Peculiar symmetry structure of some known discrete nonautonomous equations

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by  Rustem N. Garifullin

Peculiar symmetry structure of some known
discrete nonautonomous equations
R.N. Garifullin1,2, I.T. Habibullin1,2 and R.I. Yamilov1
1 Ufa Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Street, Ufa 450008, Russian Federation
2 Bashkir State University,
32 Zaki Validi Street, Ufa 450074, Russian Federation
E-mails: mailto:[email protected],
5
1 mailto:[email protected], mailto:[email protected]
0 URL: http://matem.anrb.ru/garifullinrn,
2
http://matem.anrb.ru/habibullinit, http://matem.anrb.ru/en/yamilovri

n
a
January 23, 2015
J

2
2

Abstract
]
I
S We study the generalized symmetry structure of three known discrete nonau-
. tonomous equations. One of them is the semidiscrete dressing chain of Shabat. Two
n
i others are completely discrete equations defined on the square lattice. The first one
l
n is a discrete analogue of the dressing chain introduced by Levi and Yamilov. The
[ secondone is a nonautonomous generalizationof the potential discrete KdV equation

or, in other words, the H1 equation of the well-known Adler-Bobenko-Suris list. We
1
v demonstrate that these equations have generalized symmetries in both directions if
5 and only if their coefficients, depending on the discrete variables, are periodic. The
3 order of the simplest generalized symmetry in at least one direction depends on the
4
period and may be arbitrarily high. We substantiate this picture by some theorems
5
in the case of small periods. In case of an arbitrarily large period, we show that it is
0
possibletoconstructtwohierarchiesofgeneralizedsymmetriesandconservationlaws.
.
1
The same picture should take place in case of any nonautonomous equation of the
0
Adler-Bobenko-Surislist.
5
1
:
v 1 Introduction
i
X
We demonstrate that the generalized symmetry structure of some nonautonomous
r
a equations may be quite unusual by example of three known equations.
The first equation reads:
(u −u )(u −u )=α −β . (1)
n+1,m+1 n,m n+1,m n,m+1 n m
Here u is an unknown function depending on two discrete variables n,m ∈ Z,
n,m
while α ,β are the given functions depending on one discrete variable. Eq. (1) is
n m
the H1 equation of the Adler-Bobenko-Suris list [2]. In the autonomous case, it is
nothing but the discrete potential KdV equation,which has been knownmuch earlier
together with its L−A pair, see e.g. [21].
1
The second equation is the well-known dressing chain studied, e.g., in [24,25,28]:
d
(u +u )=u2 −u2 +α −α . (2)
dx n n+1 n n+1 n n+1
Herethe unknownfunction u =u (x) depends ononecontinuousx andonediscrete
n n
nvariables. Thethirdequationisacompletelydiscreteanalogueofthedressingchain:
α (u +1)(u −1)=α (u −1)(u +1), (3)
n n,m+1 n,m n+1 n+1,m+1 n+1,m
which has been introduced in [18]. In the autonomous case see, e.g., [12,21].
The equations (1) and (3) belong to the following class of discrete equations on
the square lattice:
F (u ,u ,u ,u )=0. (4)
n,m n+1,m n,m n,m+1 n+1,m+1
In autonomous case, when the function F does not depend on n,m explicitly, all
n,m
knownintegrableequationsofthisformhavetwohierarchiesofgeneralizedsymmetries,
and this property can be used as a criterion of integrability. Generalized symmetries
in the n-direction have the form
du
n,m
=Φ (u ,u ,...,u ), (5)
n,m n+k,m n+k−1,m n−k,m
dt
k
wherek ≥1,andthenumberk canbecalledtheorderofsuchsymmetry. Generalized
symmetries of an order l≥1 in the m-direction have the form
du
n,m
=Ψ (u ,u ,...,u ). (6)
n,m n,m+l n,m+l−1 n,m−l
dτ
l
In most of autonomous integrable cases, the simplest generalized symmetries in
both directions have the orders k = l = 1, see e.g. [16,19,29]. These symmetries
correspond to integrable Volterra type equations of a complete list obtained in [31],
seealsothereviewarticle[33]. Thereareafewexampleswiththesimplestsymmetries
of orders k =l =2, see [1,20,26]. Up to now there has been known the only example
withanessentiallyasymmetricstructureofgeneralizedsymmetries. Ithasbeenfound
in [8], see also [7]. In that example, the orders of simplest symmetries are different
(k = 2 and l = 1), and examples we discuss in this article will be asymmetric in the
same sense.
Asforthenonautonomouscase,thesituationisdifferent. Weknownonautonomous
examples of the form (4) with two hierarchies of generalized symmetries [9,30]. How-
ever,there aresome knownintegrablenonautonomousequations whichhaveonly one
hierarchy of generalized symmetries or have no hierarchy at all. This is the case of
nonautonomous equations of the Adler-Bobenko-Suris list. It has been hypothesized
in [22], and this will be confirmed in the present paper, that there is no general-
ized symmetry in the n-direction when α is an arbitrary n-dependent function and
n
no generalized symmetry in the m-direction when β is an arbitrary m-dependent
m
function.
In this paper, instead of arbitrary functions α ,β in eq. (1), we consider the
n m
concrete ones. We look for functions α ,β , such that the corresponding equation
n m
(1) has two hierarchies of generalized symmetries. We prove that symmetries of the
form (5) and (6) exist if and only if α and β are the periodic functions. We do
n m
this for some low orders k and l only. The orders k and l of the simplest generalized
symmetries (5) and (6) depend on the periods of α and β and may be arbitrarily
n m
high as well as different.
For eq. (3) the picture is similar. In case of eq. (2), the form of symmetries is
different, but the results are quite analogous too.
2
In case of the periodic coefficients α and β in eqs. (1,2,3), whose periods may
n m
be arbitrarily large, we demonstrate that two hierarchies of generalized symmetries
andconservationlawscanbeconstructedbyusingknownnonautonomousL−Apairs
of these equations. We do that, using a method presented in [11].
It seemshighly probablethattwo hierarchiesofconservationlawsalsoexistif and
onlyifthecoefficientsofeqs. (1,2,3)areperiodic. Thispropertyhasbeenconfirmedin
asensein[23],whereithasbeenshownforeq. (1)thatso-calledfive-pointconservation
laws disappear when the coefficients α and β become nonconstant.
n m
As a result of our investigation we come to an opinion that eqs. (1,2,3) with
the periodic coefficients are “more integrable”. In this case we can derive, in both
directions,generalizedsymmetriesandconservationlawsfromtheirL−Apairs,while
in general case those L−A pairs seem to be much more inconvenient for use.
In Section 2 we prove a few theorems showing that two hierarchies of generalized
symmetries of eq. (1) exist only in the case of periodic coefficients. In particular, we
construct an interesting example with a simplest generalized symmetry of the second
order in one direction and of the third order in the second one. In Sections 3,4 we
proveanaloguestheoremsforeqs. (2,3). InSection5weexplainhowtoconstructtwo
hierarchies of generalized symmetries and conservation laws for eqs. (1,2,3) with the
periodic coefficients. Examples of generalized symmetries of low orders together with
their L−A pairs are constructed for such equations in Section 6. Conservation laws
of low orders are given in Section 7. The nature of some generalized symmetries of
eqs. (1,2,3) with the periodic coefficients is discussed in Section 8.
2 H1 equation
We study in this section the H1 equation (1) for which we use a natural assumption:
α 6=β (7)
n m
for any n,m ∈ Z. In opposite case the equation becomes degenerate in some points,
see e.g. [19] for the autonomous case and [9] for the nonautonomous one.
Inthe autonomouscase,generalizedsymmetriesofthe Adler-Bobenko-Surisequa-
tions and of the H1 equation, in particular, were constructed in [14,15,22,27]. Here
welookforgeneralizedsymmetriesoftheH1equationinthenonautonomouscaseand
obtain, as a result, some statements on the symmetry structure of this equation.
Due to the invariance of eq. (1) with respect to the involution n↔m, α ↔β ,
n m
we can restrict ourselves to generalized symmetries of the form (5). According to its
definition, see e.g. [18], the symmetry (5) of eq. (4) must satisfy the relation
∂F ∂F ∂F ∂F
n,m n,m n,m n,m
Φ + Φ + Φ + Φ =0 (8)
n+1,m n,m n,m+1 n+1,m+1
∂u ∂u ∂u ∂u
n+1,m n,m n,m+1 n+1,m+1
on the solutions of eq. (4) and for all n,m∈Z.
It is natural to assume for the symmetries (5) of order k that Φ depends on
n,m
both u and u at at least one point n,m. We prove theorems below under
n+k,m n−k,m
a stronger nondegeneracy condition for such generalized symmetries:
∂Φ ∂Φ
n,m 6=0 and n,m 6=0 for all n,m∈Z. (9)
∂u ∂u
n+k,m n−k,m
In all known cases, if an equation of the form (4) has a generalized symmetry (5) of
an order k ≥1, then it has the nondegenerate symmetry of the same order.
We find generalized symmetries by using a scheme developed in [6,19]. Some
annihilation operators [10] play an important role in this scheme.
3
Eq. (1) has the following point symmetry:
d
u =ν +ν (−1)n+m+ν u (−1)n+m, (10)
n,m 1 2 3 n,m
dt
0
whereν ,ν ,ν arearbitraryconstants. We write downbelowgeneralizedsymmetries
1 2 3
up to this point one.
2.1 First and second order generalized symmetries
Here we get some theoretical results in case of the first and second order generalized
symmetries. The following result has been obtained in [22], and we present it below
for completeness.
Theorem 1 Eq. (1,7) has a first order nondegenerate generalized symmetry in the
n-direction iff α ≡α .
n n−1
Sketch of proof. The compatibility condition (8) for eqs. (1) and (5) implies:
∂Φ ∂Φ
n,m n,m
(α −α ) ≡0, (α −α ) ≡0, (11)
n n−1 n n−1
∂u ∂u
n+1,m n−1,m
and we get the first part of the theorem. On the other hand, eq. (1) with α ≡α
n n−1
has, for any β , the generalized symmetry
m
d 1
u = (12)
n,m
dt u −u
1 n+1,m n−1,m
which is nondegenerate.
Remark. The same result can be obtained under a weakerassumption instead of
the nondegeneracy condition (9) with k =1. For example, we can assume that there
exists m, such that
∂Φ ∂Φ
n,m n,m
6=0 or 6=0 (13)
∂u ∂u
n+1,m n−1,m
for any n. In this case we also derive α ≡α from eq. (11).
n n−1
In the case α ≡α ≡α and β ≡β ≡β, there is one more symmetry:
n n−1 m m−1
d 2n(β−α)
u = +u . (14)
dt′ n,m u −u n,m
1 n+1,m n−1,m
Any symmetry of the order k ≤ 1 of eq. (1,7) with α ≡ α is, up to a point
n n−1
symmetry (10), the following linear combination with constant coefficients µ ,µ :
1 2
du du du
n,m n,m n,m
=µ +µ . (15)
dt′′ 1 dt 2 dt′
1 1 1
Eq. (12)isaknownintegrableequationoftheVolterratype[31,33]. Eq. (14)isits
knownmastersymmetry[3]. Itgeneratesgeneralizedsymmetriesnotonlyforeq. (12)
butalsoforthediscreteequation(1,7)withα ≡αandanarbitraryβ . Forexample,
n m
itcanbe checkedby directcalculationthat the followingequation,constructedinthe
standard way,
d d d d d
u = u − u (16)
dt n,m dt dt′ n,m dt′ dt n,m
2 1 1 1 1
is the second order generalized symmetry for both of these equations.
Below we consider the two-periodic case: α ≡α . There we have two possi-
n+1 n−1
bilities. First of them is α = α , hence α ≡ α is a constant, and we are led to the
0 1 n
previous one-periodic case. In the second case α 6=α , then α 6=α for any n.
0 1 n+1 n
4
Theorem 2 The following two statements take place:
1. If eq. (1,7) has a nondegenerate generalized symmetry of order 2 in the n-
direction, then α ≡α .
n+1 n−1
2. If α in eq. (1,7) satisfies the conditions
n
α ≡α and α 6=α , (17)
n+1 n−1 0 1
theninthen-directionthereexiststhenondegeneratesecondordersymmetryand
there is no symmetry of the first order.
Proof. We can derive from the compatibility condition (8) the following relations:
∂Φ ∂Φ
n,m n,m
(α −α ) ≡0, (α −α ) ≡0,
n+1 n−1 n n−2
∂u ∂u
n+2,m n−2,m
which provide the first part of the theorem. In case (17) we have the second order
symmetry
d c (u −u )
n n,m n+2,m
u =
n,m
dt γ +(u −u )(u −u )
2 n n−1,m n+1,m n,m n+2,m
(18)
c (u −u )
n−1 n,m n−2,m
+ ,
γ +(u −u )(u −u )
n n−1,m n+1,m n,m n−2,m
where c ≡ c is an arbitrary two-periodic function, and γ = α −α 6= 0 for
n+2 n n n+1 n
any n. This formula yields the nondegenerate symmetries of order 2 (e.g. if c ≡1 or
n
c =2+(−1)n).
n
In the case β ≡β we have an additional symmetry:
m
d n(β−α )(u −u )
n+1 n,m n+2,m
u =
dt′ n,m γ +(u −u )(u −u )
2 n n−1,m n+1,m n,m n+2,m (19)
(n−1)(β−α )(u −u )
n n,m n−2,m
+ −u .
n,m
γ +(u −u )(u −u )
n n−1,m n+1,m n,m n−2,m
Any symmetry of the order k ≤2 of eq.(1,7,17) is a linear combination of (10,18,19).
For this reason, there is no first order symmetry.
Letusnotethatifα ≡α ,theneqs. (18,19)turninto(12,14). Theformula(18)
n+1 n
provides two linear independent and commuting symmetries of the discrete equation.
Eq. (19)shouldbethemastersymmetryfor(18),providinggeneralizedsymmetries
of even orders not only for (18) but also for eq. (1,7,17) with an arbitrary β . We
m
have checked that by direct calculation in the first step, constructing a fourth order
generalized symmetry.
2.2 An example with asymmetric symmetry structure
We consider here eq. (1,7) satisfying the conditions
α ≡α , β ≡β . (20)
n+3 n m+2 m
We also require:
α 6=α , α 6=α , α 6=α , β 6=β . (21)
0 1 0 2 1 2 0 1
This provides that
α −α ≡α −α 6=0, β 6=β
n+2 n n−1 n m m−1
5
for all n,m. Taking into account the condition (7), we see that all the five numbers
α ,α ,α ,β ,β must be different.
0 1 2 0 1
There is in the n-direction the following generalized symmetry:
du (v v +γ )a
n,m n+2,m n+1,m n+1 n+1
=
dt v v v −v γ +v γ
3 n+2,m n+1,m n,m n+2,m n+2 n,m n+1
v v a
n+1,m n−1,m n
+
vn+1,mvn,mvn−1,m+vn+1,mγn+1+vn−1,mγn (22)
(v v −γ )a
n−1,m n−2,m n n−1
+ ,
v v v −v γ −v γ
n,m n−1,m n−2,m n,m n n−2,m n−1
v =u −u , γ =α −α , a ≡a .
n,m n+1,m n−1,m n n+1 n n+3 n
Itisoftheorderk=3andisnondegenerateinparticularcases,e.g. a ≡1. Thereare
n
herethreelinearindependentandcommutinggeneralizedsymmetries. Anysymmetry
in the n-direction of an order k ≤ 3 is a linear combination of eqs. (22) and (10).
That is why there is no symmetry of the orders k =1 and k =2.
There is in the m-direction the following generalized symmetry of the form (6):
du w b w b
n,m n,m+1 m+1 n,m−1 m
= + ,
dτ2 wn,m+1wn,m+δm wn,mwn,m−1−δm (23)
w =u −u , δ =β −β , b ≡b .
n,m n,m+1 n,m−1 m m+1 m m+2 m
It is of the order l = 2 and is nondegenerate in particular cases, e.g. b ≡ 1. We
m
have here two linear independent and commuting symmetries. Any symmetry in the
m-direction of an order l ≤ 2 is a linear combination of eqs. (23) and (10). For this
reason there is no symmetry of the order l =1.
The results can be formulated as follows:
Theorem 3 Eq. (1,7,20,21) has in the n-direction a nondegenerate generalized sym-
metry of the order k = 3 and has no symmetry of the orders k = 1,2. This equation
possesses in the m-direction a nondegenerate symmetry of the order l =2 and has no
symmetry of the order l =1.
3 Dressing chain
Inthissectionwediscussthedressingchain(2). Fromtheviewpointofthegeneralized
symmetryproperties,equationsoftheform(4)arethediscreteanaloguesofhyperbolic
type equations:
u =f(x,y,u,u ,u ).
xy x y
Eq. (2) belongs to the class of equations
u =f (x,u ,u ,u ) (24)
n+1,x n n n+1 n,x
whichare,inthesamesense,thesemidiscreteanaloguesofhyperbolictypeequations.
Intheautonomouscase,allintegrableequationsofthesethreeclassesshouldhavetwo
hierarchies of generalized symmetries in two different directions. In the paper [32] a
numberofautonomousexamplesofthe form(24), includingeq. (2)withthe constant
α , have been presented together with two generalized symmetries in two different
n
directions.
Eq. (2) is a nonautonomousrepresentativeof the class(24). A hierarchyof gener-
alized symmetries in the x-direction exists for any α , and the simplest equation has
n
the form
du d3u du
n = n −6(u2 +α ) n. (25)
dθ dx3 n n dx
6
For any fixed n this equation is nothing but the well-known modified Korteweg–de
Vries equation. As it will be shown below, symmetries of eq. (2) in the n-direction
disappear in the generic case, i.e. when α is an arbitrary function. We will search
n
the functions α , such that generalized symmetries in the n-direction exist.
n
Generalized symmetries of eqs. (24) in the n-direction of an order k ≥1 have the
form
du
n
=Φ (x,u ,u ,...,u ). (26)
n n+k n+k−1 n−k
dt
k
It is natural to assume for such symmetries that Φ depends on both u and u
n n+k n−k
in at least one point n. We prove theorems below under the following nondegeneracy
condition for the generalized symmetries:
∂Φ ∂Φ
n 6=0 and n 6=0 for all n∈Z. (27)
∂u ∂u
n+k n−k
In all integrable cases we know, if an equation (24) has a generalized symmetry (26)
of an order k ≥1, then it has the nondegenerate symmetry of the same order.
The generalized symmetry (26) of eq. (24) must satisfy to the compatibility con-
dition
d ∂f ∂f ∂f d
n n n
Φ = Φ + Φ + Φ (28)
n+1 n n+1 n
dx ∂u ∂u ∂u dx
n n+1 n,x
on any solution of eq. (24) and for all n.
3.1 First order generalized symmetries
Theorem 4 Eq. (2) has a first order nondegenerate generalized symmetry in the
n-direction iff α ≡α .
n n+1
Proof. We can derive from the compatibility condition (28) the relations
∂Φ ∂Φ
n n
(α −α ) ≡0, (α −α ) ≡0
n+1 n n n−1
∂u ∂u
n+1 n−1
whichprovidethefirstpartofthetheorem. Ontheotherhand,eq. (2)withα ≡α
n n+1
has the following nondegenerate generalized symmetry:
du 1 1
n
= − . (29)
dt u +u u +u
1 n+1 n n n−1
Intheautonomouscase,bothsymmetries(25)and(29)ofeq. (2)havebeenfound
in [32].
There is one more generalized symmetry:
du n n−1
n
= − , (30)
dt′ u +u u +u
1 n+1 n n n−1
andanysymmetryofanorderk ≤1ofeq. (2)withα ≡α isalinearcombination
n n+1
of (29,30). Eq. (30) provides the master symmetry for eq. (29). For example,
du d d d d 1
n
= − u =
dt dt dt′ dt′ dt n (u +u )(u +u )2
2 (cid:18) 1 1 1 1(cid:19) n+2 n+1 n+1 n (31)
u −u 1
n+1 n−1
− −
(u +u )2(u +u )2 (u +u )2(u +u )
n+1 n n n−1 n n−1 n−1 n−2
isthegeneralizedsymmetrynotonlyforeq. (29)butalsofortheautonomoussemidis-
crete equation (2).
7
3.2 Second order generalized symmetries
Theorem 5 The following two statements take place:
1. If eq. (2) has a nondegenerate generalized symmetry of the second order in the
n-direction, then α ≡α .
n+1 n−1
2. If α of eq. (2) satisfies the conditions
n
α ≡α and α 6=α , (32)
n+1 n−1 0 1
then there exists in the n-direction a nondegenerate second order symmetry and
there is no symmetry of the first order.
Proof. The compatibility condition (28) implies
∂Φ ∂Φ
n n
(α −α ) ≡0, (α −α ) ≡0,
n+2 n n n−2
∂u ∂u
n+2 n−2
and we get the first part of the theorem.
In the case (32) all symmetries of orders k ≤2 are described as follows:
du a (u +u )
n n+2 n+2 n+1
=
dt γ +(u +u )(u +u )
2 n n+2 n+1 n+1 n
(33)
a (u −u ) a (u +u )
n+1 n+1 n−1 n n−1 n−2
+ − ,
γ −(u +u )(u +u ) γ +(u +u )(u +u )
n n+1 n n n−1 n n n−1 n−1 n−2
where γ =α −α 6=0 for all n. The function a is givenby a =b +cn, where
n n+1 n n n n
b is an arbitrary two-periodic function and c is an arbitrary constant.
n
There are here nondegenerate examples of the second order, e.g. a ≡ 1 or a =
n n
2+(−1)n, but there is no symmetry of the first order.
In the case when a is the two-periodicfunction, i.e. a =b , we have in(33)two
n n n
linear independent and commuting generalized symmetries. The linear case a = cn
n
provides the master symmetry for eq. (33) with a = b . This master symmetry
n n
generates symmetries not only for eq. (33) with a =b but also for the semidiscrete
n n
equation (2,32). We have checked that on the first step, constructing a fourth order
generalized symmetry.
4 Discrete dressing chain
In this section we discuss eq. (3) with α 6=0 for any n. In [18] a complete analogue
n
of the dressing chain (2) has been introduced in the following form:
(v +d )(v −d )=(v −d )(v +d ). (34)
n+1,m m n,m m n+1,m+1 m+1 n,m+1 m+1
If d 6= 0 for all m, then after using the involution n ↔ m and an obvious rescaling
m
of v we obtain the discrete equation (3) with α =d2 6=0 for any n. This form is
n,m n n
more comfortable for further investigation.
For any α there exists a hierarchy of generalized symmetries of eq. (3) in the
n
m-direction, and its simplest representative reads [18]:
du
n,m =α (u2 −1)(u −u ). (35)
dτ n n,m n,m+1 n,m−1
1
For any fixed n it obviously is the modified Volterra equation. We will look for
functions α , such that generalized symmetries in the n-direction exist.
n
It should be remarked that an integrable generalization of eq. (3) has been pre-
sented in [9] together with one hierarchy of generalized symmetries and an L −A
pair.
8
4.1 Simplest case
Thefollowingresulthasbeenobtainedin[9],andwepresentithereforcompleteness.
Theorem 6 Eq. (3)withα 6=0foranynhasafirstordernondegenerategeneralized
n
symmetry of the form (5) iff α ≡α .
n n−1
In the case α ≡α , the following nondegenerate symmetry is known [18]:
n n−1
du 1 1
n,m =(u2 −1) − . (36)
dt n,m u +u u +u
1 (cid:18) n+1,m n,m n,m n−1,m(cid:19)
We just can add that there is one more generalized symmetry in the n-direction:
du n n−1
n,m =(u2 −1) − , (37)
dt′ n,m u +u u +u
1 (cid:18) n+1,m n,m n,m n−1,m(cid:19)
and any symmetry of an order k ≤ 1 of eq. (3) is a linear combination of (36) and
(37).
Eq. (37) is the known master symmetry of eq. (36) [4]. It provides generalized
symmetries not only for (36) but also for eq. (3).
4.2 Second order generalized symmetries
Theorem 7 The following two statements take place:
1. If eq. (3) with α 6=0 for all n has a nondegenerate generalized symmetry of the
n
second order in the n-direction, then α ≡α .
n+1 n−1
2. If α 6=0 for any n in eq. (3) and it satisfies the conditions
n
α ≡α and α 6=α , (38)
n+1 n−1 0 1
then there exists in the n-direction a nondegenerate second order symmetry of
eq. (3) and there is no symmetry of the first order.
Proof. The compatibility condition (8) implies:
∂Φ ∂Φ
n,m n,m
(α −α ) ≡0, (α −α ) ≡0,
n+2 n n n−2
∂u ∂u
n+2,m n−2,m
and we get the first part of the theorem.
In the case (38), allsymmetries of ordersk ≤2 in the n-directionaredescribed as
follows:
du a (u +u )
n,m =(u2 −1) n+2 n+2,m n+1,m
dt n,m U
2 (cid:18) n+1,m
an+1(un+1,m−un−1,m) an(un−1,m+un−2,m) (39)
− − ,
U U
n,m n−1,m (cid:19)
U =(u +u )(u +u )+(u2 −1)(β −1),
n,m n+1,m n,m n,m n−1,m n,m n−1
where β = α /α 6= 1 for all n. The function a is given by a = b +cn, where
n n+1 n n n n
b is an arbitrary two-periodic function and c is an arbitrary constant.
n
There are in (39) nondegenerate examples of the second order, e.g. a ≡ 1 or
n
a =2+(−1)n, but there is no symmetry of the first order.
n
In the case when a is the two-periodicfunction, i.e. a =b , we have in(39)two
n n n
linear independent and commuting generalized symmetries. The linear case a = cn
n
9
provides the master symmetry for eq. (39) with a =b , which generates generalized
n n
symmetriesforthisequation. Thosegeneralizedsymmetriesshouldbethesymmetries
of the discrete equation (3,38) too, as the Lie algebra of symmetries should be closed
under the operation of commutation, but the verification of this property is difficult
even on the first step.
5 Method of the construction of generalized symme-
tries and conservation laws
Amethod hasbeen developedin[11] forthe autonomousandweaklynonautonomous
discrete and semidiscrete equations, which allows one to construct generalized sym-
metries andconservationlawsby using the L−A pairs. Thatmethod is basedonthe
formal diagonalization of an L−A pair in the neighborhood of a stationary singular
point. In this section we generalize that method to the case of the nonautonomous
equations with periodic coefficients.
5.1 Formal diagonalization
Let us first discuss the formal diagonalization in the case of systems of the linear
discrete equations.
We consider a discrete linear vector equation of the form
Ψ =f (u ,λ)Ψ , k ≥1, (40)
n+k n n n
where Ψ is anunknownvector,the matrix potentialf (u ,λ)∈Cs×s is a meromor-
n n n
phic function ofλ∈C, andthe vector function u is a functional parameter. A point
n
λ=λ is calledthe point of singularityofeq. (40) if it is either a pole of f (u ,λ) or
0 n n
a solution of the equationdetf (u ,λ)=0. It is assumedthat the set of roots of the
n n
equation detf (u ,λ)=0, as well as the set of poles of f , does not depend on n.
n n n
Wesupposeherethateq. (40)withthesingularpointλ isreducedtothefollowing
0
special form:
Ψ =P (u ,λ)ZΨ , (41)
n+k n n n
where Z is a diagonal matrix Z = diag((λ−λ0)γ1,(λ−λ0)γ2,...,(λ−λ0)γs) with
integer exponents γ , such that γ <γ <...<γ .
j 1 2 s
It should be remarked that there is no proof that eq. (40) can be transformed
into the form (41). However, there is a general scheme which provides, as a rule, a
transition from eq. (40) to (41). That scheme has been presented in [11]. Besides, it
will be explained in detail in Section 5.3 for three examples under consideration how
to get the representation (41).
Let us rewrite eq. (41) as LΨ =Ψ with
n n
L=D−kP (u ,λ)Z, (42)
n n n
where D is the shift operator acting by the rule D : n → n+1. The following
n n
statement on the formal diagonalization of the operator L takes place, see [11,13].
Theorem 8 Assume that, for any integer n and for u ranging in a domain, the
n
function P (u ,λ) is analytic in a neighborhood of the point λ , and all the leading
n n 0
principal minors det of the matrix P (u ,λ ) do not vanish:
j n n 0
detP (u ,λ )6=0 for j =1,2,...,s and for all n.
n n 0
j
Then there exists a formal series T = T(i)(λ −λ )i, detT(0) 6= 0, with the
n i≥0 n 0 n
matrixcoefficients, suchthattheoperatorL =T−1LT is oftheformL =D−kh Z,
P0 n n 0 n n
10