Interrelations of discrete Painlevé equations through limiting procedures
- 10.1080/14029251.2020.1683981How to use a DOI?
- discrete Painlevé equations; affine Weyl groups; limiting procedures; canonical forms
We study the discrete Painlevé equations associated to the affine Weyl group which can be obtained by the implementation of a special limits of -associated equations. This study is motivated by the existence of two -associated discrete both having a double ternary dependence in their coefficients and which have not been related before. We show here that two equations correspond to two different limits of a -associated discrete Painlevé equation. Applying the same limiting procedures to other -associated equations we obtained several -related equations most of which have not been previously derived.
- © 2020 The Authors. Published by Atlantis and Taylor & Francis
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Integrable systems are reputed for the numerous interrelations which establish links among them. Miura and/or Bäcklund transformations are omnipresent in the integrability domain. Once a relation of this type is established between two equations, they cease to be two different ones but are, in fact, the same system under two different guises. Thus establishing relations between known integrable systems is of paramount importance. Painlevé equations, be they continuous or discrete, possess the same rich structure of relations.
This paper focuses on a particular class of discrete Painlevé equations , multiplicative ones which are associated to the affine Weyl group  . The main bulk of discrete Painlevé equations with symmetry are obtained by deautonomisation of a QRT mapping  belonging to one of three families characterised by distinct canonical forms  of the A1 QRT matrix. The first two and best known families, referred to as VI and VI′ respectively, correspond to the matrices
(The third family, recently discovered , will not be the object of the present study.) The canonical form of the mappings of the VI and VI′ families are
2. A short digression
Since in this paper we will be heavily using the QRT formalism  for the writing of mappings in symmetric and asymmetric form it is useful to summarise here the conventions that govern it and their consequences for discrete Painlevé equations.
The starting point is a QRT-symmetric mapping relating the variable w over three adjacent points, i.e. wn+1 = F(wn, wn−1). Typically a discrete Painlevé equation involves a variable introducing the secular dependence for which we can simply assume the form sn = αn + β. Moreover a discrete Painlevé equation equation can have parameters with a periodic dependence of the independent variable. To this end we have introduced two useful periodic functions ϕk and χ2k obeying the periodicity relations ϕk(n) = ϕk(n + k) (but where the constant solution of this relation is excluded) and χ2k(n) = −χ2k(n + k). Practically this gives the following forms for ϕk and χ2k:
When dealing with multiplicative equations the expressions above must be understood as defining the logarithms of the periodic functions and thus the ϕ and χ we shall encounter in section 2, 3 and 4 are the exponentials of the right-hand sides of the above expressions.
Starting from a QRT-symmetric mapping one constructs an asymmetric one by separating the evolution for even and for odd values of the indices. One has thus two mappings one giving w2n+1 as a function of w2n and w2n−1 and one giving w2n+2 as a function of w2n+1 and w2n. The QRT convention is to introduce two distinct dependent variables corresponding to indices of different parity. One has thus xm = w2n and ym = w2n+1. Similarly for the secular dependence one introduces zm = s2n and ζm = s2n+1. Given this notation one remarks that ζm = zm+1/2.
Let us illustrate this by producing the asymmetric form of the symmetric mapping
Using the conventions explained in the previous paragraph we find for n = 2m, n = 2m + 1 respectively
Up to this point the transcription is straightforward. However in the case of discrete Painlevé equations one must also be able to take care of the periodic functions. The periodic function ϕ2(n) is simply equal to γ(−1)n. Thus whenever the equation involves this function the latter will enter as γ in the even-index equation (2.2a) in the example above) and as −γ in the odd-index one (2.2b) in the example). The periodic function χ2(n) is identical to ϕ2(n). For the function ϕ3 we consider the four points ϕ3(2n − 1), ϕ3(2n), ϕ3(2n + 1), and ϕ3(2n + 2). Using the property ϕ3(n) = ϕ3(n + 3) we can rewrite these four quantities as ϕ3(2n − 4), ϕ3(2n), ϕ3(2n − 2), and ϕ3(2n + 2) and the transcription becomes now easy: φ3(m − 2), φ3(m), φ3(m − 1), and φ3(m + 1). In fact, the periodicity property can be used for all ϕs of odd periods. While ϕk(2n) and ϕk(2n + 2) are simply ϕk(m) and ϕk(m + 1), when k = 2p + 1 we have ϕk(2n + 1) = ϕk(2n + 1 − k) = ϕk(2n − 2p) which can be written as φk(m − p). The case of even k is more complicated. We start with the remark that the ϕk for even k can be decomposed in terms of a ϕ of lower index and a χ based on the identity ϕ2p(n) = ϕp(n) + χ2p(n). Since we already discussed the case of ϕ2 it suffices to study the transcription of the first few χs. The function χ4(n) is represented in the QRT-asymmetric form by two different functions χ2(m) and for the even and odd-index equation respectively. A short analysis shows that χ6(n) is transcribed as ϕ3(m) + c in the even-index equation and ϕ3(m − 1) − c in the odd-index one. The function χ8(n) becomes two different functions χ4(m) and for the even- and odd-index equation respectively. The pattern becomes now clear. For χ2(2p+1) we must introduce ϕ2p+1(m) + c in the even-index equation and ϕ2p+1(m − p) − c in the odd-index one. The χ4p on the other hand become two different functions χ2p and .
Using the procedure described above one can transcribe any QRT-symmetric discrete Painlevé equation into a QRT-asymmetric one in a straightforward way.
3. The case of the two -associated equations
The case that motivated the present study is that of two discrete Painlevé equations which have a double ternary periodicity, i.e. one involving two different periodic functions ϕ3(n) and . This is a feature appearing in equations associated to and which does not exist for any of the “lower” groups. The first equation was obtained in  under the form
Here sn has a secular dependence of the form logsn = αn + β and a periodic dependence involving a period-3 function ϕ3(n). The functions dn and gn have the same ternary periodicity involving a function as well as an even-odd dependence, which is also present in fn, hn. Neglecting the period-3 dependence we have simply dnfn = 1 and hngn = 1 and moreover gn+1 = dn.
Before addressing the question of whether (3.1) and (3.2) are related in full generality by the simple transformation presented in the previous paragraph, we can simply ask here whether it is possible to transform (3.1) into (3.2), and vice versa, in the case where the ternary dependence is neglected and we keep just the secular one. We start from (3.2) and write it in asymmetric, in the QRT sense, form as
A simple glance at the right-hand sides of (3.5) suffices to convince oneself that the terms do not match those of (3.4). Thus even when one considers just the secular dependence one cannot relate the two form-VI and form-VI′ equations.
Does the result above mean that there is no relation whatsoever between equations (3.1) and (3.2)? As we shall show in the next section this is not the case: the two equations have a common ancestor, namely an equation with symmetry from which they can be obtained through two different limiting procedures.
4. From to through limiting procedures
Our starting point will be the multiplicative -associated equation which has the form
Taking the limit x → ∞ while keeping z finite leads to the equation
In practical calculations, using the explicit form of R(x) may turn out to be cumbersome and it is preferable in this case to resort to the ancillary representation we have introduced in . As we showed there, we can introduce the ancillary dependent variable ξ by
The P(ξ) and Q(ξ) are given by
Before proceeding to the calculations aiming at relating (3.1) and (3.2) we introduce another auxiliary variable qn = q0λn which will be used in order to represent the secular dependence of the various parameters we are going to work with.
In our exhaustive investigation of symmetric -associated equations we have, among others, derived a single equation, 4.3.1 in , which has a double ternary dependence, just like equations (3.1) and (3.2). The equation was obtained in additive form but its transcription to multiplicative form is straightforward. In the latter case its parameters are:
Thus is exactly the right-hand side of (3.2) (calling x what is called there w) where sn = qnϕ3(n) (as in equation 4.3.1), , , fn = ϕ3(n)/ϕ2(n) and hn = ϕ2(n). Thus by taking the limit xn → ∞ together with b and c we obtained precisely equation (3.2). However another possibility of limit does exist. We can balance the limit xn → ∞ by taking the γ of the ϕ2(n) to infinity. In that case it is more convenient to split the equation according to the parity of the indices and write it in QRT-asymmetric form.
Taking the limit γ → ∞ we obtain for the even-index equation. In order to use the QRT notations we introduce sm = z2m and ζm = z2m+1. Since zn = qnϕ3(n) we can define φ3(m) = ϕ3(2n) and rewrite sm = q2mφ3(m) which allows us to write ζm = q2m+1ϕ3(2m + 1) = q2m+1ϕ3(2m − 2) = q2m+1φ3 = (m − 1). The right-hand side is
Thus, by taking two different limits of the same -associated equation we obtained the two -related equations with double ternary dependence.
5. More limits
Having introduced the two possible ways of taking limits in -associated equations in order to reduce them to -associated ones, one can ask the question whether the case of the two equations analysed in the previous section is unique or more similar cases do exist. It turns out that the latter is indeed the case.
We start from the equations derived in  and identified the equations where at least one constant and one ϕ2(n) were present in their parameters. Four such equations are obtained, the ones labeled 4.2.1, 4.2.5, 5.2.1 and 5.2.5 in . In what follows we shall present their different limits. (As pointed out in the previous section, the results of  concerned additive equations but their transcription to multiplicative ones is straightforward).
Equation 4.2.1 The parameters of this equation are(5.1)
Note that two different period-2 functions appear here ϕ2(n) = γ(−1)n and . In order to balance the limit xn → ∞ we must take the γ of the ϕ2(n) to infinity as well as either of the c or the δ of . It turns out that the two latter choices are perfectly equivalent leading to the same final equation.
Once the equation is VI′ form is obtained it is interesting to convert it into a form VI, using the transformations we outlined in the Introduction. The end result is a QRT-symmetric equation of the form(5.2)where φ2(n) must be understood as the limit of when both γ and δ go to infinity. This is an -associated equation which, at least to the authors knowledge, has not been previously derived.
Equation 4.2.5 Before giving the parameters of this equation it is convenient to introduce the variable Zn, a variable simply related to zn by Zn = znzn+1. We can now write the parameters as(5.3)
We remark that, here, it is possible to balance the limit xn → ∞ either by taking c to infinity or by taking the γ of the ϕ2(n) to infinity. Taking c to infinity results to an equation in a QRT-symmetric equation of form VI′. Its precise expression is(5.4)
When we take γ to infinity we obtain a QRT-asymmetric equation in VI′ form. However it is possible to cast it is form VI in which case the equation becomes QRT-symmetric. We are not going to give the intermediate calculations (which are straightforward but rather tedious) and present directly the form of the resulting equation:(5.5)
The various quantities appearing in (5.5) are given by the expressions: logZnalphan + β − 2γ(−1)n + ϕ3(n + 1) + χ6(n) + χ6(n − 1), logAn = αn + β + (−1)n(γn + δ) + ϕ3(n) + χ6(n) and Bn = ZnZn+4/An+5, Cn = Zn+2/An+3, Dn = An+2/Zn+1. (Note that in these expressions we have used parameters and functions without making any attempt to link them to those of the initial equation, lest the result become inextricably complicated.)
Both equations (5.4) and (5.5) are new, derived here for the first time (with, the usual caveat, that the novelty statement concerns the present authors and their knowledge of discrete Painlevé equations).
Equation 5.2.1 Here we have six with expressions(5.6)
Clearly there are four different ways one can balance the limit xn → ∞: taking all three a, b, c to infinity or the γ of ϕ2(n) and b, c or the δ of and a or both γ and δ to infinity. All four limits lead to the same equation which is given in a QRT-symmetric VI′ form in the case (a, b, c) → ∞ or can be converted to such a form through a simple gauge choice. Without entering into further details we present the form obtained in the straightforward case when all three a, b, c go to infinity. We find(5.7)where b, c must be understood as the ratios b/a, c/a when all three parameters go to infinity. Equation (5.7) is a discrete Painlevé equation already derived in .
Equation 5.2.5 Here again we have six . Their expressions are(5.8)
We remark that the limit xn → ∞ can be balanced either by taking both a, b to infinity or by taking to infinity the γ of ϕ2(n). In the first case we obtain a QRT-symmetric discrete Painlevé equation in form VI′:(5.9)where Φ(n) = ϕ5(n)ϕ5(n + 1) 2ϕ5(n + 2) 2. Again, this is an equation which does not appear among those previously derived. Next we turn to the limit obtained when we take the γ of ϕ2(n) to infinity. In this case the equation obtained has a QRT-asymmetric form. However by transforming it into a VI form we find that the resulting discrete Painlevé equation can be cast into a QRT-symmetric form. Without entering into the tedious (but straightforward) details we give the final form of the equation:(5.10)
Again, this is a new, to the authors’ knowledge, discrete Painlevé equation.
This paper was motivated by the existence of two -associated discrete Painlevé equations, known under different canonical forms, both having a double ternary dependence in their coefficients. The usual conversion of form-VI to form-VI′ did not suffice in order to link the two equations and thus their relation was somewhat puzzling. In this paper we have shown that these two equations correspond to two different limits of a -associated discrete Painlevé equation the coefficients of which do have a double ternary dependence.
Going from discrete Painlevé equations with symmetry to ones with can be obtained by taking the dependent variable to infinity. In order to balance this and obtain some meaningful equation necessitates taking also some of the parameters to infinity. In every case the independent variable must remain finite. In the case at hand the first of the two limits was a straightforward one, corresponding to taking the limit of a constant appearing explicitly in the equation to infinity. The second limit was somewhat more tricky since it consisted in taking the value of the parameter appearing in the period-2 function to infinity. Since we have ϕ2(n) = γ(−1)n, this means that we have to split the equation into even- and odd-index parts, i.e. write it in QRT-asymmetric form, and then apply the limit.
Having produced the complete list of QRT-symmetric -associated discrete Painlevé equations, we asked ourselves whether there existed other cases where two different limits could be implemented and whether these two limits yielded different -associated discrete Painlevé equations or the same one written under two different, but equivalent, forms. It turned out that both situations materialised: two of the equations led each to a single one at the limit (out of two and four possibilities respectively) the remaining ones yielding each two distinct equations. In all cases we were able to cast the equations into a QRT-symmetric form, either a VI form or a VI′ one. With one exception, (5.7), all the equations thus obtained had not been derived before (at least as far as the present authors can confirm). In this sense the method introduced here turned out to be a particularly fruitful one.
In our approach we took particular care of balancing the equations so as to lose just one parameter and moreover to be able to do this in at least two different ways. Thus we limited ourselves to the cases where that was possible. Of course it is also possible to implement the limit for to -associated equations in just a single way, and, in the light of the present results, derive new discrete Painlevé equations. What is potentially even more interesting is to consider the limits where one can go from a discrete Painlevé equation with symmetry to one associated to or even lower groups. Given the limiting procedure all these equations would have a canonical VI′ form. From our study in  we know that such equations do exist and deriving all of them in a systematic approach could be an interesting extension of the present work.
Cite this article
TY - JOUR AU - A. Ramani AU - B. Grammaticos AU - T. Tamizhmani PY - 2019 DA - 2019/10/25 TI - Interrelations of discrete Painlevé equations through limiting procedures JO - Journal of Nonlinear Mathematical Physics SP - 95 EP - 105 VL - 27 IS - 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1683981 DO - 10.1080/14029251.2020.1683981 ID - Ramani2019 ER -