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where
kk k
l
,
,
"
,
,
λ
are constants to be determined, and
aABC
,
,
,
are constants to
i
12
be determined,
F
(
ξ
)
satisfies elliptic equation (2).
Step 1: Inserting (4) into (1) yields an ODE for
:
(;;;) 0
u
(
ξ
)
"
. (6)
Step 2: Determine N by considering the homogeneous balance between the govern-
ing nonlinear term(s) and highest order derivatives of
Puu u
′ ′
=
()
u
ξ
in (6).
Step 3: Substituting (4) into (6), and using (2), and then the left-hand side of (6) can
be converted into a finite series in
k
()(
)
F
ξ
k M
=
0,1,
"
.
,
k
()
Step 4: Equating each coefficient of
F
ξ
to zero yields a system of algebraic
(
)
equations for
ai
=
0,1,
"
.
,
N
i
Step 5: Solving the system of algebraic equations, with the aid of Mathematica or
Maple,
a
ii
λ can be expressed by A, B, C (or the coefficients of ODE
,
,
(6)).
Step 6: Substituting these results into (4), we can obtain the general form of travel-
ling wave solutions for equation (1).
Step 7: From propositon, we can give theta function periodic solutions for equation
(1).
3 Exact Solutions
In this section, we will make use of the auxiliary equation method and symbolic com-
putation to find the exact solutions to the (2+1)-dimensional KP-type equation.
We assume that (1) has travelling wave solution in the form
uxyt u
(; ;)
=
(),
ξξ α
=
x y t
+
β
+
ω
.
(7)
Substituting (7) into (1), then (1) is transformed into the following form:
4
⎛
⎞
3
α
2
2
β
ω ξ
U
′
()
U
′′′
() 3
ξ α ξ ξ
U
() () 0
U
′
(8)
−
+
−
=
⎜
⎟
⎝
⎠
4
16
′′′
′
According to step 2 in section 2, by balancing
U
(
ξ
)
and
U
ξξ
(
)
(
)
in (8), we
obtain
n
=
2
, and suppose that Eq. (8) has the following solutions:
2
Ua
()
ξ
=+
a F
()
ξ
+
a F
(),
ξ
(9)
01
2
F
ξ .
Setting their coefficients to zero yields a set of algebraic equations for unknown pa-
rameters
Substituting (9) along with Eq. (2) into (8) yields a polynomial equation in
(
)
aaa
ω . Solving these equations, we can get the following solutions:
, , ,
012
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