1. Introduction
In recent decades, soliton equations have attracted much attention for both their physical and mathematical significance. For the soliton solutions of soliton equations, they scatter elastically with phaseshifts or amplitude changes in uniform or nonuniform media in dynamics. Many soliton equations such as KdV, mKdV, nonlinear Schrödinger equation, and sineGordon equation are viewed as the compatibility condition of the AblowitzKaupNewellSegur (AKNS) spectral problem. Recently, the nonlinear Schrödinger equation attracts much attention in many fields such as oceanics, nonlinear optics, BoseEinstein condensations [1], and atmospherics, for it interprets some freak wave (rogue wave) [2]. The propagation of the optical field complex envelope in a singlemode fiber, accounting for group velocity dispersion and Kerr nonlinearity, is governed by the nonlinear Schrödinger equation (see [3]). As the discrete version of the AKNS spectral problem, the AblowitzLadik (AL) spectral problem has been studied for a few decades. The nonlinear selfdual network and Toda lattice equations are important equations related to the AL spectral problem. The Toda lattice is a system of unit masses, connected by nonlinear springs subject to an exponential restoring force [4]. The corresponding AblowitzLadik hierarchy is researched for its symmetries [5] and its decompositions [6] in mathematical sense. It is interesting to study the differentialdifference equation in the AblowitzLadik hierarchy and find their soliton and rational solutions (see [5, 6] and references therein).
Many methods such as the inverse scattering transformation [7], Hirota method [8], Bäcklund transformation (e.g., [9]), dressing method (e.g., [10]), and Wronskian/Casoratian technique [11] can be used for finding their solutions of continuous [12] and discrete soliton equations [13] and soliton equations with selfconsistent sources [14]. It is well known that the Wronskian/Casoratian technique has been used to construct various types of exact solutions of soliton equations, such as soliton solutions (e.g., [15]) and rational solutions [16]. Nsoliton solution of the discretetime relativistic Toda lattice equation is explicitly constructed in the form of the Casorati determinant [17]. The resulting solutions can be verified by direct substitution into the corresponding linear equation. By using the technique, the solitons and the rational solutions are introduced and constructed. In the paper, we would like to consider the differentialdifference equation related to the AblowitzLadik spectral problem [18]:
(1)
Q
n
R
n
t
=
Q
n
+
1
1

Q
n
R
n

R
n
+
1
1

Q
n
R
n
,
which corresponding Lax pair is
(2a)
ϕ
1
,
n
+
1
ϕ
2
,
n
+
1
=
z
Q
n
R
n
1
z
ϕ
1
,
n
ϕ
2
,
n
,
(2b)
ϕ
1
,
n
ϕ
2
,
n
t
=
1
2
z
2

Q
n
R
n

1
z
Q
n
z
R
n

1

1
2
z
2
ϕ
1
,
n
ϕ
1
,
n
,
where
Q
n
=
Q
(
t
,
n
)
,
R
n
=
R
(
t
,
n
)
,
ϕ
1
,
n
=
ϕ
1
(
t
,
n
)
, and
ϕ
2
,
n
=
ϕ
2
(
t
,
n
)
are functions dependent on variable
t
,
n
(
n
∈
Z
)
and decrease rapidly when
n
tend to
∞
and
z
is a spectral parameter. In general, the distinct discrete eigenvalues
{
z
k
}
are included in the scattering data of direct scattering problem in the frame of the inverse scattering transform.
{
z
k
}
usually influence the shape and propagation of the multisolitons.
The paper is organized as follows. Section 2 gives the double Casoratian solution of (1). In Section 3 soliton solutions and “rationallike” solutions are derived from general solutions directly.
2. The Double Casoratian Solution
By the dependent variable transformation
(3)
Q
n
=
g
n
f
n
,
R
n
=
h
n
f
n
,
(1) can be transformed to bilinear form
(4a)
e
D
n

1
f
n
·
f
n
=

g
n
h
n
,
(4b)
D
t
g
n
·
f
n
=
g
n
+
1
f
n

1
,
(4c)
D
t
h
n
·
f
n
=

h
n

1
f
n
+
1
,
where the operators
D
t
and
e
D
n
are defined by
(5a)
D
t
f
t
,
n
·
g
t
,
n
=
∂
t

∂
t
′
f
t
,
n
g
t
,
n
t
=
t
′
,
∂
t
=
∂
∂
t
,
(5b)
e
D
n
f
t
,
n
·
g
t
,
n
=
f
t
,
n
+
1
g
t
,
n

1
.
To use the Casoratian technique, we give the compact notation by Freeman and Nimmo [11]:
(6)
Cas
N
,
M
ϕ
n
,
ψ
n
=
ϕ
n
,
E
ϕ
n
,
…
,
E
N

1
ϕ
n
;
ψ
n
,
E
ψ
n
,
…
,
E
M

1
ψ
n
=
N

1
^
;
M

1
^
,
where
(7)
ϕ
n
=
ϕ
1
,
n
,
ϕ
2
,
n
,
…
,
ϕ
N
+
M
,
n
T
,
ψ
n
=
ψ
1
,
n
,
ψ
2
,
n
,
…
,
ψ
N
+
M
,
n
T
.
E
is shift operator defined by
E
ϕ
n
=
ϕ
n
+
1
. We also denote that
(8a)

1
,
N

1
^
;

1
,
M

1
^
=
E

1
ϕ
n
,
ϕ
n
,
E
ϕ
n
,
…
,
E
N

1
ϕ
n
;
E

1
ψ
n
,
ψ
n
,
E
ψ
n
,
…
,
E
M

1
ψ
n
,
(8b)
2
N

2
^
;
2
M

2
^
=
ϕ
n
,
E
2
ϕ
n
,
…
,
E
2
N

2
ϕ
n
;
ψ
n
,
E
2
ψ
n
,
…
,
E
2
M

2
ψ
n
,
(8c)
2
N

1
^
;
2
M

1
^
=
E
ϕ
n
,
E
3
ϕ
n
,
…
,
E
2
N

1
ϕ
n
;
E
ψ
n
,
E
3
ψ
n
,
…
,
E
2
M

1
ψ
n
,
where
E

1
ϕ
n
=
ϕ
n

1
. It is easy to get the following lemma.
Lemma 1.
Consider the following:
(9)
Q
,
a
,
b
Q
,
c
,
d

Q
,
a
,
c
Q
,
b
,
d
+
Q
,
a
,
d
Q
,
b
,
c
=
0
,
where
Q
is an
N
×
(
N

2
)
matrix and
a
,
b
,
c
, and
d
represent
N
dimensional column vectors.
Theorem 2.
Equation (1) has the double Casoratian solution,
(10a)
f
n
=
2
N

2
^
;
2
M

2
^
,
(10b)
g
n
=

1
,
2
N

1
^
;
2
M

3
^
,
(10c)
h
n
=

2
N

3
^
;

1
,
2
M

1
^
,
for the condition equations,
(11a)
ϕ
j
,
n
+
1
=
e
k
j
ϕ
j
,
n
,
ψ
j
,
n

1
=
e
k
j
ψ
j
,
n
,
(11b)
ϕ
j
,
n
,
t
=
1
2
ϕ
j
,
n
+
2
,
ψ
j
,
n
,
t
=

1
2
ψ
j
,
n

2
j
=
1,2
,
…
,
N
+
M
,
and in general
(12a)
ϕ
n
+
1
=
A
ϕ
n
,
ψ
n

1
=
A
ψ
n
,
(12b)
ϕ
n
,
t
=
1
2
ϕ
n
+
2
,
ψ
n
,
t
=

1
2
ψ
n

2
,
where
A
=
(
a
i
j
)
(
N
+
M
)
×
(
N
+
M
)
is an arbitrary matrix independent of
t
and
n
. Thus the corresponding solution of (1) can be expressed as
(13)
Q
n
=

1
,
2
N

1
^
;
2
M

3
^
2
N

2
^
;
2
M

2
^
,
R
n
=

2
N

3
^
;

1
,
2
M

1
^
2
N

2
^
;
2
M

2
^
.
Proof.
Here we give two notations.
(
2
N

2
~
;
2
M

2
~
)
denotes the columns
(
E
2
ϕ
n
,
E
4
ϕ
n
,
…
,
E
2
N

2
ϕ
n
;
E
2
ψ
n
,
E
4
ψ
n
,
…
,
E
2
M

2
ψ
n
)
, which is related to the evenorder derivatives.
(
2
N

1
~
;
2
M

1
~
)
denotes the columns
(
E
3
ϕ
n
,
E
5
ϕ
n
,
…
,
E
2
N

1
ϕ
n
;
E
3
ψ
n
,
E
5
ψ
n
,
…
,
E
2
M

1
ψ
n
)
, which is related to the oddorder derivatives. First, by use of the Casoratian technique, it is easy to get
(14a)
f
n
+
1
=
2
N

1
^
;
2
M

1
^
,
f
n

1
=

1
,
2
N

3
^
;

1
,
2
M

3
^
,
(14b)
g
n
+
1
=
2
N
^
;
2
M

2
~
,
h
n

1
=

2
N

4
^
;

2
,
2
M

2
^
,
(14c)
f
n
,
t
=
1
2
2
N

4
^
,
2
N
;
2
M

2
^

2
N

2
^
;

2
,
2
M

2
~
,
(14d)
g
n
,
t
=
1
2

1
,
2
N

3
^
,
2
N
+
1
;
2
M

3
^


1
,
2
N

1
^
;

1
,
2
M

3
~
,
(14e)
h
n
,
t
=
1
2
2
N

3
^
;

3
,
2
M

1
^

2
N

5
^
,
2
N

1
;

1
,
2
M

1
^
.
For (4a), note that
(15)
2
N

2
^
;
2
M

2
^
2
=
∏
j
=
1
N
+
M
e
k
j
2
N

2
^
;
2
M

2
^
∏
j
=
1
N
+
M
e

k
j
2
N

2
^
;
2
M

2
^
=
2
N

1
^
;

1
,
2
M

3
^

1
,
2
N

3
^
;
2
M

1
^
and taking
Q
=
(
2
N

3
^
,
2
M

3
^
)
in (9) yields
(16)
2
N

3
^
,
2
N

1
;
2
M

3
^
,
2
M

1

1
,
2
N

3
^
;

1
,
2
M

3
^

2
N

3
^
,
2
N

1
;

1
,
2
M

3
^

1
,
2
N

3
^
;
2
M

3
^
,
2
M

1


1
,
2
N

3
^
,
2
N

1
;
2
M

3
^
2
N

3
^
;

1
,
2
M

3
^
,
2
M

1
=
0
.
It follows that
(17)
f
n
2

g
n
h
n

f
n
+
1
f
n

1
=
2
N

1
^
;

1
,
2
N

3
^

1
,
2
N

3
^
;
2
M

1
^
+

1
,
2
N

1
^
;
2
M

3
^

1
,
2
N

3
^
;
2
M

1
^

2
N

1
^
;
2
M

1
^

1
,
2
N

3
^
;

1
,
2
M

3
^
=
0
.
Thus, ((10a), (10b), (10c)) solves (4a).
For (4b), from (9) and noting the following identities,
(18a)
∏
j
=
1
N
+
M
e
k
j
2
N

2
^
;
2
M

2
^
=
2
N

1
^
;

1
,
2
M

3
^
,
(18b)
∏
j
=
1
N
+
M
e
k
j
2
N

4
^
,
2
N
;
2
M

2
^
=
2
N

3
^
,
2
N
+
1
;

1
,
2
M

3
^
,
(18c)
∏
j
=
1
N
+
M
e
k
j
2
N
^
;
2
M

2
~
=
2
N
+
1
^
;
2
M

3
^
,
we have
(19)

1
,
2
N

3
^
,
2
N
+
1
;
2
M

3
^
2
N

2
^
;
2
M

2
^


1
,
2
N

1
^
;
2
M

3
^
2
N

4
^
,
2
N
;
2
M

2
^

2
N
^
;
2
M

2
~

1
,
2
N

3
^
;

1
,
2
M

3
^
=
0
.
Similarly, we can get
(20)


1
,
2
N

1
^
;

1
,
2
M

3
~
2
N

2
^
;
2
M

2
^
+

1
,
2
N

1
^
;
2
M

3
^
2
N

2
^
;

2
,
2
M

2
~

2
N
^
;
2
M

2
~

1
,
2
N

3
^
;

1
,
2
M

3
^
=
0
.
By combining (19) with (20) and from ((14a), (14b), (14c), (14d), (14e)), one can have that ((10a), (10b), (10c)) solves (4b). In a similar way, we can also verify that ((10a), (10b), (10c)) solves (4c).
Now from ((10a), (10b), (10c)) and using the Casoratian technique, we show that ((10a), (10b), (10c)) solves ((4a), (4b), (4c)) for the condition equations ((12a), (12b)). In fact, we only need to verify (15), which can be obtained by the following identities:
(21a)
2
N

1
^
;

1
,
2
M

3
^
=
A
2
N

2
^
;
2
M

2
^
,
(21b)

1
,
2
N

3
^
;
2
M

1
^
=
A

1
2
N

2
^
;
2
M

2
^
,
where
A

1
is the inverse matrix of
A
.
3. Soliton Solutions and RationalLike Solutions
From ((12a), (12b)), we can get the general solution
(22)
ϕ
n
=
C
e
1
/
2
e
2
B
t
+
n
B
,
ψ
n
=
D
e

1
/
2
e
2
B
t

n
B
,
where
A
=
e
B
,
C
=
(
c
1
,
c
2
,
…
,
c
N
+
M
)
T
, and
D
=
(
d
1
,
d
2
,
…
,
d
N
+
M
)
T
are real constant vectors. In order to get the soliton solutions and rationallike solutions, we take matrix
B
in canonical form
Γ
in (22) and expand the functions
ϕ
n
and
ψ
n
:
(23a)
ϕ
n
=
C
e
1
/
2
e
2
Γ
t
+
n
Γ
=
∑
s
=
0
∞
P
s
n
,
t
e
1
/
2
t
I
Γ
s
C
,
(23b)
ψ
n
=
D
e

1
/
2
e
2
Γ
t

n
Γ
=
∑
s
=
0
∞
P
s

n
,

t
e

1
/
2
t
I
Γ
s
D
,
where
(24a)
P
s
n
,
t
=
∑
j
=
0
s
n
j
j
!
p
s

j
t
~
,
p
s
t
~
=
∑
γ
=
s
2
γ
γ
!
t
~
γ
,
(24b)
γ
=
γ
1
,
γ
2
,
γ
3
,
…
,
γ
j
≥
0
,
j
=
1,2
,
3
,
…
,
γ
=
γ
1
+
2
γ
2
+
3
γ
3
+
⋯
,
(24c)
γ
!
=
γ
1
!
γ
2
!
γ
3
!
⋯
,
t
~
=
t
1
2
,
t
2
2
,
t
3
2
,
…
,
t
j
=
t
j
!
,
t
~
γ
=
t
1
2
γ
1
t
2
2
γ
2
t
3
2
γ
3
⋯
and
I
is the unit matrix. If
(25)
Γ
=
k
1
0
k
2
⋱
0
k
N
+
M
,
k
i
≠
k
j
i
≠
j
,
then we can obtain the soliton solutions (13) from (22) (substituting
B
for
Γ
), where
(26)
ϕ
j
,
n
=
c
j
e
1
/
2
e
2
k
j
t
+
n
k
j
,
ψ
j
,
n
=
d
j
e

1
/
2
e
2
k
j
t

n
k
j
j
=
1,2
,
…
,
N
+
M
.
If
(27)
Γ
=
0
0
1
0
⋱
⋱
0
1
0
N
+
M
,
then it follows from ((23a), (23b)) and
Γ
N
+
M
=
0
that
(28a)
ϕ
n
=
∑
s
=
0
N
+
M

1
P
s
n
,
t
e
1
/
2
t
I
Γ
s
C
,
(28b)
ψ
n
=
∑
s
=
0
N
+
M

1
P
s

n
,

t
e

1
/
2
t
I
Γ
s
D
.
We specially give some rationallike solutions in the following. Taking
(
N
,
M
)
=
(
1,1
)
,
Γ
=
0
0
1
0
, and
C
=
D
=
(
1,0
)
T
in ((28a), (28b)) leads to
(29)
ϕ
n
=
e
1
/
2
t
1
n
+
t
,
ψ
n
=
e

1
/
2
t
1

n

t
.
Hence, the solution ((10a), (10b), (10c)) reads
(30)
Q
n
=

e
t
n
+
t
,
R
n
=

e

t
n
+
t
.
Similarly, we can calculate the following rationallike solutions with respect to
n
from ((28a), (28b)),
(31a)
Q
n
=
2
e
t
n
+
n
2
+
2
n
t
+
t
2
,
R
n
=

e

t
n
+
n
2
+
2
n
t
+
2
t
+
t
2
n
+
n
2
+
2
n
t
+
t
2
;
(31b)
Q
n
=

e
t
n
+
n
2
+
2
n
t
+
t
2
n
+
n
2
+
2
n
t
+
2
t
+
t
2
,
R
n
=

2
e

t
n
+
n
2
+
2
n
t
+
2
t
+
t
2
;
(31c)
Q
n
=

2
e
t
n
3
+
3
n
2
+
3
n
2
t
+
t
3
+
n
2
+
3
t
+
3
t
2
n
4
+
4
n
3
t
+
1
+
t
2
6
+
4
t
+
t
2
+
n
2
5
+
12
t
+
6
t
2
+
2
n
1
+
4
t
+
6
t
2
+
2
t
3
,
R
n
=
2
e

t
n
3
+
3
n
2
t
+
1
+
t
6
+
6
t
+
t
2
+
n
2
+
9
t
+
3
t
2
n
4
+
4
n
3
t
+
1
+
t
2
6
+
4
t
+
t
2
+
n
2
5
+
12
t
+
6
t
2
+
2
n
1
+
4
t
+
6
t
2
+
2
t
3
;
(31d)
Q
n
=

6
e
t
n
3
+
t
3
+
3
n
2
t
+
1
+
n
2
+
3
t
+
3
t
2
,
R
n
=

e

t
n
4
+
n
3
t
+
1
+
n
2
5
+
12
t
+
6
t
2
+
t
2
6
+
4
t
+
t
2
+
2
n
1
+
4
t
+
6
t
2
+
2
t
3
2
n
3
+
t
3
+
3
n
2
t
+
1
+
n
2
+
3
t
+
3
t
2
;
(31e)
Q
n
=

e
t
n
4
+
n
3
t
+
1
+
n
2
5
+
12
t
+
6
t
2
+
t
2
6
+
4
t
+
t
2
+
2
n
1
+
4
t
+
6
t
2
+
2
t
3
n
3
+
t
6
+
6
t
+
t
2
+
3
n
2
t
+
1
+
n
2
+
9
t
+
3
t
2
,
R
n
=

6
e

t
n
3
+
t
6
+
6
t
+
t
2
+
3
n
2
t
+
1
+
n
2
+
9
t
+
3
t
2
for
(
N
,
M
)
=
(
2,1
)
,
(
N
,
M
)
=
(
1,2
)
,
(
N
,
M
)
=
(
2,2
)
,
(
N
,
M
)
=
(
3,1
)
, and
(
N
,
M
)
=
(
1,3
)
, respectively.