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

124 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 2012
Thus
where
ϕ(x, t; p) = ϕ(x, t; 0) +
ψ(x, t; p) = ψ(x, t; 0) +
∞∑
m=1
∞∑
m=1
φ(x, t; p) = u 0 (x, t) +
ϕ(x, t; p) = v 0 (x, t) +
ψ(x, t; p) = z 0 (x, t) +
p m ∂ m ϕ(x, t; p)
m! ∂p m | p=0 ,
p m ∂ m ψ(x, t; p)
m! ∂p m | p=0 .
∞∑
u m (x, t)p m , (6)
m=1
∞∑
v m (x, t)p m , (7)
m=1
∞∑
z m (x, t)p m , (8)
m=1
u m (x, t) = 1 ∂ m φ(x, t; p)
m! ∂p m | p=0 , (9)
v m (x, t) = 1 ∂ m ϕ(x, t; p)
m! ∂p m | p=0 , (10)
z m (x, t) = 1 ∂ m ψ(x, t; p)
m! ∂p m | p=0 . (11)
According to [4], the suitable choice of the initial condition u 0 (x, t), v 0 (x, t) and z 0 (x, t), the
auxiliary linear parameter L u , L v and L z , the non-zero auxiliary parameter and the auxiliary
function H 1 (x, t), H 2 (x, t) and H 3 (x, t) will guarantee that
(1) The solution φ(x, t; p), ϕ(x, t; p) ψ(x, t; p) of the zero-order deformation equation (3)-(5)
exists for all p ∈ [0, 1].
(2) The deformation derivative ∂m φ(x, t; p)
∂p m | p=0 , ∂m ϕ(x, t; p)
∂p m | p=0 and ∂m ψ(x, t; p)
∂p m | p=0 exists
for m = 1, 2, . . . .
(3) The power series (6)-(8) converges at p = 1.
If p = 1
φ(x, t; 1) = u(x, t) = u 0 (x, t) +
ϕ(x, t; 1) = v(x, t) = v 0 (x, t) +
ψ(x, t; 1) = z(x, t) = z 0 (x, t) +
∞∑
u m (x, t), (12)
m=1
∞∑
v m (x, t), (13)
m=1
∞∑
z m (x, t), (14)
which must be one of the solutions of new coupled MKdV system (1). According to the definition
(9)-(11), the governing equations can be deduced from the (zero-order deformation) equations
(3)-(5). For further analysis, the vectors
m=1
⃗u n (x, t) = {u 0 (x, t), u 1 (x, t), . . . , u n (x, t)},
⃗v n (x, t) = {v 0 (x, t), v 1 (x, t), . . . , v n (x, t)},
⃗z n (x, t) = {z 0 (x, t), z 1 (x, t), . . . , z n (x, t)},