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and get some traveling wave solutions in terms of theta functions with the aid of sym-
bolic computation for the first time. The (2+1)-dimensional KP-type eqation (1) arises
from the soliton hierarchy associated with spectral problem [8]:
−+
λ
u
2
v
φ
=
UU
φ
,
=
(3)
x
2(ln
vr v u
+
) /
λ
1
(, ) T
, r is constant.
This paper is arranged as follows. In section 2, we shall illustrate the auxiliary equa-
tion method; In section 3, we apply auxiliary method [12] to seek exact solutions of
(2+1)-dimensional KP-type equation; Some conclusions are given in section 4.
φ =
qp
where
2 Method of Solution
For the elliptic equation (2), the following fact is needed to realize the aim of this
paper.
2
2
2
2
CA
=−
=−
ϑϑ
(0)
(0)
B
=
ϑ
(0)
ϑ
(0),
Proposition: If we take
and
then
2
4
2
4
Fz
()
=
ϑϑ
(0)
(0)
satisfies the elliptic equation (2), where theta functions are
1
3
defined as following
⎝⎠
ε
ε
ε
ε
ϑ
[
](
z
τ
)
=
exp{
π τ
i n
(
+
)
2
+
2(
n
+
)(
z
+
)}
ε
2
2
2
n
=−∞
ϑ
( )
z
ϑ
(
z
τ
)
=
ϑ ε
[
](
z
τ
),
i
i
i
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
1
1
0
0
ε
=
,
ε
=
,
ε
=
,
ε
=
1
2
3
4
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1, 2, 3, 4
1
0
0
1
.
In the following, we seek traveling wave solution in the formal solution for the
equation (1)
i
=
N
n
u
()
ξ
=+
a
=
a F
()
ξ
(
a
0)
(4)
i
0
N
i
1
by taking
ux x x t u
l
xkx kx t
(; ; ; ;)
"
=
( ,
ξ
12
(5)
ξ
=+
++
"
+
λ
.
122
ll
 
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