THE SOLUTION OF COUPLED MODIFIED KDV SYSTEM BY THE ...

130 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 2012
ħcurve for v t 0,0
ħcurve for z t 0,0
0
0
2
2
4
4
v t 0,0
6
8
z t 0,0
6
8
10
10
12
12
2.0 1.5 1.0 0.5 0.0 0.5 1.0
ħ
2.0 1.5 1.0 0.5 0.0 0.5 1.0
ħ
Figure 6. The -curve for v t(0, 0) and z t(0, 0) given by the 5-order HAM approximation solution when
H 2 (x, t) = H 3 (x, t) = 1.
Example 3.2. Consider the coupled MKdV equation (2) with the initial conditions [8, 13]
u(x, 0) = k tanh(kx),
v(x, 0) = 1 2 (4k2 + λ) − 2k 2 tanh 2 (kx).
It can be verified that the exact solutions of this system are
(
u(x, t) = k tanh kx − k )
2 (2k2 + 3λ)t ,
v(x, t) = 1 2 (4k2 + λ) − 2k 2 tanh
(kx 2 − k )
2 (2k2 + 3λ)t .
We start with the initial conditions
u 0 (x, t) = k tanh(kx), (35)
v 0 (x, t) = 1 2 (4k2 + λ) − 2k 2 tanh 2 (kx). (36)
According to section 2, we can define the operators N 1 and N 2 as
N 1 [φ] = φ t − 1 2 φ xxx + 3φ 2 φ x − 3 2 (ϕ) xx − 3(φϕ) x + 3λφ x (37)
N 2 [ϕ] = ϕ t + ϕ xxx + 3ϕϕ x + 3φ x ϕ x − 3φ 2 ϕ x − 3λϕ x , (38)
where φ = φ(x, t; p) and ϕ = ϕ(x, t; p).
equation (15) and (16) that are
Now, we can obtain the zeroth-order deformation
R 1m (⃗u m−1 ) = (u m−1 ) t − 1 2 (u m−1) xxx + 3
− 3 2 (v m−1) xx − 3
m−1
∑
i=0
(
∑
u i
m−1−i
k=0
( m−1
)
∑
(u i v m−1−i )
i=0
u k (u m−1−i−k ) x
)
−
x
+ 3λ(u m−1 ) x ,
(39)