Exact Solutions of a Generalized KdV-mKdV Equation - IJNS, The ...

ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.13(2012) No.1,pp.94-98
Exact Solutions of a Generalized KdV-mKdV Equation
Cesar A. Gómez Sierra 1 , Motlatsi Molati 2 ∗ , Motlatsi P. Ramollo 2
1 Department of Mathematics, Universidad Nacional de Colombia, Bogotá Colombia.
2 Department of Mathematics and Computer Science, National University of Lesotho, PO Roma 180, Lesotho.
(Received 26 January 2011 , accepted 9 August 2011)
Abstract : We study the combined Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations.
By using symmetries we construct the exact solutions for the underlying equation. Exact solutions in
explicit form are obtained for two particular cases. Other soliton solutions are derived by using the improved
tanh-coth method.
Keywords : KdV-mKdV equation ; Lie point symmetry ; exact solution ; solitons
1 Introduction
In practical applications finding the exact solutions for nonlinear partial differential equations (NLPDEs) is one of the
most important task. Until this end, several direct and computational methods have been implemented. Among the direct
methods often used are the Hirota method [1], inverse scattering method [2], and analysis by Lie group theory [3],[4], [5].
On the other hand, the tanh-coth [6] method is one of the frequently used computational method.
The combined KdV equation and modified KdV equation (KdV-mKdV) [7] that we study reads
ut + auux + āu 2 ux + buxxx = 0, (1)
with constants a, ā and b. Particular cases of (1) and similar models have been studied extensively using various approaches
[8],[9] and the references therein. The main goal in this work is obtain the traveling wave solutions to (1) using two
approaches. First, we use Lie point symmetries of the underlying equation to obtain the most general traveling wave
solution and then we consider explicit solutions for two particular cases of (1). Thereafter, we use the improved tanh-coth
method to obtain other exact solutions to (1). The paper is organized as follows : In Sec. 2, we use the Lie symmetry
approach to derive the traveling wave solution to (1) in an implicit form. As a consequence the soliton solution for two
particular cases are analyzed. In Sec. 3, we use the improved tanh-coth method to obtain the exact traveling wave solutions
to (1) in explicit form. Finally, some conclusions are given.
2 Lie symmetry approach
According to Lie’s algorithm [3],[4], [5], the infinitesimal generator of the maximal symmetry group admitted by (1)
is given by
X = ξ 1 (t, x, u) ∂
∂t + ξ2 (t, x, u) ∂
∂
+ η(t, x, u) , (2)
∂x ∂u
if and only if the invariance condition of (1) is
X [3] ( ut + auux + āu 2 )
ux + buxxx = 0, (3)
where
X [3] ∂
= X + ζt
∂ut
∗. Corresponding author. E-mail address : m.molati@gmail.com
+ ζx
∂
∂ux
∂
+ ζxxx
∂uxxx
Copyright c⃝World Academic Press, World Academic Union
IJNS.2012.02.15/580
, (4)

C. A. Gómez Sierra, M. Molati, M. P. Ramollo: Exact Solutions of a Generalized KdV-mKdV Equation 95
is the prolongation of the vector field (2). The variables ζis are given by the formulae :
⎧
ζ
⎪⎨
α i = Di(ηα ) − uα j Di(ξj ),
ζα ij = Dj(ζ α i ) − uα ilDj(ξ l ),
.
⎪⎩
ζα i1···ik = Dik (ζα i1···ik−1 ) − uα i1···ik−1lDik (ξl ).
Executing the Lie’s algorithm, we obtain the Lie point symmetries of (1) given by
X1 = ∂
∂t , X2 = ∂
∂x , X3 = −3t ∂
∂t +
( ) 2 a t ∂
− x
2ā ∂x +
We look for solutions invariant under the linear combination X1 + λX2 where λ is a constant.
Solving the characteristic system for the invariants of the linear combination, we obtain
where v is an arbitrary function of ξ. The substitution of (7) into (1) yields
(5)
(
a
)
∂
+ u . (6)
2ā ∂u
ξ = x − λt, u(t, x) = v(ξ), (7)
−λ v ′ (ξ) + av(ξ)v ′ (ξ) + āv 2 (ξ)v ′ (ξ) + bv ′′′ (ξ) = 0, (8)
where a prime denotes differentiation with respect to ξ and λ represents the wave speed. Integration of (8) once leads to
Multiplying both sides of (9) by v ′ and integrating once more we obtain
−λv + a
2 v2 + ā
3 v3 + bv ′′ = k1. (9)
− λ
2 v2 + a
6 v3 + ā
12 v4 + b
2 (v′ ) 2 = k1v + k2. (10)
Finally, from (10) we have
√
√
b dv
2k1v + 2k2 + λv2 − a
3 v3 − ā
6 v4
= dξ (11)
where k1 and k2 are arbitrary constants.
Integrating both sides of this last equation we obtain the general traveling wave solution to (1) in implicit form
∫
√
b dv
√
2k1v + 2k2 + λv2 − a
3 v3 − ā
6 v4
= ξ + k3 = x − λ t + k3
for an arbitrary constant k3.
2.1 First particular case
Setting a = 0 in (1) we obtain the following KdV of higher order
(12)
ut + āu 2 ux + buxxx = 0. (13)
If we assume that v → 0, v ′ → 0, v ′′ → 0 when ξ → ∞ in the analysis presented in the previous section, then (12)
reduces to
With the substitution v =
√ 6λ
ā
where k4 is a constant of integration.
√ ∫
b
λ
dv
√
v 1 − ā
6λv2 = ξ + k3. (14)
sechw finally we obtain the following soliton solution to (13)
√
6λ
u(t, x) =
ā sech
[√ ]
λ
(x − λ t + k4) , (15)
b
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96 International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 94-98
2.2 Second particular case
If we set ā = 0, then the equation under study (1) reduces to
ut + auux + buxxx = 0, (16)
which is the generalized KdV equation. Proceeding as in the first case, from (12) we obtain
√ ∫
b
λ
dv
v √ 1 − a
3λ v = ξ + k3. (17)
Using the substitution v = 3λ
a sech 2 w, and after simplifications, finally we obtain the following traveling wave
solution to (16)
u(t, x) = 3λ
a sech2
3 The improved tanh-coth method
[ √ ]
1 λ
(x − λ t + k4) . (18)
2 b
In this section we consider the equation (1) with variable coefficients (depending of t), that is,
ut + a(t)uux + ā(t)u 2 ux + b(t)uxxx = 0. (19)
Our goal is to obtain the exact solutions of (19) using the improved tanh-coth method [10]. As before, we seek a solution
in the form u(t, x) = v(ξ), being ξ = x − λ t. In this case, (19) reduces to
−λ v ′ (ξ) + a(t)v(ξ)v ′ (ξ) + ā(t)v 2 (ξ)v ′ (ξ) + b(t)v ′′′ (ξ) = 0. (20)
Using the idea of the tanh-coth method introduced by Wazwaz [6], we assume the solution of (20) in the form
where ϕ(ξ) satisfies the Riccati equation
M∑
ai(t)ϕ(ξ) i +
i=0
2M∑
M+1
ai(t)ϕ(ξ) M−i , (21)
ϕ ′ (ξ) = γ(t)ϕ 2 (ξ) + β(t)ϕ(ξ) + α(t). (22)
The coefficients ai(t) are unknown functions to be determined later and α(t), γ(t), β(t) are independent of variable ξ. It
is well known that the solution of (22) is given by [11]
⎧
1
(
−
⎪⎨ γ(t)
ϕ(ξ) =
⎪⎩
1
)
β(t)
ξ − 2 , β2 = 4γ(t)α(t)
( √ [√ ]
1 β2 (t)−4α(t)γ(t)
β2 (t)−4α(t)γ(t)
−
γ(t)
2 tanh
2 ξ − β(t)
)
2 , β2 ̸= 4γ(t)α(t)
(23)
Balancing the linear term of highest order with the nonlinear term in (20) we have
so that M = 1. Therefore, (21) takes the form
M + 3 = 2M + (M + 1),
a0(t) + a1(t)ϕ(ξ) + a2(t)ϕ(ξ) −1 . (24)
Substitution of (24) along with (22) into (20) yields a set of algebraic equations for λ, ai(t), α(t), β(t) and γ(t).
Solving the algebraic system with aid of Mathematica we consider only the following non-trivial solutions :
1. ⎧⎨
λ = −a(t)2 +4ā(t) 2 √
+a1(t)a2(t)
ā(t)
6ā(t) , α(t) = ±i a2(t) 6b(t)
⎩a0(t)
= − a(t)
ā(t) .
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√
ā(t)
ia(t)
, γ(t) = ±i a1(t) 6b(t) , β(t) = ± √ ,
6ā(t)b(t)

C. A. Gómez Sierra, M. Molati, M. P. Ramollo: Exact Solutions of a Generalized KdV-mKdV Equation 97
2. ⎧ ⎨
3. ⎧ ⎨
4. ⎧ ⎨
5. ⎧ ⎨
6. ⎧ ⎨
7. ⎧ ⎨
λ = −a(t)2
√
ā(t)
6ā(t) , α(t) = 0, γ(t) = ±i a1(t) 6b(t)
⎩
a0(t) = 0, a2(t) = 0.
λ = −a(t)2
√
ā(t)
6ā(t) , α(t) = 0, γ(t) = ±i a1(t) 6b(t)
⎩a0(t)
= − a(t)
ā(t) , a2(t) = 0.
λ = −a(t)2
√
ā(t)
6ā(t) , α(t) = ±i a2(t) 6b(t)
⎩
a0(t) = 0, a1(t) = 0.
λ = −a(t)2
√
ā(t)
6ā(t) , α(t) = ±i a2(t) 6b(t)
⎩a0
= − a(t)
ā(t) , a1(t) = 0.
λ = −a(t)2
6ā(t)
, α(t) = ±
⎩
a0(t) = − a(t)
2ā(t) , a2(t) =
λ = −a(t)2
6ā(t)
, α(t) = ±
⎩
a0 = − a(t)
2ā(t) , a2(t) = 0.
ia1(t) 2
16a1(t) 5 2
2
a1(t)
16ā(t) 2a1(t) .
ia1(t) 2
8a1(t)ā(t) 3 2
8.
⎧
⎨λ
=
⎩
−a(t)2 +a(t)ā(t)a0(t)+2ā(t) 2 a0(t) 2
6ā(t) , α(t) =
a2(t) = − a(t)a0(t)
4ā(t)a1(t) .
ia(t)
, β(t) = ± √ ,
6ā(t)b(t)
ia(t)
, β(t) = ∓√ ,
6ā(t)b(t)
ia(t)
, γ(t) = 0, β(t) = ± √ ,
6ā(t)b(t)
ia(t)
, γ(t) = 0, β(t) = ∓√ ,
6ā(t)b(t)
√ 6b(t) , γ(t) = ∓i a1(t)
√ 6b(t) , γ(t) = ∓i a1(t)
−i a(t)a0(t)
4 √ 6ā(t)b(t)a1(t)
√
ā(t)
6b(t)
√
ā(t)
6b(t)
, β(t) = 0,
, β(t) = 0,
√
ā(t)
i[a(t)+2ā(t)a0(t)]
, γ(t) = i a1(t) 6b(t) , β(t) = √ ,
6ā(t)b(t)
In all cases, substituting the respective values of α(t), β(t), γ(t) into (23) and using (24) and taking into account that
u(t, x) = v(ξ) with ξ = x − λ t, we obtain traveling wave solutions to (19). By conserving the space, we consider the
solutions for the values given by (1), (2) and (8) :
From (1) we have
i sech
u(t, x) = −
2 ( 1
2ξ √ (
a(t) 2−4ā(t) 2a1(t)a2(t) 1
2i
ā(t) tanh
[ ]
2 2 −a(t) + 4ā(t) + a1(t)a2(t)
ξ = x −
t.
6ā(t)
According to (2) we obtain
Finally, from (8) we have the following solution
√
2ā(t)a1(t)a2(t)
3b(t) − a(t)2
6ā(t)b(t)
2ξ √
2ā(t)a1(t)a2(t)
3b(t) − a(t)2
6ā(t)b(t)
u(t, x) = − [
a(t)
] ,
ξ =
ia(t)ξ
exp √
6ā(t) b(t)
[ ] 2 −a(t)
x − t.
6ā(t)
ā(t) + ā(t)
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) [a(t) 2 − 4ā(t) 2 a1(t)a2(t) ]
)
ā(t) 3/2 ,
+ 2ia(t)ā(t)

98 International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 94-98
u(t, x) =
a(t)a0(t)
2a(t) + 4ā(t)a0(t) + 2 √ [a(t) + ā(t)a0(t)][a(t) + 4ā(t)a0(t)] tanh
a(t) +
−
√ [a(t) + ā(t)a0(t)][a(t) + 4ā(t)a0(t)] tanh
2ā(t)
[ 2 −a(t) + a(t)ā(t)a0(t) + 2ā(t)
ξ = x −
2a0(t) 2 ]
t.
6ā(t)
4 Conclusion
( iξ
2
( √
iξ [a(t)+ā(t)a0(t)][a(t)+4ā(t)a0(t)]
2
6ā(t) b(t)
√ )
[a(t)+ā(t)a0(t)][a(t)+4ā(t)a0(t)]
6ā(t) b(t)
We have used two approaches to determine the exact solutions of the equation under study (combined KdV-mKdV
equation). The first direct method was based upon the use of Lie symmetries. The most general traveling wave solution in
implicit form was derived. With the aim to obtain solutions in explicit form we have used the improved tanh-coth method
and derived new traveling wave solutions.It should be noted that other solutions beside the travelling wave solutions can
be constructed using Lie point symmetry X3 and the linear combination of X3 with Lie point symmetries X1 and X2.
Acknowledgments
M. Molati acknowledges financial support by the National University of Lesotho and IMU.
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[5] L. V. Ovsiannikov. Group Analysis of Differential Equations, Academic, 1982.
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Upon Tyne, 1994.
[8] E. Yasar. On the conservation laws and invariant solutions of the mKdV equation, J. Math. Anal. Appl. 363
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[10] C.A. Gómez and A. Salas. The Cole Hopf transformation and improved tanh-coth method applied to new integrable
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