Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with E₆⁽¹⁾ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlev...

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Дата:	2014
Автор:	Ormerod, C.M.
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Опубліковано:	Інститут математики НАН України 2014
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation / C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 55 назв. — англ.

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spelling	irk-123456789-1468502019-02-12T01:24:13Z Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation Ormerod, C.M. We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with E₆⁽¹⁾ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation. 2014 Article Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation / C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 55 назв. — англ. DOI:10.3842/SIGMA.2014.002 1815-0659 2010 Mathematics Subject Classification: 39A10; 37K15; 33C05 http://dspace.nbuv.gov.ua/handle/123456789/146850 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with E₆⁽¹⁾ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation.
format	Article
author	Ormerod, C.M.
spellingShingle	Ormerod, C.M. Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ormerod, C.M.
author_sort	Ormerod, C.M.
title	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
title_short	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
title_full	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
title_fullStr	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
title_full_unstemmed	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation
title_sort	symmetries and special solutions of reductions of the lattice potential kdv equation
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146850
citation_txt	Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation / C.M. Ormerod // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 55 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ormerodcm symmetriesandspecialsolutionsofreductionsofthelatticepotentialkdvequation
first_indexed	2025-07-11T00:46:27Z
last_indexed	2025-07-11T00:46:27Z
_version_	1837309421668007936
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 002, 19 pages Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation? Christopher M. ORMEROD Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA E-mail: christopher.ormerod@gmail.com URL: http://www.math.caltech.edu/~cormerod/ Received September 19, 2013, in final form December 28, 2013; Published online January 03, 2014 http://dx.doi.org/10.3842/SIGMA.2014.002 Abstract. We identify a periodic reduction of the non-autonomous lattice potential Korte- weg–de Vries equation with the additive discrete Painlevé equation with E (1) 6 symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation. Key words: difference equations; integrability; reduction; isomonodromy 2010 Mathematics Subject Classification: 39A10; 37K15; 33C05 1 Introduction Finding explicit solutions to integrable partial differential equations in terms of solutions of ordinary differential equations, such as the Painlevé equations, is a topic that is of interest to many researchers [1, 7, 10, 28]. Finding explicit solutions to the discrete analogues of integrable partial differential equations, integrable lattice equations, in terms of known ordinary difference equations, such as the discrete Painlevé equations, has recently been a hot topic [12, 13, 14, 37, 38, 39, 45]. The symmetries of the Painlevé equations are well known to be realizations affine Weyl groups [22]. The work of Sakai provides a geometric framework for these realizations [47]. An- other approach to symmetries of discrete Painlevé equations are discrete Schlesinger transforma- tions, which can be derived by the framework of connection preserving deformations [5, 17, 35]. This article presents a description of the symmetries of periodic reductions of quad equations. A discussion of the symmetries of reductions is a necessary step towards identifying reductions with the full parameter versions of some of the higher discrete Painlevé equations. Our second aim is to present a novel method of finding special solutions of reductions in terms of the lattice equations. It is well known that the Painlevé equations admit rational and hypergeometric solutions [6]. It is even more surprising that the discrete Painlevé equations also admit rational and hypergeometric solutions [34], basic hypergeometric solutions [21] and even elliptic hypergeometric solutions [20, 44]. There are many approaches to finding special solutions, such as a direct approach [17], bilinear approaches [19], orthogonal polynomial ap- proaches [36, 40, 52] and geometric approaches [21, 26]. Our approach could be called a reductive approach. ?This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html mailto:christopher.ormerod@gmail.com http://www.math.caltech.edu/~cormerod/ http://dx.doi.org/10.3842/SIGMA.2014.002 http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 C.M. Ormerod To demonstrate the general symmetry structure of reductions and our approach to find special solutions, we will consider an identification of a periodic reduction of the nonautonomous lattice potential Korteweg–de Vries equation, (wl,m − wl+1,m+1)(wl+1,m − wl,m+1) = pl − qm, (1.1) with the additive Painlevé equation with E (1) 6 symmetry, given by (z̃ + y)(y + z) = (y − a3)(y − a4)(y − a5)(y − a6) (y − a1 + t)(y − a2 + t) , (1.2a) (ỹ + z̃)(y + z̃) = (z̃ + a3)(z̃ + a4)(z̃ + a5)(z̃ + a6) (z̃ + a7 + t)(z̃ + a8 + t) , (1.2b) where t̃ = t+ δ and f̃ = f̃(t) = f(t+ δ) and (a1 + a2 + a7 + a8)− (a3 + a4 + a5 + a6) = δ. (1.3) This is sometimes known as the asymmetric d-PIV [46], the difference PVI [2], the additive discrete Painlevé equation with E (1) 6 symmetry or d-P ( A (1)∗ 2 ) [9, 24, 47]. The Riccati solutions were found relatively early by Ramani et al. [46] and their expressions in terms of 3F2(1) functions was presented by Kajiwara [18]. There are very good reasons to consider this equation. Firstly, this equation possesses the sixth Painlevé equation as a continuum limit and is the lowest member of the additive type discrete Painlevé equations that does not arise as a contiguous relation for a continuous Painlevé equation [34]. Secondly, it is very interesting that this equation arises from two characteristically dis- tinct Lax pairs; a difference-difference Lax pair, by Arinkin and Borodin [2], and a recent differential-difference Lax pair, by Dzhamay et al. [9]. These two Lax pairs also arise from two distinct notions of isomonodromy. The first known Lax pair was derived as a discrete ana- logue of an isomonodromic deformation in that it preserves a connection matrix, in the sense of Birkhoff [3, 5]. The second was derived as a discrete isomonodoromic deformation of a 3× 3 system, i.e., a Schlesinger transformation, such as those considered by Jimbo and Miwa [16]. Finally, while the q-Painlevé equation with E (1) 6 -symmetry was recently identified as a re- duction of the lattice Schwarzian Korteweg–de Vries equation [39], the relation between (1.2) and any lattice equation is not known at this point. These two systems have a rich enough symmetry structure to extrapolate the general symmetry structure for more general reductions of lattice equations. This work on symmetries and special solutions is applicable to a wide class of reductions, and includes those presented in [39] and degenerations thereof. The outline of the paper is as follows: in Section 2 we will specify the reduction and its Lax representation, in Section 3 we will discuss the correspondence between the reduction and (1.2), in Section 4 we engage in a discussion of the symmetries of the reduction, which are described in terms of (1.1), and their relation to the symmetries of (1.2) and in Section 5 we will discuss the way in which the reduction leads us to a fundamental set of rational and Riccati solutions. 2 Reduction The discrete potential KdV equation, given by (1.1), was one of the first integrable lattice equations to be derived [27, 31]. Autonomous reductions of this equation have been considered by many authors [15, 32, 41, 50, 51], and there have been some studies of non-autonomous similarity reductions [30, 39]. In [39], the equation was treated as a test case, where we obtained a difference version of d-PIV. Here, we will explore a much more involved reduction. Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 3 Figure 1. The labeling of lattice points in accordance with the prescribed periodicity and definition of n = 2m− l. We will consider periodic (4, 2)-reductions, which are special solutions satisfying the con- straint wl+4,m+2 = wl,m. (2.1) A special case of these solutions are (1, 2)-reductions, which may be expressed in terms of the solution of a discrete analogue of the first Painlevé equation and a version of the discrete fourth Painlevé equation [39]. In order to specify a (4, 2)-reduction, we are required to specify six initial conditions, which are periodically continued in both directions via (2.1), making this a six-dimensional mapping that we will find a sufficient number of transformations and integrals to express as a two-dimensional mapping. We follow [39] by defining an evolution variable, n, by n = 2m− l. (2.2) For every value of n, up to periodicity, we have two distinct lattice values, w0,n and w1,n. In this way, every point (l,m) ∈ Z2, is associated with either a value, w0,n or w1,n. For convenience, we will use the notation wi,n = wi and wi,n+1 = w̄i. We have shown this is Fig. 1. In order for the constraint to define the iterates of w0 and w1 consistently, we require the equation defining the evolution at (l,m) to coincide with the equation at (l + 4,m+ 2), hence, we require pl+4 − qm+2 = pl − qm, which, by a separation of variables argument, defines a constant in l and m, which we label h, given by pl+4 − pl = qm+2 − qm := h. This difference equation defines 6 additional constants. That is to say that the difference equation for pl is of degree 4, defining 4 constants in general, and the difference equation for qm is of degree 2, defining 2 new constants. We label these constants a1, . . . , a6, which enter the system via qm and ql by letting qm =  mh 2 − a1 if m = 0 mod 2, mh 2 − a2 if m = 1 mod 2, (2.3) 4 C.M. Ormerod pl =  lh 4 − a3 if l = 0 mod 4, lh 4 − a4 if l = 1 mod 4, lh 4 − a5 if l = 2 mod 4, lh 4 − a6 if l = 3 mod 4. (2.4) The mapping that brings n→ n+1 is called the generating shift, as any shift on the lattice is some power of the generating shift. The generating shift is equivalent to the shift (l,m)→ (l+1,m+1), hence, the mapping corresponding to n → n + 1 permutes the roles of the ai in the following way ( ā1 ā2 ā3 ā4 ā5 ā6 ) = a2 + h 2 a1 − h 2 a4 + h 4 a5 + h 4 a6 + h 4 a3 − 3h 4  . (2.5a) In particular, notice that ¯̄̄̄ai = ai. If we assume l and m are 0 mod 4, under this correspondence, the equations governing the lattice variables are (w̄0 − ¯̄w0)(w1 − ¯̄̄w1) = a1 − a6 − nh 4 − h, (2.5b) (w̄1 − ¯̄w1)(w0 − ¯̄̄w0) = a2 − a4 − nh 4 . (2.5c) Let us simplify these equations by introducing variable u and v by u = w̄0 − w0, v = w̄1 − w1, hence, (2.5b) and (2.5c) become the degree 4 mapping ū(v + v̄ + ¯̄v) = a1 − a6 − nh 4 − h, (2.6a) v̄(u+ ū+ ¯̄u) = a2 − a4 − nh 4 . (2.6b) We use the second equation to obtain v in terms of u and its iterates, which then gives us an equation which may be written solely in terms of ν = ū/u, reducing this fourth order system to a third order system a1 − a5 − n+4 h 1 + ν̄ + ν̄ ¯̄ν + ν ( a2 − a4 − nh 4 ) 1 + ν + νν̄ + νν ( a1 − a3 − nh 4 ) 1 + ν + νν = a1 − a6 − (n+ 4)h 4 . It is not completely trivial to see that this is a third order system with an invariant, d1, which defines our second order evolution νν ( a1 − a3 − nh 4 ) 1 + ν + νν − a2 − a4 − nh 4 1 + ν + νν̄ = d1 + a1 + a4 + a5. (2.7) It is worth reminding the reader that we are primarily interested in the fourth power of this map, because the parameters change with each power of the map in accordance with (2.5a), in a similar manner to the asymmetric forms of discrete Painlevé equations provided by Kruskal et al. [23]. To avoid any confusion, we use a common notation to describe this map: ( a1 a2 a3 d1 a4 a5 a6 d2 ;n, ν, ν ) → a2 + h 2 a1 − h 2 a4 + h 4 d1 a5 + h 4 a6 + h 4 a3 − 3h 4 d2 ;n+ 1, ν, ν  . The fourth power of this mapping fixes the ai variables and sends n to n+ 4. Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 5 We now describe a Lax pair for this system. A general method for determining the Lax representation of an autonomous reduction has been known for some time. While there were examples of derivations of Lax pairs for non-autonomous lattice equations [11, 12], a direct method for determining a Lax representation was presented only recently [37, 39]. One of the interesting consequences of this theory is that the resulting Lax matrices factorize in a novel manner. The starting point for the Lax pair for the reduction is the Lax pair for the lattice equation. In this case, the Lax representation is given by Ψl+1,m = Ll,mΨl,m, Ψl,m+1 = Ml,mΨl,m, where Ll,m = ( −wl+1,m 1 −κ− wl,mwl+1,m + pl wl,m ) , (2.8a) Ml,m = ( −wl,m+1 1 −κ− wl,mwl,m+1 + qm wl,m ) , (2.8b) where κ is a spectral parameter. The consistency of this system, in the calculation of Ψl+1,m+1, gives us the condition Ll,m+1Ml,m = Ml+1,mLl,m, which is equivalent to imposing (1.1). We introduce a new non-autonomous spectral paramater, related to κ by x = lh 4 − κ. (2.9) Using (2.2), (2.9), (2.3) and (2.4), we can represent linear problems in l, m and κ in terms of x, n and the ai variables. Let us form two operators, An(x) and Bn(x), that are equivalent to the shifts (l,m)→ (l+ 4,m+ 2) and (l+ 1,m+ 1) respectively. This gives us a linear system satisfying the equations Yn(x+ h) = An(x)Yn(x), (2.10a) Yn+1 ( x+ h 4 ) = Bn(x)Yn(x). (2.10b) These matrices are given by the products An(x) ∼= Ll+3,m+2Ll+2,m+2Ml+2,m+1Ll+1,m+1Ll,m+1Ml,m, Bn(x) ∼= Ll,m+1Ml,m. Explicitly, this is Bn(x) = ( −w̄1 1 x− a3 − ¯̄w0w̄1 ¯̄w2 )( − ¯̄w0 1 x+ nh 4 − a1 − ¯̄w0w1 w1 ) , An(x) = ( −w1 1 x− a6 − w̄0w1 w̄0 )( −w̄0 1 x− a5 − w̄0 ¯̄w1 ¯̄w1 ) × ( − ¯̄w1 1 x+ nh 4 − a2 − w0 ¯̄w1 w0 )( −w0 1 x− a4 − w0w̄1 w̄1 ) Bn(x), (2.11) 6 C.M. Ormerod whose compatibility, given by An+1 ( x+ h 4 ) Bn(x)−Bn(x+ h)An(x) ≡ 0, gives (2.5). Through gauge transformations, we could express this in terms of the u and v, or ν, but we leave this out for succision. 3 Correspondence with d-P ( A (1)∗ 2 ) For very simple reductions, it is often straightforward to write down a correspondence between the variables on the lattice and the variables of the corresponding discrete Painlevé equation. For higher equations, the correspondences may be highly non-trivial and requires some auxilia- ry information as a guide. In the case of q-P ( A (1) 2 ) , the auxiliary information was the Lax pair, which was found by Sakai [49] and Yamada [55]. Nicholas Witte and this author showed these two Lax pairs were related, furthermore that the associated linear problem for the spe- cial hypergeometric solutions may be expressible in terms of a certain orthogonal polynomial ensemble [53]. Our guide in this case is the Lax pair of Arinkin and Borodin [2]. While the result of Arinkin and Borodin was indeed of interest to many in the field, the Lax pair presented was not as explicitly presented as other Lax pairs in the literature (see for example [17, 25, 42]). We use this opportunity to present an explicit parameterization of this Lax pair. Before doing so, we first examine some of the properties of the Lax pair we have thus far. The matrix An(x) is of degree 3, and may be written An(x) = A0,n +A1,nx+A2,nx 2 +A3,nx 3, where A3,n = I. The other property that is important is that, by taking the determinant of (2.11), we see that detAn(x) = ( x− a1 + nh 4 )( x− a2 + nh 4 ) (x− a3)(x− a4)(x− a5)(x− a6). (3.1) What is crucial to making the correspondence is A2,n, which is of the form A2,n = d1 + nh 4 0 ρ21 d2 + nh 4  , where the d1 is defined by (2.7) and d2 is determined by d1 + d2 + a1 + a2 + a3 + a4 + a5 + a6 = 0. (3.2) The additional variable, labeled ρ12, is ρ21 = ū2v + ū ( a3 − a1 + nh 4 + uv ) + u ( a3 + a4 + a6 + d1 + nh 4 ) + (d1 − d2)w0, however, given that A2,n is a lower triangular matrix, and that A3,n = I, we may transform our linear problem, Yn(x), by multiplication on the left by a simple lower triangular matrix so that A2,n is may be taken to be diagonal, hence, An(x) takes the general form An(x) = x3I +  ( d1+ nh 4 ) ((x− α)(x− y) + z1) ( d1+ nh 4 ) ω(x− yn)( d2+ nh 4 ) (δx− ε) ω ( d2+ nh 4 ) ((x− β)(x− y) + z2) , (3.3) Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 7 where the function ω is related to the gauge freedom. The functions, α, β, δ and ε are specified by conditions (3.2) and (3.1). There is also a relation between z1 and z2, which means that z1 and z2 may be written in terms of a single variable, z, chosen later to simplify the evolution equations. In the interest of being explicit, we give expressions for these functions; we define the notation 6∑ k=0 µix k = detAn(x), then the functions α, β, γ and δ are given, in terms of the µi, as α = h2n2 16 (d1 − d2) + 4 ( (d2 − d1) ( y2 − z2 ) + µ3 + µ4y ) (d1 − d2) (4d1 + nh) + nh (d1 + d2 − y) 4 (d1 − d2) − d1 (y − d2) + µ4 − 2y2 + z1 + z2 d1 − d2 , (3.4a) β = h2n2 16 (d2 − d1) − 4 ( (d1 − d2) ( y2 − z1 ) + µ3 + µ4y ) (d1 − d2) (4d2 + nh) − nh (d1 + d2 − y) 4 (d1 − d2) + y (d2 − 2y)− d1d2 + µ4 + z1 + z2 d1 − d2 , (3.4b) δ = αβ − 4 (µ0 + µ1y) (d1 − d2) y2 (4d1 + nh) + 4 (µ0 + µ1y) (d1 − d2) y2 (4d2 + nh) − z1z2 y2 + y(α+ β) + z1 + z2, (3.4c) ε = 16µ0 − (4d1 + nh) (4d2 + nh) (αy + z1) (βy + z2) y (4d1 + nh) (4d2 + nh) . (3.4d) The relationship between z1 and z2 is expressed as the determinant detA(y) = ( y3 + z1 ( d1 + nh 4 ))( y3 + z2 ( d2 + nh 4 )) , which we specify by letting the (1, 1) and (2, 2) entries of An(yn) be y3 + z1 ( d1 + nh 4 ) = (y − a3) (y − a4) (y − a5) (y − a6) z + y , (3.5a) y3 + z2 ( d2 + nh 4 ) = (z + y) ( y − a1 + nh 4 )( y − a2 + nh 4 ) . (3.5b) This specifies z in a manner that simplifies the resulting evolution equations. The equations (3.3), (3.4) and (3.5) specify a parameterization of the Lax matrix described in the work of Arinkin and Borodin [2] (written as A(z) in [2]). We are now in a better position to explicitly relate the system defined by (2.6) with (1.2). The variable y in (1.2) is y = vū+ a3 + uv − v (vū+ uv + a3 − a4) (uv̄ − vū− a3 + a5) v ( 2uv̄ − a2 − a3 + a4 + a5 + nh 4 ) + v̄ ( uv̄ − a2 + a5 + nh 4 ) − v2ū , (3.6) and the variable z in (1.2) is most succinctly expressed in terms of y as z + y = v (y − a4) (y − a5) (y − a6)( y − a1 + nh 4 ) ( v ( uv̄ − a2 + hn 4 + y ) + v̄ ( uv̄ − a2 + a5 + nh 4 )) . (3.7) 8 C.M. Ormerod With these variables, defined1 in terms of u and v, we can verify that the y and z satisfy (¯̄̄̄z + y)(y + z) = (y − a3)(y − a4)(y − a5)(y − a6)( y − a1 + nh 4 ) ( y − a2 + nh 4 ) , (3.8a) (¯̄̄̄y + ¯̄̄̄z)(y + ¯̄̄̄z) = (¯̄̄̄z + a3)( ¯̄̄̄z + a4)( ¯̄̄̄z + a5)( ¯̄̄̄z + a6)( ¯̄̄̄z − (d1 + a1 + a2) + nh 4 )( ¯̄̄̄z + (d1 + a3 + a4 + a5 + a6 + h) + nh 4 ) , (3.8b) which may be identified with (1.2) when δ = h, f̃ = ¯̄̄̄ f , t = nh/4 and a7 = −d1 − a1 − a2, a8 = d1 + a3 + a4 + a5 + a6 + h. The constraint, (1.3), is trivially satisfied by this choice. At this point, we note that it is somewhat remarkable that this choice, coupled with the definition of the evolution of u and v, given by (2.6), are sufficient to check that (1.2) is satisfied. To actually derive (1.2), we require a little more work. This is a system of linear difference equations admitting two fundamental solutions, Y∞ and Y−∞, which are analytic throughout the complex plane, except for possible poles to the left and right of the ai respectively. The solutions are asymptotically represented, á la Birkhoff [3], by expansions of the form Y±∞(x) = x3x/he−3x/h ( I + Y1 x + Y2 x2 + · · · ) diag ( x n−6 4 + d1 h , x n−6 4 + d2 h ) . A more current exposition detailing the existence of a more general class of solutions were obtained by [43]. The connection matrix is defined by P (x) = (Y∞(x))−1Y−∞(x), which is h-periodic in x (i.e., P (x + h) = P (x)). Borodin specified a canonical group of trans- formations, specified by a general form Ỹn(x) = Rn(x)Yn(x), that preserves this matrix [5] which he used with Arinkin to formulate the original Lax pair in [2]. Since the shift n → n + 1 has an undesired non-trivial effect on the parameters, we need to consider a transformation that sends n → n + 4, which we label B′n(x). A Lax matrix that represents the shift (l,m)→ (l,m+ 2) may be given by B′n(x) ∼= Ml,m+1Ml,m, (3.9) which, is of the general form B′n(x) =  x+ r1 r2 − ( nh 4 − a1 − r1 ) ( nh 4 − a2 − r1 ) r2 nh 2 + x− a1 − a2 − r1  . This means that our new auxiliary linear equation is Yn+4(x) = B′n(x)Yn(x), 1Equivalent formulations in terms of ν are easy to derive, but they were not succinct enough for presentation. Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 9 where the new compatibility condition is B′n(x+ h)An(x)−An+4(x)B′n(x) ≡ 0. (3.10) One may derive the entries, r1 and r2, by performing the required change of variables, from w0 and w1 to functions in y and z to (3.9), however, it is much more convenient to use the (3.10) directly, to give r1 = 1 h ( d1 + nh 4 ) (α̃− α+ ỹ − y + 1) + α̃+ ỹ, r2 = ( d1 + nh 4 ) w − ( d1 + nh 4 + h ) w̃ d1 − d2 + h . Using the definition of y and z, the evolution, and the expressions for y and z in terms of w0 and w1 recovers the expression for B′n(x) in terms of w0 and w1. Using these values for r1 and r2 in (3.10) gives us (1.2), meaning that the expressions, (3.6) and (3.7), in terms of u and v solve (3.8). 4 Symmetries The Bäcklund transformations of (1.2) form a realization of an affine Weyl group of type E (1) 6 [47]. It is interesting and possible to explore how the Schlesinger transformations, such as those explored by Borodin [5], are tied to the concept of consistency around a cube [33]. That is, we seek to relate the Bäcklund transformations for lattice equation, defined by the mul- tilinear function, (1.1), to the discrete isomonodromic (or connection preserving) deformations of Borodin [5]. Naturally, the connection preserving deformations form sub-groups of the full affine Weyl groups of symmetries. The q-analogues of connection preserving deformations appeared in the work of Jimbo and Sakai [17], and there have been but a few steps towards a general system of Schlesinger transformations that describe symmetries for the discrete Painlevé equations [35, 48]. The first set of symmetries we wish to describe is permutations on the set of parameters {a1, a2} and the permutations on the set {a3, a4, a5, a6}. Let us first describe the symmetry s34, which permutes a3 and a4, which we will generalize to provide the symmetry for the remaining symmetry group on the set of parameters. It is important to also note that these symmetries commute with the fourth power of the generating shift, not the generating shift itself. Let us describe the symmetry s34 as a map that sends the lattice variables, wi and their iterates, to ŵi. The path described in 1 assumes a certain labeling where, left to right, the parameters cycle through a3 to a6. The action of s34 on the wi has a trivial effect on almost all the lattice variables except for w̄1, which is related to ˆ̄w1 by a quad centered at w̄1 with variables a3 and a4 on the edges, as in Fig. 2. That is to say that the symmetry is described by the equation Q( ¯̄w0, w̄1, ˆ̄w1, w0, a4, a3) = 0, which means that the s34 has the effect s34 : w̄1 → ˆ̄w1 = w̄1 + a3 − a4 w0 − ¯̄w0 , s34 : a3 → a4, s34 : a4 → a3. This may be easily lifted to a symmetry of the u and v variables, where we note that the u and ū are unchanged, and the v variables become s34 : v → v̂ = v + a3 − a4 u+ ū , s34 : v̄ → ˆ̄v = v̄ − a3 − a4 u+ ū . This allows us to easily show that ŷ = y and ẑ = z. 10 C.M. Ormerod Figure 2. A pictorial representation of the symmetry s34 which permutes the roles of a3 and a4. Figure 3. A pictorial representation of the symmetry s45 which permutes the roles of a4 and a5. Similarly, to obtain the transformation, s56, on the lattice variables, we place a quad centered at w̄0, which will permute a5 and a6 in the same way. That is to say that s56 has a trivial effect on all the lattice variables except for w̄0, which is related to the transformed variable, ˆ̄w0 via the relation Q( ¯̄w1, ˆ̄w0, w̄0, w1, a5, a6) = 0, which means s56 is specified by s56 : w̄0 → ˆ̄w0 = w̄0 + a5 − a6 w1 − ¯̄w1 , s56 : a5 → a6, s56 : a6 → a5. We lift this to the variables u and v to give a trivial effect on v and an action on u and ū described by s56 : u→ û = u+ a5 − a6 v + v̄ , s56 : ū→ ˆ̄u = ū− a5 − a6 v + v̄ , while, once again, this change has a trivial effect on y and z. To extrapolate this principle to obtain s12 and s45, we remark that the while the path chosen above gives us natural places to insert quads that permute the parameters, the path chosen is still arbitrary. That is to say that because Q is multilinear, specifying the six initial conditions on any path is in one-to-one correspondence with the specification of variables along any other path. In this sense, while the standard staircase of [41, 51] is useful in describing the particular form of the evolution equations, (2.5), it is not actually that important in building a correspondence between (2.5) or (2.6) with (1.2). With this in mind, let us consider a small deviation of our original path that passes through w1 instead of ¯̄w1, as depicted in Fig. 3. Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 11 The quad centered at w0 in Fig. 3 has a trivial effect on each of the variables except for w0. Knowing ŵ0 and ŵ1 = w1 is sufficient to obtain ˆ̄̄w1, i.e., one can recover the transformation of the variables along our original path by using the quad equation. This gives us that the symmetry that permutes the a4 and a5 is trivial for all variables except for w0 and ¯̄w1, which is related to the transformed variables by the two quad equations Q(w̄1, ŵ0, w0, w1, a5, a4) = 0, Q ( ŵ0, w1, ˆ̄̄w1, w̄0, a2 − nh 4 , a4 ) = 0, where the nh/4 contribution comes from the difference between an pl and a qm and w1 is known from knowledge of the original initial conditions, which gives us s45 : w0 → ŵ0 = w0 + (a4 − a5) (w0 − w̄0) a5 − a2 + nh 4 + (w0 − w̄0) (w̄1 − ¯̄w1) , s45 : w̄1 → ˆ̄w1 = w̄1 + (w0 − w̄0)(w̄1 − ¯̄w1) w̄0 − ŵ0 , s45 : a4 → a5, s45 : a5 → a4. The transformation fixes ū and v, and changes u and v̄ in the following manner, s45 : u→ û = u+ u(a4 − a5) nh 4 + uv̄ − a2 + a5 , s45 : v̄ → ˆ̄v = v̄ + v̄(a4 − a5) nh 4 + uv̄ − a2 + a4 , which also has a trivial effect on y and z. To obtain the transformation that permutes the a1 and a2 variables, the simplest choice of path passes through w̃1 (i.e., w1(n + 4)) and ¯̄̄w3. By placing a quad centered at ¯̄w0, the transformation fixes all the variables, except for ¯̄w0 on this path, meaning the transformation, changes w0, w̄1 and ¯̄w1 in accordance with Q ( w1, ¯̄w0, ˆ̄̄w0, w̃1, a2, a1 ) = 0, Q ( ˆ̄̄w0, ˆ̄w1, w̃1, ¯̄̄w0, a1 − nh 4 , a3 ) = 0, Q ( ˆ̄w1, ¯̄̄w0, ŵ1, ŵ0, a1 − nh 4 , a4 ) = 0, which we will write implicitly as s12 : ¯̄w0 → ˆ̄̄w0 = ¯̄w0 + a2 − a1 w̃1 − w1 , s12 : w̄1 → ˆ̄w1 = w̃1 + nh 4 − a1 + a3 ¯̄̄w0 − ˆ̄̄w0 , s12 : w0 → ŵ0 = ¯̄̄w0 + nh 4 − a1 + a4 ¯̄w1 − ˆ̄w1 , s12 : a1 → a2, s12 : a2 → a1, which again, may be expressed in terms of transformations in u and v as s12 : u→ û = un + nh 4 − a2 + a4 v̄ − nh 4 − a1 + a4 ˆ̄v , s12 : v → v̂ = v + λ, s12 : ū→ ˆ̄u = ū+ nh 4 − a1 + a3 v − nh 4 − a2 + a3 v̂ , s12 : v → ˆ̄v = v̄ − λ, where λ = (a1 − a2) vv̄ v ( (ū+ u) v̄ − a2 + a4 + nh 4 ) + v̄ ( a3 − a1 + nh 4 ) . This transformation has a trivial effect on d1, d2, y and z. 12 C.M. Ormerod Let us now describe two transformations, Ta1 and Ta3 , which, in combination with the sym- metries described above, will be sufficient to describe a complete set of connection preserving deformations described by Borodin [5]. These transformations will necessarily act non-trivially on the variable y and z and may be used to obtain infinite families of rational and Riccati solutions from the seed solutions in Section 5. The first transformation, Ta1 , may be considered to be the image of the shift in the positive m direction. This may also be seen to be induced by a lift of the Ml,m matrix, (2.8b), to the level of the reduction. This also represents the discrete analogue of a fundamental Schlesinger transformation in the work of Jimbo and Miwa [16] since (2.8b) induces the transformation of the linear system Ŷn(x) = Rn,a1(x)Yn(x), (4.1) where Rn,a1(x) is linear in x and detRn,a1(x) = x− a1 − nh 4 . This transformation may be described on the level of the lattice variables by Ta1 : w1 → ŵ1 = ¯̄w0, Ta1 : w0 → ŵ0 = ¯̄w1, Q ( w̄1, w0, ˆ̄w1, ¯̄w1, a2 − nh 4 , a4 ) = 0, Q ( w̄0, w1, ˆ̄w0, ¯̄w1, a1 + h− nh 4 , a5 ) = 0, Q ( ¯̄w0, w̄1, ˆ̄̄w0, ˆ̄w1, a2 − nh 4 , a3 ) = 0, Q ( ¯̄w1, w̄0, ˆ̄̄w1, ˆ̄w0, a1 + h− nh 4 , a6 ) = 0, where the parameters, a1 and a2 change via the rule Ta1 : a1 → a2, Ta1 : a2 → a1 − h. When lifted to the level of the u and v variables, the transformation is described by Ta1 : u→ û = −v − v̄ − h(n+4) 4 − a1 + a6 ū , Ta1 : v → v̂ = −u− ū− nh 4 − a2 + a4 v̄ , Ta1 : ū→ ˆ̄u = v + h(n+4) 4 − a1 + a6 ū − nh 4 − a2 + a3 ˆ̄v , Ta1 : v̄ → ˆ̄v = u− h(n+4) 4 − a1 + a5 û + nh 4 − a2 + a4 v̄ , which is equivalent to a double shift in n. The first implication of this is that our invariant, d1 and d2 change, in accordance with the rule Ta1 : d1 → d2, Ta1 : d2 → d1 + h. To calculate the effect of Ta1 on y and z, we use the compatibility between (4.1) and (2.10a), where An(x) is defined by (3.3). The effects of Ta1 on y and z are given by Ta1 : y → ŷ = y + 1 d1 − d2 y2 + δ ( d1 + nh 4 ) ( d2 + nh 4 ) ( d1 + nh 4 ) ( α− a1 + nh 4 ) − (nh 4 −a1) 3−z1(d1+nh 4 ) y−a1+nh 4 Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 13 − z1 ( d1 + nh 4 ) + y3 y − a1 + nh 4 + a1 ( d2 + a1 − nh 4 + y ) − d2 ( β + nh 4 ) − (β + y)nh 4 ) , Ta1 : z → ẑ, where the relation between z and ẑ is given implicitly by the relation ẑ1 = y − ŷ d2 + nh 4 ( d1 ( α− a1 + nh 4 ) − a1nh 4 + (β − ŷ) ( d2 + nh 4 ) + h2n2 16 + αnh 4 − ( nh 4 − a1 ) 3 − z1 ( d1 + nh 4 )( −a1 + nh 4 + y ) + z2 ( d2 + nh 4 ) (y − ŷ) ) . Similarly, if we were to lift the Ll,m to the reduced linear problem, we obtain a linear system Ŷn(x) = Rn,a3(x)Yn(x), (4.2) where Rn,a3(x) is also linear in x and detRn,a3(x) = x− a3. This is a fundamental Schlesinger transformation that induces a transformation, which we la- bel Ta3 , that has the effect of fixing a1 and a2, permuting the other variables, a3 to a6, as follows Ta3 : a3 → a4, Ta3 : a4 → a5, Ta3 : a5 → a6, Ta3 : a6 → a3 − h. The effect on the lattice variable is given in terms of the multilinear function, Q, as Ta3 : w̄0 → ˆ̄w0 = w1, Ta3 : w̄1 → ˆ̄w1 = w0, Ta3 : ¯̄w0 → ˆ̄̄w0 = w̄1, Ta3 : ¯̄w0 → ˆ̄̄w1 = w̄0, Q ( w1, ŵ1, ¯̄w0, w̄1, a3, a1 − nh 4 ) = 0, Q ( w0, ŵ0, ¯̄w1, w̄0, a5, a2 − nh 4 ) = 0. The effect of Ta3 on u and v is simply given by the inverse of (2.6). In the same way as for Ta1 , we use (4.2), (2.10a) and (3.3) to find that Ta3 : d1 → d2, Ta3 : d2 → d1 + h, and the effect on y and z is given by Ta3 : y → ŷ = y + 1 d1 − d2 ( z1 ( d1 + nh 4 ) + y3 (a3 − y) + y2 − βd2 + a3 ( d2 + nh 4 + y ) − βnh 4 + a23 + δ (y − a3) ( d1 + nh 4 ) ( d2 + nh 4 )(( d1 + nh 4 ) (y (α− a3)− αa3 + z1) + a23 ( a3 + d1 + nh 4 ))) , Ta3 : z → ẑ, where we give the relation implicitly by stating that ẑ1 is specified by ẑ1 = y − ŷ d2 + nh 4 ( z1 ( d1 + nh 4 ) + y3 (y − a3) ( d2 + nh 4 ) − a3 ( d1 + nh 4 + y ) d2 + nh 4 − a23 d2 + nh 4 + β + α ( d1 + nh 4 ) − y2 d2 + nh 4 + z2 y − ŷ − ŷ ) . 14 C.M. Ormerod What is important in these calculations is that the general symmetry structure of reduc- tions of quad-equations apparent. In fact, the way in which the general symmetry structure is described by quad-equations is much more natural as a higher-dimensional reduction than a two-dimensional reduction. In this case, this is more naturally a 6-dimensional reduction. This work, in combination with [39], is sufficient to give a symmetry structure to reductions that are equivalent to all the systems below those with A (1) 2 surfaces in the Sakai classification [47]. It should be noted that the above constitutes a set of symmetries that have been derived by an application of Q as a symmetry. There are other equations that are consistent with Q, which can be found in the work of Boll [4], however, none of the tested equations that are consistent with (1.1) seemed to produce a non-trivial transformation of the Painlevé variables, y and z. Perhaps some further investigation is required in this direction. 5 Special solutions Lastly, this brings us to the method we wish to use to solve (1.2). There are numerous studies that show that the Painlevé equations admit solutions expressible in terms of hypergeometric and confluent hypergeometric functions (see [34] for a review). Many of the simplest discrete Riccati solutions were, including those for (1.2), were presented in [46]. A more geometric approach was taken for the q-Painlevé equations by Kajiwara et al. [21]. The work of Nicholas Witte has shown that is possible to find many hypergeometric solutions, including those for (1.2), in terms of the moments of a certain semi-classically deformed orthogonal polynomial ensembles [52], which is closely related to the Padé approximation method of Yamada et al. [54]. For an equation such as (1.2), it is not immediately obvious how to even obtain very basic rational solutions. Our work on the reduction gives us a number of different equivalent forms of the equation that we may solve explicitly. If we take (2.6) in the special case in which the parameters of the lattice equation, pl and qm, are linear pl = a+ hl 4 , qm = b+ hm 2 , corresponding to the choice of parameters a1 = −b, a2 = −b− h 2 , a3 = −a, a4 = −a− h 4 , a5 = −a− h 2 , a6 = −a− 3h 4 , the evolution equations simplify to v̄(u+ ū+ ¯̄u) = a− b− (n+ 1)h 4 , ū(v + v̄ + ¯̄v) = a− b− (n+ 1)h 4 . If u = v, this becomes a version of d-PI that has no rational solutions. This equation does possess a relatively simple one parameter family of rational solutions, in the case that u is constant and v is linear, explicitly we have u = 1 3λ , v = λ ( a− b+ nh 4 ) , or equivalently, this choice prescribes a two parameter family of solutions to the system in w0 and w1, given by w0 = n 3λ + θ1, w1 = λn ( a− b+ h 8 ) + λn2h 8 + θ2, Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 15 where θ1 and θ2 are arbitrary constants. Substituting these solutions into y, z, d1 and d2 gives d1 = 2a+ b+ 5h 6 , d1 = 2a+ b+ 7h 6 , y = 1 24 ( 7h2(2nh− h− 8a+ 8b) 64a2 − 16a(8b+ h(2n− 1)) + 64b2 + 16bh(2n− 1) + h2(4(n− 1)n− 1) − 8a− 16b− 4hn− 7h ) , z = 1 24 ( h ( 14h −8a+ 8b+ 2nh− 5h − 4n+ 19 ) + 40a− 16b ) . Alternatively, by swapping the roles of u and v we obtain the solution d1 = 2a+ b+ 2h 3 , d2 = d2 = 2a+ b+ 4h 3 , y = 1 24 ( h ( − 5h 8a− 8b− 2nh+ h − 4n− 7 ) − 8a− 16b ) , z = 1 24 (40a− 16b+ h(19− 4n)). To consider the hypergeometric solutions, let us first review the work of Ramani et al. [46] who present solutions of the form ỹ = σ1z̃ + σ2 σ3z̃ + σ4 , z̃ = τ1y + τ2 τ3z + τ4 , where the σi’s and τi’s are functions on n alone. There are a multitude of values the σ and τ variables may take in order for the evolution to coincide with (1.2). We choose just one, which yields ỹ = z̃ ( a2 − 1 4h(n+ 4) ) + a3a4 z̃ − a2 + a3 + a4 + hn 4 + h , z̃ = −y ( a6 + nh 4 ) + a1y + a5 (a6 − y) y − a1 + nh 4 , where we have one of two alternative constraints, either d1 + a1 + a3 + a4 + h = 0 or d1 + a2 + a3 + a4 + h = 0. Combining these two equations gives us a single second order difference equation which also admits the hypergeometric differential equation in a continuum limit [46]. These solutions may be identified as hypergeometric functions by relating these recurrence relations to contiguous relations for hypergeometric functions of type 3F2(1), as done by Kajiwara [18]. We can now consider the possibility of the evolution of ν coinciding with a fractional linear transformation of the form ν̄ = c11ν + c12 c21ν + c22 , (5.1) which may be inverted so calculate ν in terms of ν. Substituting (5.1), and its inverse, into (2.7) gives us a number of conditions in the variable ν, which we solve to give the equation ν̄ = (1 + ν) ( nh 4 + d1 + a3 + a4 + a5 ) a2 − a4 − nh 4 − ν ( nh 4 + d1 + a3 + a4 + a5 ) , 16 C.M. Ormerod which coincides with the evolution of (2.7) so long as 2d1 + a1 + a2 + a3 + a4 + a5 + a6 = 0. In the language of the affine Weyl symmetries, this condition ensures that the parameters correspond to a point on a wall of an affine Weyl chamber. To obtain the corresponding solution for y and z, we first take the fourth power of this mapping to find ñu is given by ν̃ = C11ν + C12 C21ν + C22 , where C11 = ( nh 2 − a1 − a2 + a3 + a4 + a5 − a6 )( h(n+ 4) 2 − a1 − a2 + a3 + a4 − a5 + a6 ) , C12 = − (2h− a1 + a2 − a3 − a4 + a5 + a6) ( h(n+ 4) 2 − a1 − a2 + a3 + a4 − a5 + a6 ) , C21 = (a1 − a2 − a3 − a4 + a5 + a6) ( nh 2 − a1 − a2 + a3 + a4 + a5 − a6 ) , C22 = h2(n+ 4)2 4 − nh (a1 + a2 − a5 − a6) + 2 (a3h+ a4h+ a5h+ a6 (a3 + a4 + h) − a2 (a5 + a6 + h)− a1 (a5 + a6 + 3h) + a21 + a22 − a23 − a24 + a3a5 + a4a5 ) . The constraint seems characteristically different from the solutions above, which means this is a hypergeometric solution for a different choice of parameters. We do not know how these two solutions are related. Using the group of symmetries derived in Section 4 one is able to generate an infinite num- ber of hypergeometric solutions, however, we do note that the one cannot apply the full set of Bäcklund transformations to the hypergeometric solutions. One characterization of the hy- pergeometric solutions is that they are solutions that are singular on a translational Bäcklund transformation. That is to say that these solutions are not just on a wall of the affine Weyl chamber, but they are on a barrier which one cannot pass through using Bäcklund transforma- tions. It would be interesting to consider determinantal structures, either from a bilinear ap- proach [19] or an orthogonal polynomial approach [40]. Determinantal formulations are beyond the scope of this paper. 6 Conclusion We have provided a reduction from one of the most degenerate lattice equations to an equation that sits above the sixth Painlevé equation in Sakai’s classification. While these reductions are somewhat complicated, their very existence suggests that the solutions to all the additive Painlevé equations, and many higher-dimensional additive equations, may be expressed in terms of the solutions of the lattice potential Korteweg–de Vries equation. A similar statement could be made about multiplicative type equations and the discrete Schwarzian Korteweg–de Vries equation. Furthermore, while we know of a reduction from the Schwarzian Korteweg–de Vries equation to the sixth Painlevé equation [28, 29], we believe there may exist a more complicated reduction from the Korteweg–de Vries equation to the sixth Painlevé equation that is yet to be found. There is still a sense in which these reductions may be more naturally approached from a geometric perspective, where one is interested in mapping between biquadratic curves for the Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation 17 lattice equations and the (moving) biquadratics for the (non-autonomous) reductions. While this work suggests that the symmetries are most naturally viewed in terms of quad-equations on higher-dimensional lattices, Doliwa notes in [8] that it might be more natural to consider reductions from higher-dimensional lattice equations. There are still also issues regarding the definitions and framework of connection preserving deformations. These connection preserving deformations should manifest themselves as auto- morphisms of the Galois group associated with the system of difference equations. This could present a natural extension of the connection preserving deformations that could be large enough to describe the full affine Weyl group of Bäcklund transformations, rather than just a restriction to the translational elements of the affine Weyl group. This work on the symmetries of reduced equations applies a wide class of periodic reductions that have appeared in the literature. It is not entirely clear at this point how, or even whether, this procedure may be applied to the so-called twisted reductions explored by recent work [13, 38]. This work suggests that the symmetry group of a (s1, s2)-reduction is at least a lattice of dimension s1 + s2, hence, we suspect it would take a (4, 4)-reduction of Q4 to be able to be identified with the full parameter version of the elliptic Painlevé equation. Acknowledgements This research is supported by Australian Research Council Discovery Grant #DP110100077. 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Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

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