Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations

0.26 MB·English

by  C. M. Ormerod

2 On Lax representations of reductions of integrable lattice
1
equations
0
2
C. M. Ormerod, P. H. van der Kamp, and G.R.W. Quispel
p
e
S
Abstract. We present a method of determining a Lax representation for

1 similarity reductions of autonomous and non-autonomous partial difference
2 equations. This method may be used to obtain Lax representations that are
generalenoughtoprovidetheLaxintegrabilityforentirehierarchiesofreduc-

] tions. Amainresultis,asanexampleofthisframework,howwemayobtain
I
S theq-Painlev´eequationwhosegroupofB¨acklundtransformationsisanaffine
. Weyl group of type E6(1) as a similarityreduction of the discrete Schwarzian
n Korteweg-deVriesequation.
i
l
n
[
Integrablepartialdifferenceequationsarediscretetimeanddiscretespaceana-

1 logues of integrable partial differential equations [1, 2, 3, 4]. Integrable partial
v
difference equations admit classicalintegrable partial differential equations as con-
1
tinuum limits [5, 6, 7]. Integrable ordinary difference equations are discrete ana-
2
7 logues of integrable ordinary differential equations. Integrable ordinary difference
4 equations admit integrable ordinary differential equations as continuum limits [8].
9. Integrable ordinary and partial difference equations possess discrete analogues of
0 many of the properties associated to the integrability of their continuous counter-
2 parts [8, 9, 10, 11].
1
We consider partialdifference equations whose evolutionon a lattice ofpoints,
:
v wl,m, is determined by the equation
i
X
(0.1) Q(w ,w ,w ,w ;α,β)=0,
l,m l+1,m l,m+1 l+1,m+1
r
a where α and β are parameters associated with the horizontal and vertical edges
respectively. The equation is imposed on each square on the space of independent
variables,(l,m)∈Z2 [3, 4, 12, 13]. From a suitable staircase of initial conditions
[13],onemaydeterminew forall(l,m)∈Z2. Imposingthesimilarityconstraint,
l,m
that
(0.2) w =w ,
l+s1,m+s2 l,m
definesa(periodic)similarityreduction[13, 14, 15]. Wewillassumeforsimplicity
that s ands areboth positive. In ananalogouswayto how similarity reductions
1 2
ofpartialdifferentialequationsyield ordinarydifferential equations [16], similarity
reductions given by (0.2) yield ordinary difference equations [12, 13, 15].
2010 Mathematics Subject Classification. 39A14; 37K15;35Q51.
1
2 C.M.ORMEROD,P.H.VANDERKAMP,ANDG.R.W.QUISPEL
Given a partial differential equation with some similarity reduction, there is
a procedure that allows one to obtain a Lax representation of the resulting ordi-
nary differential equation from the Lax representation of the partial differential
equation. This holds for autonomous and non-autonomous reductions [17]. The
discrete analogue of this procedure is fairly straightforwardfor autonomous reduc-
tions [10, 11, 18], however, the process of determining the Lax representation for
non-autonomous reductions has been somewhat ad hoc [19, 20].
Given a reduction, another task is to determine whether the reduction is a
known system of difference equations. For autonomous reductions, one may be
abletofindacertainparameterisationwhichidentifiesthesystemasaknownQRT
mapping [21, 22], which may be classified in terms of elliptic surfaces [23]. For
nonautonomousreductions,onemaybeablefindaparameterisationoftheequation
that identifies the system as one of the Painlev´e equations, which are classified by
the group of symmetries of their surface of initial conditions [24].
The aim of this note is to demonstrate a method, which we outline in §1, by
which we may directly obtain a Lax representation of both autonomous and non-
autonomousreductionsfromaLaxrepresentationofpartialdifference equationsin
an algorithmic manner. The method gives Lax representations in a manner that
is general and concise enough to directly provide the Lax integrability of entire
hierarchiesof reductions. As anapplicationof this method, we presenta reduction
ofthe non-autonomousdiscreteSchwarzianKorteweg-deVriesequation(whichisa
non-autonomous version of Qδ=0 in the classification of Adler, Bobenko and Suris
1
[3, 4])totheq-Painlev´eequationassociatedwithasurfacewithA(1) symmetry(or
2
q-P(A(1))):
2
(a y′−1)(a y′−1)(a y′−1)(a y′−1)
(0.3a) (y′z−1)(y′z′−1)= 1 2 3 4 ,
(b q4ty′−1)(b q4ty′−1)
1 2
θ (z−a )(z−a )(z−a )(z−a )
(0.3b) (yz−1)(y′z−1)= 1 1 2 3 4 ,
(b b tz+θ )(a a a a +θ q4tz)
1 2 1 1 2 3 4 1
wheret′ =q4t,thea ,b andθ arefixedparametersandqissomecomplexnumber
i i 1
whose modulus is not 1. This is the q-Painlev´e equation whose group of B¨acklund
transformations is an affine Weyl group of type E(1) [24].
6
Ourmethodstandsincontrasttotwomethodsofperformingreductionsofpar-
tial difference equations in the literature, namely the method of Hydon et al. [25]
whichisbasedonthe existence ofcertainLie pointsymmetries,andthe method of
Grammaticos and Ramani, who perform autonomous reductions, then deautono-
mizetheequationviasingularityconfinement[26]. Whilethefirstmethodseemsto
relyonasimilarapproachtoours,neithermethodgivesrisetotheassociatedlinear
problem for the reduced equation. The approach most similar to our method has
been discussed by Hay et al. [20], in which the form of the monodromy matrix for
autonomous reductions, and its properties, are used as an ansatz for an associated
linearproblemofthenon-autonomousreductionsofthe latticemodifiedKorteweg-
de Vries equation. A further extension to this work successfully determined the
associated linear problem for a hierarchy of systems [19].
To demonstrate our method we first provide some simple examples in §2. We
presentanautonomousreductionofthe discretepotentialKorteweg-deVriesequa-
tion (dKdV) [7], then present the non-autonomous generalization of this example.
ON LAX REPS. OF REDUCTIONS OF INTEGRABLE LATTICE EQUATIONS 3
In§3wefirstpresenttheq-Painlev´eequationassociatedwiththeA(1) surface(oth-
3
erwise knownas q-P [27]) as a reduction of (3.1) before going to the higher case
VI
wherewepresentthe above-mentionedreductionof (3.1)to(0.3),whichwebelieve
to be the first known reduction to this equation.
1. The method
We start by imposing (0.2) as a constraint on our initial conditions, then the
periodicity gives us that there are s +s independent initial conditions to define.
1 2
We solve this periodicity constraint by a specific labelling following [13, 15]; let
s = ag and s = bg where g = gcd(s ,s ), then the direction of the generating
1 2 1 2
shift, (c,d), associated with the increment n→n+1 is chosen so that
a b
det =1.
c d
(cid:18) (cid:19)
We specify an n∈Z and a p∈Z by letting
g
a b l m
n=det , p≡det mod g,
l m c d
(cid:18) (cid:19) (cid:18) (cid:19)
where the labelling of variables is specified by
(1.1) w 7→wp.
l,m n
In the case in which g = 1 the superscript will be omitted. The reduction in the
autonomous case is a system of g equations given by
(1.2) Q(wp,wp+d,wp−c,wp−c+d;α,β)=0, p=0,1,...,g−1,
n n−b n+a n+a−b
where α and β are constants. In the nonautonomous setting, we have
(1.3) Q(wp,wp+d,wp−c,wp−c+d;α ,β )=0, p=0,1,...,g−1,
n n−b n+a n+a−b l m
whereα andβ willbe,aposteriori,constrainedfunctionsoflandm. Wewillnow
l m
outlinehowtoobtainLaxrepresentationsforthe autonomousandnonautonomous
reductions respectively.
1.1. Autonomous reductions. It is known that multilinear partial differ-
enceequationsthatareconsistentaroundacubeare,inasense,theirownLaxpair
[3, 4]. For a generic multilinear equation, (0.1), that is consistent around a cube,
a Lax pair may be written as
(1.4a) φ =L φ ,
l+1,m l,m l,m
(1.4b) φ =M φ ,
l,m+1 l,m l,m
where
∂Q(x,u,v,0;α,γ)
− −Q(x,u,0,0;α,γ)
∂v
(1.5a) Ll,m = λl,m∂2Q(x,u,v,y;α,γ) ∂Q(x,u,0,y;α,γ)(cid:12) ,
(cid:12)
∂v∂y ∂y (cid:12) x=wl,m
 (cid:12)
 (cid:12)(cid:12) u=wl+1,m
∂Q(x,u,v,0;β,γ) (cid:12)
− −Q(x,0,v,0;β,γ)
∂u
(1.5b) Ml,m = µl,m∂2Q(x,u,v,y;β,γ) ∂Q(x,0,v,y;β,γ)(cid:12) ,
(cid:12)
∂u∂y ∂y (cid:12) x=wl,m
 (cid:12)
 (cid:12)(cid:12) v =wl,m+1
(cid:12)
4 C.M.ORMEROD,P.H.VANDERKAMP,ANDG.R.W.QUISPEL
where γ is a spectral parameter. The compatibility condition is
(1.6) M L =L M ,
l+1,m l,m l,m+1 l,m
forcing the prefactors, λ and µ , to be chosen in a manner that satisfies the
l,m l,m
equation
detL detM
l,m+1 l+1,m
= .
detL detM
l,m l,m
Whentheprefactorsareappropriatelychosen,imposing(1.6)isequivalentto(0.1).
Inpractice,itisoftencomputationallyconvenienttodealwithsometransformation
of this Lax pair.
ToobtainaLaxrepresentationforthe systemofordinarydifferenceequations,
(1.2), we define two operators, A and B , associated with the shifts (l,m) →
n n
(l+s ,m+s )andthe generatingshift,(l,m)→(l+c,m+d),respectively. These
1 2
operators have the effect
(1.7a) φ =A φ ,
n n n
(1.7b) φ =B φ ,
n+1 n n
where one representation1, that is simple to write, is as follows:
s2−1 s1−1
A ←[ M L ,
n l+s1,m+j l+i,m
j=0 i=0
Y Y
d−1 c−1
B ←[ M L ,
n l+c,m+j l+i,m
j=0 i=0
Y Y
where the dependence on n and p is specified by
L (w ,w ;γ)7→Lp(γ)=Lp(wp,wp+d;γ),
l,m l,m l+1,m n n n n−b
M (w ,w ;γ)7→Mp(γ)=Mp(wp,wp−c;γ).
l,m l,m l+1,m n n n n+a
The compatibility condition,
(1.8) A B −B A =0,
n+1 n n n
isequivalenttoimposing(1.2). WecallA themonodromymatrixforthefollowing
n
reason: by identifying all points in Z2 that are multiples of (s ,s ) apart, we may
1 2
consider the space in which the new system exists as being cylindrical. We wrap
around in a manner that connects the points that are identified by the similarity
reduction. Themonodromymatrix,ratherthanpresentingatrivialactionas(1.7a)
suggests, expresses the action of wrapping around the cylinder, as in figure 1.
The monodromy matrix can be expressed as a function of the s +s initial
1 2
conditions, (w0,w0 ,...,wg−1 ), by following the standard staircase. Geomet-
n n+1 a+b−1
rically, the standard staircase is the path between two lines which squeeze a set of
squares with the same values, i.e., a set of squares shifted by (s ,s ) [15].
1 2
One advantage of the generating shift is that every other shift in n may be
expressedassomepowerofthe generatingshiftbyconstruction[28]. Furthermore,
this generating shift allows us to constrain the non-linear component, where we
need to use (1.2), to just g places. We have illustrated the standard staircase and
generating shift in figure 2.
1Inpractice, theproductfollowsthepathofastandardstaircase[28].
ON LAX REPS. OF REDUCTIONS OF INTEGRABLE LATTICE EQUATIONS 5
Figure 1. A pictorial representation of the way in which the
monodromy matrix wraps around to similar points for a (12,4)-
reduction.
wn0
wn2+1
wn0−1
wn2
wn1+1
wn2−1
wn1
wn0+1
wn1−1
wn0
wn0−1
Figure 2. A (9,6)-reduction and the labelling of variables. In
this example, the shift (p,n)→(p+1,n) corresponds to the shift
(a,b) = (3,2) and the shift (p,n) → (p,n+1) corresponds to the
shift (c,d)=(1,1)
Intheexampledefinedbyfigure2,ifweallowourmonodromymatrixtofollow
the standard staircase, the monodromy matrix is
A ←[L M L L M L M
n l+8,m+6 l+8,m+5 l+7,m+5 l+6,m+5 l+6,m+4 l+5,m+4 l+5,m+3
L L M L M L L M ,
l+4,m+3 l+3,m+3 l+3,m+2 l+2,m+2 l+2,m+1 l+1,m+1 l,m+1 l,m
and the other half of the Lax pair is
B ←[M L .
n l+1,m l,m
6 C.M.ORMEROD,P.H.VANDERKAMP,ANDG.R.W.QUISPEL
The resulting compatibility condition, (1.8), gives the evolution equations for the
wi , i=0,1,2:
n+1
Q(w1 ,w2 ,w0 ,w1 ;α,β)=0,
n−2 n−4 n+1 n−1
Q(w2 ,w0 ,w1 ,w2 ;α,β)=0,
n−2 n−4 n+1 n−1
Q(w0 ,w1 ,w2 ,w0 ;α,β)=0.
n−2 n−4 n+1 n−1
In general, this procedure gives us an s +s dimensional mapping,
1 2
φ:Cs1+s2 →Cs1+s2,
which,applied to (w0,w0 ,...,wg−1 ), gives(w0 ,w0 ,...,wg−1 ). This
n n+1 n+a+b−1 n+1 n+2 n+a+b
new set of values forms a new standard staircase. As a matter of fact, this new
standard staircase is the old one translated by the generating shift.
1.2. Nonautonomous reductions. To deautonomize this theory, we con-
sider the α and β to be functions of l and m. As L and M are shifted in
l,m l,m
only m and l respectively in the compatibility condition, (1.6), replacing α and β
with α and β , which are arbitrary functions of l and m respectively, preserves
l m
the Lax integrability. Hence, our basic non-autonomous lattice equations may be
considered to be of the form
(1.9) Q(w ,w ,w ,w ;α,β )=0,
l,m l+1,m l,m+1 l+1,m+1 l m
where the Lax representationis specified by (1.4) where
(1.10a)
∂Q(x,u,v,0;α,γ)
l
− −Q(x,u,0,0;α,γ)
l
∂v
Ll,m = λl,m∂2Q(x,u,v,y;α,γ) ∂Q(x,u,0,y;α,γ)(cid:12) ,
l l (cid:12)
∂v∂y ∂y (cid:12) x=wl,m
 (cid:12)
 (cid:12)(cid:12) u=wl+1,m
(1.10b) (cid:12)
∂Q(x,u,v,0;β ,γ)
− m −Q(x,0,v,0;β ,γ)
m
∂u
Ml,m = µl,m∂2Q(x,u,v,y;β ,γ) ∂Q(x,0,v,y;β ,γ)(cid:12) ,
m m (cid:12)
∂u∂y ∂y (cid:12) x=wl,m
 (cid:12)
 (cid:12)(cid:12) v =wl,m+1
(cid:12)
where γ is a spectral parameter and the prefactors, λ and µ , are chosen to
l,m l,m
satisfy the compatibility conditions, in an analogous manner to the autonomous
case.
Ifoneassumesthattheαandβ arefunctionsofbothlandm,i.e.,α=α and
l,m
β = β , then demanding that α is independent of m and β is independent
l,m l,m l,m
ofl hasalsobeenshowntobe anecessaryconditionforsingularityconfinementfor
equations in the ABS list [29]. The above constitutes a Lax pair interpretation of
this constraint.
Letusnowspecialiseourchoiceofsystemstothose thatadmitrepresentations
of the additive form
Q(w ,w ,w ,w ;α −β )=0,
l,m l+1,m l,m+1 l+1,m+1 l m
or the multiplicative form
α
l
Q w ,w ,w ,w ; =0,
l,m l+1,m l,m+1 l+1,m+1
β
(cid:18) m(cid:19)
ON LAX REPS. OF REDUCTIONS OF INTEGRABLE LATTICE EQUATIONS 7
with a possible additional dependence on α −α and β −β in the additive
l+1 l m+1 m
case, or α /α and β /β in the multiplicative case. A list of transformed
l+1 l m+1 m
equationsappearsintable1,wheresubscriptsmandadenotethosefunctions,(0.1),
dependent on a multiplicative or additive combination of α and β respectively.
l m
ABS Q(x,u,v,y;α,β )
l m
H1 (w −w )(w −w )+β −α
a l,m l+1,m+1 l+1,m l,m+1 m l
β β α2
H1 w − m+1w w − m+1w +1− l
m l,m β l+1,m+1 l+1,m β l,m+1 β2
(cid:18) m (cid:19)(cid:18) m (cid:19) m
β β α2
H2 w − m+1w w − m+1w − l
m l,m β l+1,m+1 l+1,m β l,m+1 β2
(cid:18) m (cid:19)(cid:18) m (cid:19) m
α β
+ 1− l w +w + m+1 (w +w ) +1
l,m l+1,m l,m+1 l+1,m+1
β β
(cid:18) m(cid:19)(cid:18) m (cid:19)
α
H3δ=0 l (w w +w w )−(w w +w w )
m β l,m l+1,m l,m+1 l+1,m+1 l,m l,m+1 l+1,m l+1,m+1
m
α2 β2
H3δ6=0 l w w + m+1w w
m β2 l,m l+1,m β2 l,m+1 l+1,m+1
m (cid:18) m (cid:19)
β α4
− m+1 (w w +w w )+δ l −1
β l,m l,m+1 l+1,m l+1,m+1 β4
m (cid:18) m (cid:19)
α
Q1δ=0 l (w −w )(w −w )
m β l,m l,m+1 l+1,m l+1,m+1
m
−(w −w )(w −w )
l,m l+1,m l,m+1 l+1,m+1
α2 β β
Q1δ6=0 l w − m+1w w − m+1w
m β2 l,m β l,m+1 l+1,m β l+1,m+1
m (cid:18) m (cid:19)(cid:18) m (cid:19)
β δα2 α2
− m+1 (w −w )(w −w )+ l l −1
β l,m l+1,m l,m+1 l+1,m+1 β2 β2
m m (cid:18) m (cid:19)
α β2 β2
Q2 l w − m+1w w − m+1w
m β l+1,m β2 l+1,m+1 l,m β2 l,m+1
m (cid:18) m (cid:19)(cid:18) m (cid:19)
β2
− m+1 (w −w )(w −w )
β2 l,m l+1,m l,m+1 l+1,m+1
m
α α β2 β2
− l l −1 w +w + m+1w + m+1w
β β l,m l+1,m β2 l,m+1 β2 l+1,m+1
m (cid:18) m (cid:19)(cid:18) m m (cid:19)
α α α2 α
− l l −1 l − l +1
β β β2 β
m (cid:18) m (cid:19)(cid:18) m m (cid:19)
Table 1. A list of various lattice equations (taken from [3, 4]) in
a suitable form for non-autonomous reductions.
Witheachlatticeequationwrittenintermsofα −β orα /β , thenecessary
l m l m
requirement for (0.2) to be consistent is the requirement that
α −β =α −β ,
l m l+s1 m+s2
in the additive case, and
α α
l = l+s1 ,
β β
m m+s2
8 C.M.ORMEROD,P.H.VANDERKAMP,ANDG.R.W.QUISPEL
in the multiplicative case. By a separation of variables argument, we define h and
q by letting
(1.11a) α −α =β −β :=habg,
l+s1 l m+s2 m
α β
(1.11b) l+s1 = m+s2 :=qabg,
α β
l m
in the additive and multiplicative cases respectively. Although it is not a technical
requirement, we will assume that h is not 0 and that q is not a root of unity. We
solve the additive and multiplicative case by letting
α =hlb+a , β =hma+b ,
l l m m
α =a qbl, β =b qam,
l l m m
wherea andb aresequencesthatareperiodicoforders ands respectively(not
l m 1 2
relatedtotheconstants,aandb). Thischoiceofα andβ ensurestheconsistency
l m
of the reduction with as many degrees of freedom as the sum of the orders of the
difference equations satisfied by α and β , (1.11a) and (1.11b), i.e., s +s .
l m 1 2
To provide a non-autonomous Lax pair, we need to choose a spectral variable,
x, in a manner that couples a linearly independent direction with the spectral
variable, γ. While any linearly independent direction may be considered a valid
choice,we presenta simple choice. Our choice ofspectralparameteris specified by
introducing the variable k =l and x=hbk−γ in the additive case and x=qbk/γ
in the multiplicative case. In the additive case
L =L (α −γ)7→L (a +x),
l,m l,m l n l
M =M ((β −α )+(α −γ))7→M (x+hn+b ),
l,m l,m m l l n m
and in the multiplicative case
L =L (α /γ)=L (a x),
l,m l,m l l,m l
M =M ((β /α)(α /γ))=M (b xqn).
l,m l,m m l l l,m m
This gives us a non-standard Lax pair, which, in the additive case reads
Y (x+abgh)=A (x)Y (x),
n n n
Y (x+cbh)=B (x)Y (x),
n+1 n n
and in the multiplicative case reads
Y (qabgx)=A (x)Y (x),
n n n
Y (qcbx)=B (x)Y (x),
n+1 n n
where
s2−1 s1−1
(1.12a) A (x)←[ M L ,
n l+s1,m+j l+i,m
j=0 i=0
Y Y
d−1 c−1
(1.12b) B (x)←[ M L .
n l+c,m+j l+i,m
j=0 i=0
Y Y
The compatibility conditions,
A (x+cbh)B (x)=B (x+abgh)A (x),
n+1 n n n
A (qcbx)B (x)=B (qabgx)A (x),
n+1 n n n
ON LAX REPS. OF REDUCTIONS OF INTEGRABLE LATTICE EQUATIONS 9
in the additive and multiplicative cases respectively, gives us (1.3). This choice
of spectral variable has the advantage that the spectral matrix and deformation
matrix, A (x) and B (x), have a simple dependence on the independent variable,
n n
n.
2. Some simple examples
Inthissection,wepresentsomeexamplesofthetheoryabove. Anexamplethat
hasappearedrecentlyistheexampleofq-P asareductionofthediscretemodified
VI
Korteweg-deVriesequation[30],whichalsogaverise,viaultradiscretization,tothe
first known Lax representation of u-P .
VI
2.1. Autonomousexample. Weconsidersomeadditiveexamples,inpartic-
ular, we will consider reductions of the discrete potential Korteweg-de Vries equa-
tion,
(2.1) (w −w )(w −w )=α−β,
l,m l+1,m+1 l+1,m l,m+1
labelled as H1 in table 1, which possesses a Lax representation of the form (1.4)
a
where L and M are specified by
l,m l,m
w α−γ−w w
(2.2a) L = l,m l,m l+1,m ,
l,m 1 −w
l+1,m
(cid:18) (cid:19)
w β−γ−w w
(2.2b) M = l,m l,m l,m+1 .
l,m 1 −w
l,m+1
(cid:18) (cid:19)
Let us consider a reduction, (0.2), where s = 2 and s = 1, with a labelling
1 2
indicated in figure 3. This gives us g =1, a=2 and b=1, hence n=2m−l, and
the direction that characterises the generating shift, (c,d), is chosen to be (1,1).
w w
4 3
w w w w
4 3 2 1
w w w w
3 2 1 0
w w
1 0
Figure 3. Thelabellingofinitialconditionswith(2,1)periodicity
and an evolution in the (1,1)-direction.
The product formula for the monodromy matrix, A , and the matrix that is
n
related to the generating shift, B , are
n
A =M L L
n l+2,m l+1,m l,m
w β−γ−w w w α−γ−w w
= n n n+2 n+1 n n+1
1 −w 1 −w
n+2 n
(cid:18) (cid:19)(cid:18) (cid:19)
w α−γ−w w
n+2 n+1 n+2 ,
1 −w
n+1
(cid:18) (cid:19)
B =M L
n l+1,m l,m
w β−γ−w w w α−γ−w w
= n+1 n+1 n+3 n+2 n+1 n+2 .
1 −w 1 −w
n+3 n+1
(cid:18) (cid:19)(cid:18) (cid:19)
10 C.M.ORMEROD,P.H.VANDERKAMP,ANDG.R.W.QUISPEL
The compatibility condition, given by (1.8), reads
M L L M L
l+3,m+1 l+2,m+1 l+1,m+1 l+1,m l,m
=M L M L L .
l+3,m+1 l+2,m+1 l+2,m l+1,m l,m
This simplifies to
L M =M L ,
l+1,m+1 l+1,m l+2,m l+1,m
w α−γ−w w w β−γ−w w
n+3 n+2 n+3 n+1 n+1 n+3
1 −w 1 −w
n+2 n+3
(cid:18) (cid:19)(cid:18) (cid:19)
w β−γ−w w w α−γ−w w
= n n n+2 n+1 n n+1 ,
1 −w 1 −w
n+2 n
(cid:18) (cid:19)(cid:18) (cid:19)
whichdefinestheevolutionofthisautonomousreductiontobegivenbytheequation
(2.3) (w −w )(w −w )=α−β.
n n+3 n+1 n+2
If we let y =w −w , this equation is equivalent to
n n n+1
α−β
(2.4) y +y +y = ,
n−1 n n+1
y
n
which is a well known example of a second order difference equation of QRT type.
2.2. Nonautonomous example. The autonomous equation and Lax repre-
sentationgeneralisenaturallytothe non-autonomouscaseby replacingαandβ by
α and β respectively. Furthermore, we may satisfy the periodicity constraint,
l m
α −α =β −β :=2h.
l+2 l m+1 m
We solve this constraint by letting
α =hl+a , β =2hm+b ,
l l m m
where a is periodic of order two and b is constant, and hence, may be taken to
l m
be 0 without loss of generality. The evolution equation for this system may be
represented as an application of the nonautonomous version of (2.1) translated by
the vector (1,1);
(2.5) (w −w )(w −w )=α −β .
n n+3 n+1 n+2 l+1 m+1
recalling that n = 2m−l. The increment in l and m by 1 directly corresponds to
the increment in n by 1. In the simplest case where a = a is constant (rather
l 1
than periodic), letting
y =w −w ,
n n n+1
results in the evolution equation
α −β −hn−h+a
l+1 m+1 1
y +y +y = = ,
n n+1 n+2
y y
n+1 n+1
or alternatively
−hn+a
1
(2.6) y +y +y = .
n−1 n n+1
y
n
To form the Lax pair for this reduction, we choose a spectral variable to be
x=hl−γ.