,w1,α1,α2)𝒢(w0)(−α1w0w1k1k2−α2w1w0),(A.2)

M(w0,w2,β1,β2)=𝒢−1(w2)M1(w0,w2,β1,β2)𝒢(w0)(−β1w0w2k1k2−β2w2w0).(A.3)

Now we replace k₁ and k₂ in (A.2) and (A.3) with l1x and l2x, respectively. For simplicity, we multiply by x, which leads to the desired results in equations 3.2.

Acknowledgment

This research is supported by Australian Research Council (ARC) grant FL120100094 and by a grant from the “Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.

Footnotes

a

We would like to thank the referee for this observation.

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Journal of Nonlinear Mathematical Physics

Hierarchies of q-discrete Painlevé equations

1. Introduction

1.1. Background

1.2. Main Result

Theorem 1.1

Remark 1.1

Remark 1.2

Remark 1.3

1.3. Outline of the paper

2. Second- and Fourth-order members of the hierarchies

2.1. The case n= 2

2.2. The case n = 3

Remark 2.1

2.3. The case n = 4

3. Proof of Theorem 1.1

3.1. qPII hierarchy

3.2. qPIII hierarchy

Remark 3.1

4. Bäcklund Transformations of the hierarchies

4.1. Bäcklund transformation of the qPII hierarchy

Proposition 4.1

Example 4.1

Proposition 4.2

Corollary 4.1

Example 4.2

4.2. Bäcklund transformation of the qPIII hierarchy

Proposition 4.3

Corollary 4.2

Example 4.3

Corollary 4.3

Example 4.4

5. Exact solution of the Lax pairs for q-discrete second and third Painlevé equations

Proposition 5.1

Proof

Remark 5.1

Proposition 5.2

6. Conclusion

A. Derivation of the Lax pair (3.2)

Acknowledgment

Footnotes

References

Cite this article

3.1. qP_II hierarchy

3.2. qP_III hierarchy

4.1. Bäcklund transformation of the qP_II hierarchy

4.2. Bäcklund transformation of the qP_III hierarchy