1. Introduction
The theory, reasoning, and algebraic structures dealing with the construction of symmetries for differential equations (DEs) are now well established and documented. Moreover, the application of these in the analysis of DEs, in particular, for finding exact solutions, is widely used in a variety of areas from relativity to fluid mechanics (see [1–4]). Secondly, the relationship between symmetries and conservation laws has been a subject of interest since Noether’s celebrated work [5] for variational DEs. The extension of this relationship to DEs which may not be variational has been done more recently [6, 7]. The first consequence of this interplay has led to the double reduction of DEs [8–10].
A vast amount of work has been done to extend the ideas and applications of symmetries to difference equations (
Δ
Es) in a number of ways—see [11–15] and references therein. In some cases, the
Δ
Es are constructed from the DEs in such a way that the algebra of Lie symmetries remains the same [16]. As far as conservation laws of
Δ
Es go, the work is more recent—see [12, 17]. Here, we construct symmetries and conservation laws for some ordinary
Δ
Es, utilise the symmetries to obtain reductions of the equations, and show, in fact, that the notion of “association” between these concepts can be analogously extended to ordinary
Δ
Es. That is, an association between a symmetry and first integral exists if and only if the first integral is invariant under the symmetry. Thus, a “double reduction” of the
Δ
E is possible.
2. Preliminaries and Definitions
Consider the following
N
th-order O
Δ
E:
(1)
u
n
+
N
=
ω
(
n
,
u
n
,
u
n
+
1
,
…
,
u
n
+
N
-
1
)
,
where
ω
is a smooth function such that
(
∂
ω
/
∂
u
n
)
≠
0
and integer
n
is an independent variable. The general solution of (1) can be written in the form
(2)
u
n
=
F
(
n
,
c
1
,
…
,
c
N
)
and depends on
N
arbitrary independent constants
c
i
.
Definition 1.
We define
S
to be the shift operator acting on
n
as follows:
(3)
S
:
n
⟼
n
+
1
.
That is, if
u
n
=
F
(
n
,
c
1
,
…
,
c
N
)
then
(4)
S
(
u
n
)
=
u
n
+
1
.
In the same way,
(5)
S
(
u
n
+
k
)
=
u
n
+
k
+
1
,
k
=
0
,
…
,
N
-
2
.
Definition 2.
A symmetry generator,
X
, of (1) is given by
(6)
X
=
Q
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
∂
∂
u
n
+
(
S
Q
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
)
∂
∂
u
n
+
1
+
⋯
+
(
S
N
-
1
Q
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
)
∂
∂
u
n
+
N
-
1
and satisfies the symmetry condition
(7)
S
N
Q
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
-
X
ω
=
0
,
where
Q
=
Q
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
is a function called the characteristic of the one-parameter group.
Definition 3.
If
ϕ
is a first integral, then it is constant on the solutions of the O
Δ
E and hence satisfies
(8)
S
(
ϕ
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
)
=
ϕ
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
,
ϕ
(
n
+
1
,
u
n
+
1
,
…
,
ω
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
)
=
ϕ
(
n
,
u
n
,
…
,
u
n
+
N
-
1
)
,
where
S
is the shift operator defined in (3).
2.1. First Integral
In [11], Hydon presents a methodology to construct the first integrals of O
Δ
Es directly. For this method, the symmetries of the O
Δ
E need not be known. Here, we will only consider second-order O
Δ
E’s.
We construct first integrals using (8) and an additional condition; that is,
(9)
ϕ
(
n
+
1
,
u
n
+
1
,
ω
(
n
,
u
n
,
u
n
+
1
)
)
=
ϕ
(
n
,
u
n
,
u
n
+
1
)
,
∂
ϕ
∂
u
n
+
1
≠
0
.
Now let
(10)
P
1
(
n
,
u
n
,
u
n
+
1
)
=
∂
ϕ
∂
u
n
(
n
,
u
n
,
u
n
+
1
)
,
P
2
(
n
,
u
n
,
u
n
+
1
)
=
∂
ϕ
∂
u
n
+
1
.
Next we differentiate (9) with respect to
u
n
; we obtain
(11)
P
1
=
S
P
2
∂
ω
∂
u
n
.
Differentiating (9) with respect to
u
n
+
1
we get
(12)
P
2
=
S
P
1
+
∂
ω
∂
u
n
+
1
S
P
2
.
Thus,
P
2
satisfies the second-order linear functional equation or first integral condition,
(13)
S
(
∂
ω
∂
u
n
)
S
2
P
2
+
∂
ω
∂
u
n
+
1
S
P
2
-
P
2
=
0
.
After solving for
P
2
and constructing
P
1
, we need to check that the integrability condition
(14)
∂
P
1
∂
u
n
+
1
=
∂
P
2
∂
u
n
is satisfied. Hence if (14) holds, the first integral takes the form
(15)
ϕ
=
∫
(
P
1
d
u
n
+
P
2
d
u
n
+
1
)
+
G
(
n
)
.
To solve for
G
(
n
)
, we substitute (15) into (9) and solve for the resulting first-order O
Δ
E.
2.2. Using Symmetries to Obtain the General Solution of an O
Δ
E
We begin this section by providing some useful definitions. We consider the theory and example provided by Hydon in [11].
Definition 4.
The commutator of two symmetry generators
X
N
and
X
M
is denoted by
[
X
N
,
X
M
]
and defined by
(16)
[
X
N
,
X
M
]
=
X
N
X
M
-
X
M
X
N
=
-
[
X
M
,
X
N
]
.
Definition 5.
Given a symmetry generator for a second-order O
Δ
E,
(17)
X
=
Q
(
n
,
u
n
,
u
n
+
1
)
∂
∂
u
n
+
Q
(
n
+
1
,
u
n
+
1
,
ω
(
n
,
u
n
,
u
n
+
1
)
)
×
∂
∂
u
n
+
1
;
there exists an invariant,
(18)
v
n
=
v
(
n
,
u
n
,
u
n
+
1
)
,
satisfying
(19)
X
v
n
=
0
,
∂
v
n
∂
u
n
+
1
≠
0
.
To determine the invariant, we use the method of characteristics. Note that the invariant satisfies
(20)
[
Q
∂
∂
u
n
+
S
Q
∂
∂
u
n
+
1
]
v
n
=
0
.
We make the assumption that (18) can be inverted to obtain
(21)
u
n
+
1
=
ω
(
n
,
u
n
,
v
n
)
for some function
ω
. Solving (21) requires finding a canonical coordinate
(22)
s
n
=
s
(
n
,
u
n
)
which satisfies
X
s
n
=
1
. The most obvious choice [11] of canonical coordinate is
(23)
s
(
n
,
u
n
)
=
∫
d
u
n
Q
(
n
,
u
n
,
ω
(
n
,
u
n
,
f
(
n
;
c
1
)
)
)
with a general solution of the form
(24)
s
n
=
c
2
+
∑
k
=
n
0
n
-
1
g
(
k
,
f
(
k
;
c
1
)
)
,
where
n
0
is any integer.