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We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all 2x2 Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems that can be

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Symmetry, Integrability and Geometry: Methods and Applications SIGMA
7
(2011), 089, 12 pages
A Completeness Study on Certain 2
×
2 Lax PairsIncluding Zero Terms
Mike C. HAY Institute of Mathematics for Industry, Kyushu University, Ito Campus,744 motooka, nishi-ku, Fukuoka, 819-0395, Japan
E-mail:
[email protected]
Received April 09, 2011, in final form September 04, 2011; Published online September 14, 2011http://dx.doi.org/10.3842/SIGMA.2011.089
Abstract.
We expand the completeness study instigated in [
J. Math. Phys.
50
(2009),103516, 29 pages] which found all 2
×
2 Lax pairs with non-zero, separable terms in each entryof each Lax matrix, along with the most general nonlinear systems that can be associatedwith them. Here we allow some of the terms within the Lax matrices to be zero. We coverall possible Lax pairs of this type and find a new third order equation that can be reducedto special cases of the non-autonomous lattice KdV and lattice modified KdV equationsamong others.
Key words:
discrete integrable systems; Lax pairs
2010 Mathematics Subject Classiﬁcation:
37K15; 39A14; 70H06
1 Introduction
Because this article expands on the notions introduced in [1], we refer the reader to that paperfor a more in depth introduction and problem description. The difference between the earlierwork and this one is that in the earlier study all of the terms in each entry of the Lax matriceswere assumed to be non-zero. Here we extend the analysis on 2
×
2 Lax pairs with a separableterm in each entry of each matrix by allowing terms to be zero. Including zero terms covers someasymmetric Lax pairs that did not arise in the previous study, some of which are associated withquad graph equations that fall outside the Adler–Bobenko–Suris (ABS) list[2]because they do
not possess the tetrahedron property, or are not even multidimensionally consistent.Briefly, the procedure used to analyze the Lax pairs runs as follows. The compatibilitycondition produces four expression, one for each entry of the 2
×
2 Lax matrices, see (3.1) below.Each of these expressions can be split into smaller equations in various ways depending on howthe spectral parameter is chosen. Thus, the compatibility condition defines various systems of equations that we subsequently solve, in a manner that preserves their full generality, up toa point where a nonlinear evolution equation is apparent, or it has been shown that the systemcannot be associated with a nonlinear system. Testing all combinations of terms, we therebysurvey the complete set of Lax pairs of the type described.The paper is organized as follows. Section2summarizes the chief results including a newthird order system and a theorem covering the relevant Lax pairs. Section3describes themethod used to identify and analyze all of the Lax pairs considered. Section4places the resultsfound here in the context of those in the literature. And a discussion rounds out the paper.
This paper is a contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)” (June 14–18, 2010, Varna, Bulgaria). The full collection is available athttp://www.emis.de/journals/SIGMA/SIDE-9.html
2 M.C. Hay
2 Results
There are two principal results reported in this paper. The first is an apparently new partialdifference equation which can be conveniently written as a coupled pair as followsˆ¯
x
ˆ
x
+
y
=ˆ¯
x
¯
x,
¯
xx
+ ¯
y
=ˆ
xx,
(2.1)where
x
(
l,m
) and
y
(
l,m
) are the dependent variables over the discrete independent variables
l
and
m
, shifts in which are represented by¯
·
andˆ
·
respectively. One may easily isolate
y
fromeither equation and use it to form a single equation in
x
which is second order in
l
and first orderin
m
. The derivation of (2.1) is given in Section3.1.1. We prefer to write the system as a pair of
equations because from this form it is easier to see how it can be reduced to scalar equations onquad graphs, see Section4. Note that no parameters or arbitrary functions, which are usuallyassociated with the lattice spacing, are present in (2.1) because they can all be removed bya gauge. However, they do appear in the reduced equations on quad graphs in Section4.A Lax pair for (2.1) is
L
=
¯
x/x ν ν
0
, M
=
ˆ
x/x ν ν y
,
(2.2)where
ν
is the spectral parameter.Also relevant to this article are the higher order analogues of the lattice modified KdVequation (LMKdV) and lattice sine-Gordon equation (LSG), called LMKdV
2
and LSG
2
, foundin[1].
This article’s second principal result is the following theorem.
Theorem 1.
The system of equations that arise via the compatibility condition of any
2
×
2
Lax pair with separable terms in each entry of each matrix
(
as described in
[1])
is either trivial,underdetermined, overdetermined, or can be reduced to one of
LMKdV
2
or
LSG
2
, a degenerate form of those equations, or a transformation from one of those.
The transformations we refer to in Theorem1are M¨obius, Miura or B¨acklund transforma-
tions, which allow one to use the same Lax pair for the new equation, after the change of variables has been applied to it as well.For the cases when none of the entries in the Lax matrices are zero, the proof of Theorem1was carried out in [1] and depends on considering all of the possible sets of equations that canarise from the compatibility condition of such Lax pairs. Section3of this paper details the proof for the cases when at least one term in at least one of the Lax matrices takes the value zero.
3 Analysis of all relevant Lax pairs
This paper relies heavily on the methodology reported in [1], see Section 2 of that paper fora detailed description of the techniques used here.As the compatibility condition (CC) is central to the entire thread of this paper, we write itout here and refer back to it throughout. Taking the general form of the Lax matrices to be
L
=
aA bBcC dD
, M
=
α
Λ
β
Ξ
γ
Γ
δ
∆
,
where, as throughout this paper, lower case letters depend on the lattice variables only,
a
=
a
(
l,m
), and upper case letters depend on the spectral variable only,
A
=
A
(
ν
).
A Completeness Study on Certain 2
×
2 Lax Pairs Including Zero Terms 3Separating the CC,
LM
=
ML
, into each of its entries gives:(1
,
1) ˆ
aαA
Λ +ˆ
bγB
Γ =
a
¯
αA
Λ +
c
¯
βC
Ξ
,
(3.1a)(1
,
2) ˆ
aβA
Ξ +ˆ
bδB
∆ =
b
¯
αB
Λ +
d
¯
βD
Ξ
,
(3.1b)(2
,
1)ˆ
dγD
Γ + ˆ
cαC
Λ =
c
¯
δC
∆ +
a
¯
γA
Γ
,
(3.1c)(2
,
2)ˆ
dδD
∆ + ˆ
cβC
Ξ =
d
¯
δD
∆ +
b
¯
γB
Γ
.
(3.1d)To solve the systems of equations that arise via the CC of each Lax pair, we use the methoddescribed in [1, Section III].We will examine Lax pairs on a case by case basis based on the number of their entries whichare zero. The following proposition is therefore useful.
Proposition 1.
If there are two or more zero entries in either Lax matrix then the resulting compatibility condition is either linear or underdetermined.
Proof.
Let the required two zero terms reside in the matrix
L
. Then there are three cases thatneed to be considered.
Case 1.
Diagonal entries of
L
are zero. Applying the compatibility condition to a Lax pairwith zero diagonals is achieved by setting
A
=
D
= 0 in (3.1). This shows that all the resultingconditions are linear. Thus, this Lax pair cannot be associated with a nonlinear system, unlessthe nonlinearity is arbitrarily introduced into a Lax pair that possesses excess freedom. Wecall such Lax pairs with excess freedom ‘false’ (see the argument in AppendixA,case 4, or
Section 4.3 of [1]). We remark that two linear equations may result in a nonlinear equation solong as one is additively linear and the other is multiplicative. An example of this can be seenin[3,equation (7)]. However, in this case, all equations are linear in a multiplicative sense.
Case 2.
Off diagonal entries are zero. Like the previous case, this also leads to linearconditions only.
Case 3.
One row or column is zero. Let the right column of
L
be zero, i.e. set
B
and
D
to zero in (3.1). Then the (1
,
2) entry of (3.1) is
aβA
Ξ = 0, which we solve by taking Ξ = 0because allowing
A
= 0 would cause the remaining conditions to all be linear. Now the (1
,
2)and (2
,
2) entries are both identities while the (1
,
1) entry gives a condition that can be solvedby introducing the new dependent variable
v
(
l,m
) as follows(1
,
1) ˆ
aα
=
a
¯
α
⇒
a
=
λ
¯
v/v, α
=
µ
ˆ
v/v,
where, as throughout this paper,
λ
=
λ
(
l
) and
µ
=
µ
(
m
) are arbitrary functions. Finally,consider the (2
,
1) entry of the compatibility conditionˆ
cαC
Λ =
a
¯
γA
Γ +
c
¯
δC
∆
.
We are able to choose the spectral terms such that all of the lattice terms remain together inone equation, or such that they are separated into more than one equation. The former caseyields the sole conditionˆ
cαc
¯
δ
=
a
¯
γ c
¯
δ
+ 1
,
but freedom remains here to the extent that any equation can be written into this form bychoosing the remaining lattice terms appropriately. Thus, this is a false Lax pair. The lattercase yields more equations, which may appear to solve the problem of having excess freedom,however, the resulting equations must all be linear, resulting in a trivial system.
We are now in a position to analyze all of the possible Lax pairs of the category describedin the introduction. Since the more interesting results occur when there are less zero entries inthe Lax matrices, we start with just a single zero in one matrix and work our way toward thesimpler systems.
4 M.C. Hay
3.1 A single zero term in one Lax matrix only
We assume that the zero lies in the
L
matrix. There are only two cases to consider, one withthe zero in a diagonal entry and one in an off-diagonal entry.
3.1.1 A zero in a diagonal entry
L
=
aA bBcC
0
,
M
=
α
Λ
β
Ξ
γ
Γ
δ
∆
. Setting
D
= 0 in (3.1) shows that the spectral terms prod-ucts in the (2,2) entry must be equal, i.e.
B
Γ =
C
Ξ. This can be seen by making considerationssimilar to those in[1,Section II]. As such, there are four cases that require further consideration,
these are listed in Table1.The analysis is similar for all four cases listed but only the first
yields a major result, so cases 2, 3 and 4 are dealt with in AppendixA, while case 1 is describedbelow.
Table 1.
The four cases that require consideration when
D
= 0 in (3.1). The lines represent propor-tionality between the spectral term products. In cases 3 and 4,
A
Ξ and
A
Γ, respectively, are split intotwo terms not proportional to one another.
Case(1
,
1)
A
Λ
B
Γ(1
,
2)
A
Ξ
B
Λ
B
∆
(2
,
1)
A
Γ
C
Λ
C
∆
Result1 not
∝
new nontrivial system2
∝
underdetermined3 not
∝
false Lax pair4
∝
false Lax pair
Case 1.
Starting with the CC (3.1), setting
D
= 0 and separating the (1,1) entry into twoequations yields the following(1
,
1)
1
ˆ
aα
=
a
¯
α,
(1
,
2) ˆ
aβ
+ˆ
bδ
=
b
¯
α,
(1
,
1)
2
ˆ
bγ
=
c
¯
β,
(2
,
1) ˆ
cα
=
a
¯
γ
+
c
¯
δ,
(2
,
2) ˆ
cβ
=
b
¯
γ.
(3.2)Solving the diagonal entries first, which are linear in this case, we find that (1,1)
1
leads to
a
=
λ
1
¯
v/v, α
=
µ
1
ˆ
v/v,
(3.3)where we have introduced the new unknown
v
(
l,m
) and the arbitrary functions
λ
1
(
l
) and
µ
1
(
m
).Equations (1,1)
2
and (2
,
2) from (3.2) are solved in tandem leading to
b
=
λ
2
ρ
¯
tu, c
=
λ
2
ρ
¯
ut, β
=
µ
2
σ
ˆ
tu, γ
=
µ
2
σ
ˆ
ut,
(3.4)where we have introduced the new unknowns
t
(
l,m
) and
u
(
l,m
), as well as the arbitrary func-tions
λ
2
(
l
),
µ
2
(
m
),
ρ
=
λ
3
(
l
)
(
−
1)
m
and
σ
=
µ
3
(
m
)
(
−
1)
l
. Substituting (3.3) and (3.4) into the
off diagonal entries of (3.2) shows that certain combinations of variables are repeated allowingus to set
t
≡
u
≡
1, w.l.o.g. We also rename
v
=
x
,
δ
=
y
, and use gauge transformations toremove all arbitrary functions. This leads to (2.1).
A Completeness Study on Certain 2
×
2 Lax Pairs Including Zero Terms 5
3.1.2 A zero in an off-diagonal entry
L
=
aA
0
cC dD
,
M
=
α
Λ
β
Ξ
γ
Γ
δ
∆
. The equations from the CC are obtained by setting
B
= 0in (3.1), these are(1
,
1) (ˆ
aα
−
a
¯
α
)
A
Λ =
c
¯
βC
Ξ
,
(1
,
2) ˆ
aβA
Ξ =
d
¯
βD
Ξ
,
(2
,
1)ˆ
dγD
Γ + ˆ
cαC
Λ =
a
¯
γA
Γ +
c
¯
δC
∆
,
(2
,
2) (ˆ
dδ
−
d
¯
δ
)
D
∆ + ˆ
cβC
Ξ = 0
.
Since any extra zero entries in
L
is covered by Proposition1, and in
M
will bring us to oneof the Lax pairs considered in the other sections below (or Proposition1), we can make thefollowing observations regarding the spectral terms. From the (1,2) entry we have
A
=
D
. Fromthe diagonal entries we have
A
Λ =
C
Ξ =
D
∆ which implies that Λ = ∆. Turning to the (2,1)entry last, we find two pairs of repeated spectral terms
A
Γ and
C
Λ. These can either be chosento be proportional to one another, so that only one lattice term equation arises in this entry, orseparated to construct two equations.Among the lattice terms there are seven unknowns, two of which are redundant after gaugetransformations, suggesting that five is the appropriate number of equations, and that the (2,1)entry should be separated into two equations. However, we must check both cases. Let us firstexamine the case where the (2,1) entry is split up, the case where the (2,1) entry is left as a singleequation is considered at the end of this subsection.Dealing with the case where
A
Γ is not proportional to
C
∆ first, we have the following set of lattice term equations:(1
,
1) ˆ
aα
−
a
¯
α
=
c
¯
β,
(1
,
2) ˆ
aβ
=
d
¯
β,
(2
,
1)
1
ˆ
dγ
=
a
¯
γ,
(2
,
2)ˆ
dδ
−
d
¯
δ
+ ˆ
cβ
= 0
,
(2
,
1)
2
ˆ
cα
=
c
¯
δ.
The three equations in the off-diagonal entries are linear and have a general solution expressibleas
a
=
λρ
¯
uu, d
=
λρ
¯
vv, α
=¯
δ c
ˆ
c, β
=
µ
1
ˆ
uv, γ
=
µ
2
ˆ
vu,
where all of the quantities are similar to those described below (3.4). However, when these valuesare substituted into the remaining equations coming from the diagonal entries, we find that
c
and
δ
are redundant, so we set these to unity. Once again the arbitrary functions are removableby gauge transformations, so we rename
v
=
x
,
u
=
z
and the final two equations are¯
x
ˆ¯
x
−
x
ˆ
x
=ˆ
z
ˆ¯
x,
(3.5a)
z
ˆ
z
−
¯
z
ˆ¯
z
=
z
¯
x,
(3.5b)where the dependent variables,
x
and
z
, may be likened to
x
and
y
, respectively, in equation (1.2)from [1], which shows that this is a degenerate form of LSG
2
. Obviously, one may isolate either
x
or
z
from the appropriate equation in (3.5), substitute the result into the other equation andobtain an expression for one variable alone.In fact, (3.5) is linearizable as follows. Use (3.5a) in (3.5b) to obtain
ˆ
z
ˆˆ
z
=ˆ¯
x
ˆˆ¯
x
·
¯
x/
ˆ¯
x
−
x/
ˆ
x
ˆ¯
x/
ˆˆ¯
x
−
ˆ
x/
ˆˆ
x.
(3.6)

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