Download Exact Traveling Wave Solutions of Some Nonlinear Equations Using

ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.14(2012) No.4,pp.416-424
Exact Traveling Wave Solutions of Some Nonlinear Equations Using
′
( GG )-expansion Method methods
Fakir Chand ∗ , Anand K Malik
Department of Physics, Kurukshetra University, Kurukshetra-136119, India
(Received 28 June 2009 , accepted 26 September 2012)
′
Abstract: In this paper, we find the exact solutions of some nonlinear evolution equations by using ( GG )expansion method. Four nonlinear models of physical significance i.e. the symmetric regularized longwave equation, the Klein-Gordon-Zakharov equations, the Burgers-Kadomtsev-Petviashvili equation and the
nonlinear Schrödinger equation with a cubic nonlinearity are considered and obtained their exact solutions.
From the general solutions, other well known results are also derived.
Keywords: Nonlinear evolution equations; Solitons; Traveling wave solutions; Nonlinear Schrödinger equation
PACS No.: 02.30.Jr, 05.45.Yv.
1
Introduction
Nonlinear evolution equations are widely used in a variety of fields such as fluid mechanics, quantum mechanics, solid
state physics, plasma physics, population dynamics, chemical kinetics, nonlinear optics etc. [1–4]. Exact analytic solutions of nonlinear equations are always in great demand to explain complex dynamics of the underlying physical systems.
In past, considerable efforts have been made to obtain exact analytical solutions of nonlinear equations with varying degree
of success. For this purpose, a number of methods have been developed for obtaining explicit traveling wave solutions of
nonlinear evolution equations [5–19].
′
Recently, in a seminal paper, Wang et al. [17] introduced a new technique called ( GG )-expansion method for an effective
dealing of nonlinear wave equations. Thereafter, many researches reported a number of new applications of this method
′
[21-30], and developed new variants of the method such as generalized and extended ( GG )-expansion method [25, 26].
′
Here, in the present work, to expand the domain of applications of ( GG )-expansion method, we solve four nonlinear equations namely, the symmetric regularized long-wave (SRLW) equation, the Klein-Gordon-Zakharov (KGZ) equations, the
Burgers-Kadomtsev-Petviashvili (BKP) equation and the Schrödinger equation with a cubic nonlinearity.
The SRLW equation arises in several physical applications, including ion sound waves in a plasma [27]. Recently, Biswas
and coworkers [8, 9] studied the RLW and its generalized version R(m, n) equations using solitary wave ansatz and
obtained 1-soliton solutions and integrals of motions. The KGZ equations present a classical model for describing the
interaction between the Langmuir wave and the ion acoustic wave in plasma [30]. Analytic and numerical solutions of
(1+1) and (2+1) dimensional KGZ equation with power law nonlinearity are discussed in [12]. The BKP equation is used
to model shallow-water waves with weakly non-linear restoring forces and governs the dust acoustic waves propagating
in a collisionless un-magnetized plasma whose constituents are Boltzmann distributed electrons, ions, and massive high
negatively charged warm adiabatic dust grains [31]. Similarly the nonlinear Schrödinger (NLS) equation appears in the
study of nonlinear wave propagation in dispersive and inhomogeneous media such as plasma phenomena and nonuniform
dielectric media [32]. The dynamics of soliton propagation through optical fibers with a variety of nonlinearities can
readily be understood by solving the NLS equation [3, 4].
′
The organization of the paper is as follows: In section 2, a description of the main steps of the ( GG )-expansion method for
finding the traveling wave solutions of nonlinear equations are given. Four illustrative examples, solved by this method,
are given in sections 3, 4, 5 and 6. Finally concluding remarks are presented in section 7.
∗ Corresponding
author.
E-mail address: [email protected] (FC); [email protected] (AKM)
c
Copyright⃝World
Academic Press, World Academic Union
IJNS.2012.12.30/686
′
F. Chand,A. K Malik: Exact traveling wave solutions of some nonlinear equations using ( GG )-expansion method
2
417
′
The ( GG )-expansion method
′
Here we briefly describe the main working steps of the ( GG )-expansion method. For that, consider a nonlinear partial
differential equation (PDE) of the form
P (u, ut , ux , utt , uxt , uxx , ...) = 0,
(1)
where u = u(x, t) is an unknown function and P is polynomial in u(x, t) and its partial derivatives, and contains higher
order derivatives and nonlinear terms.
Step 1: To find the traveling wave solution to Eq.(1), introduce the wave variable
ξ = (x − ct),
(2)
u(x, t) = u(ξ).
(3)
which implies
Based on this, we can easily derive
∂
∂
= −c ,
∂t
∂ξ
2
∂2
2 ∂
=
c
,
∂t2
∂ξ 2
∂
∂
=
,
∂x
∂ξ
∂2
∂2
=
,
∂x2
∂ξ 2
(4)
and so on for other derivatives. Eq.(4) changes the PDE, Eq.(1), to an ODE as
P (u, uξ , uξξ , uξξξ , ...) = 0,
(5)
where uξ , uξξ etc. denote derivatives of u with respect to ξ.
Next integrate Eq.(5) as many times as possible, and set the constants of integration to be zero.
′
Step 2: The solution of Eq.(5) can be expressed by a polynomial in ( GG ) i.e.
u(ξ) = αn (
G′ n
G′
) + αn−1 ( )n−1 + .....,
G
G
(6)
where G = G(ξ) satisfies the second order linear ODE of the form
G′′ + λG′ + µG = 0,
(7)
with αn , αn−1 ,.....,α0 , λ and µ are constants to be determined later and αn ̸= 0. The positive integer n can be determined
by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq.(5).
′
Step 3: Substituting Eq.(6) into Eq.(5) and using Eq.(7), collecting all terms with the same order of ( GG ) together, and
then equating each coefficient of the resulting polynomial to zero yields a set of algebraic equations for αn , αn−1 ,.....,α0 ,
c, λ and µ.
Step 4: Since the general solutions of Eq.(7) have been well known, then substituting αn , αn−1 ,.....,α0 and c and the
general solutions of Eq.(7) into Eq.(6), we obtain more traveling wave solutions of Eq.(1).
Now in the following four sections, we employ the above recipe to develop analytic solutions of four nonlinear equations.
3
The SRLW equation
Now we start with the SRLW equation
utt − uxx +
( u2 )
2
xt
− uxxtt = 0,
(8)
which, after using Eqs.(2)-(4), transforms into the following nonlinear ODE
(c2 − 1)u′′ − c
( u2 )′′
2
− c2 u′′′′ = 0.
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(9)
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International Journal of Nonlinear Science, Vol.14(2012), No.4, pp. 416-424
On integrating Eq.(9) with respect to ξ twice and setting constants of integration to zero, we obtain
c
(c2 − 1)u − u2 − c2 u′′ = 0.
2
(10)
where primes on u denote derivatives with respect to ξ. Further, for the solutions of Eq.(10), we make an ansatz
u(ξ) =
n
( G ′ )i
∑
,
ai
G
i=0
(11)
where G = G(ξ) satisfies the second order linear ODE, Eq.(7). Here, n is a positive integer which is determined by
balancing the highest order derivative term with the highest order nonlinear term and comes out 2 for the present case.
This suggests the choice of u(ξ) in Eq.(11) as
u(ξ) = a0 + a1
( G′ )
G
+ a2
( G′ )2
G
.
(12)
′
The use of Eq.(12) in Eq.(10) and the rationalization of the resultant expression with respect to the powers of ( GG ) yields
the following set of algebraic equations
c
(c2 − 1)a0 − a20 − c2 (2a2 µ2 + a1 µλ) = 0,
2
(c2 − 1)a1 − ca1 a0 − c2 (6a2 µλ + 2a1 µ + a1 λ2 ) = 0,
c
(c2 − 1)a2 − (a21 + 2a2 a0 ) − c2 (8a2 µ + 3a1 λ + 4a2 λ2 ) = 0,
2
−ca2 a1 − c2 (2a1 + 10a2 λ) = 0,
c
− a22 − 6c2 a2 = 0.
2
(13a)
(13b)
(13c)
(13d)
(13e)
These equations can be easily solved for the constants a0 , a1 and a2 as
a0 = −12cµ, − 2c(λ2 + 2µ), a1 = −12cλ,
a2 = −12c.
(14)
By using Eq.(14) in Eq.(12), we get
( G ′ )2
( G′ )
u1 (ξ) = −12c
− 12cλ
− 12cµ,
G
G
( G ′ )2
( G′ )
u2 (ξ) = −12c
− 12cλ
− 2c(λ2 + 2µ).
G
G
The general solution of Eq.(7) can be written as
( √
)
( √
)
1
1
2
2
G = c1 e− 2 λ− λ −4µ ξ + c2 e− 2 λ+ λ −4µ ξ .
(15)
(16)
(17)
Now, substitute this general solution in Eqs.(15) and (16), we obtain the traveling wave solutions of Eq.(8) as
12cc1 c2 (λ2 − 4µ)
√ 2
u1 (ξ) = ( 1 √ 2
)2 ,
1
c1 e 2 (λ −4µ)ξ + c2 e− 2 (λ −4µ)ξ
where (λ2 − 4µ) =
(18)
( c2 −1 )
, and
c2
[
u2 (ξ) = (λ2 − 4µ) − 2c + (
√
1
c1 e 2
12cc1 c2
(λ2 −4µ)ξ
]
,
)
(λ2 −4µ)ξ 2
√
1
+ c2 e− 2
(19)
(
2)
where (λ2 − 4µ) = 1−c
.
c2
Special Cases:
The traveling wave solutions in Eqs.(18) and (19) can further be analyzed under various conditions imposed on c1 and c2
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′
F. Chand,A. K Malik: Exact traveling wave solutions of some nonlinear equations using ( GG )-expansion method
as
Case 1: For c1 = c2 and c2 > 1. Eq.(18) provides soliton solution as
( c2 − 1 )
( √c2 − 1 )
2
u1 (ξ) = 3
sech
ξ,
c
2c
which is same as given in [33]. However, Eq.(19) gives periodic solution in the form
( c2 − 1 )[
( √c2 − 1 ) ]
2
u2 (ξ) =
2 − 3sec
ξ .
c
2c
Case 2: For c1 = c2 and c2 < 1. We obtain periodic solution from Eq.(18) as
( c2 − 1 )
( √1 − c2 )
2
u1 (ξ) = 3
ξ.
sec
c
2c
This solution is same as obtained in [33]. However, from Eq.(19) we obtain soliton solution in the form
( √1 − c2 ) ]
( c2 − 1 )[
2
u2 (ξ) =
2 − 3sech
ξ .
c
2c
419
(20)
(21)
(22)
(23)
Similarly by choosing c1 = -c2 with c2 > 1 and c2 < 1, one can recover the other results of [33] alongwith additional
solutions.
4
The KGZ equations
Next we consider the KGZ equations
utt − uxx + u + uv + |u|2 u = 0,
vtt − vxx = (|u|2 )xx .
(24)
This system describes interaction between Langmuir waves and ion sound waves. These are apparently coupled equations
by two functions u(x, t) and v(x, t) where the function u(x, t) is complex and denotes the fast time scale component
of electric field raised by electrons and the function v(x, t) is real and denotes the deviation of ion density from its
equilibrium.
Thus, again using Eqs.(2)-(4), the above PDEs are transformed into ODEs as
(c2 − 1)u′′ + u + uv + uv + |u|2 u = 0,
(c2 − 1)v ′′ = (|u|2 )′′ .
(25)
On integrating the second equation and for simplicity taking constant of integration to zero, we get
v=
|u|2
.
c2 − 1
(26)
Substitution of Eq.(26) in the first equation of Eq.(25) provides us
(c2 − 1)2 u′′ + (c2 − 1)u + c2 u3 = 0.
As before, using the balancing procedure one obtains n = 1 and thus the ansatz Eq.(11) for u(ξ) is written as
( G′ )
u(ξ) = a0 + a1
.
G
(27)
(28)
Now on inserting Eq.(28) into Eq.(27) and the rationalization of the resultant expression with respect to the powers of
′
( GG ) provides us a set of algebraic equations
2a1 (c2 − 1)2 + c2 a31 = 0,
3λa1 (c2 − 1)2 + 3c2 a21 a0 = 0,
a1 (2µ + λ2 )(c2 − 1)2 + a1 (c2 − 1) + 3c2 a1 a20 = 0,
(29a)
(29b)
(29c)
a1 µλ(c2 − 1)2 + a0 (c2 − 1) + a30 c2 = 0.
(29d)
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International Journal of Nonlinear Science, Vol.14(2012), No.4, pp. 416-424
On solving the above set of algebraic equations, we have
√ ( c2 − 1 )
a1 = ± −2
,
c
λ(c2 − 1)
a0 = ∓ √
, c=±
−2c
√
λ2 − 4µ + 2
.
λ2 − 4µ
(30)
So, with reference to the above solutions, Eq.(28) becomes
√ ( c2 − 1 )( G′ ) λ(c2 − 1)
u(ξ) = ± −2
∓ √
.
c
G
−2c
Finally, the traveling wave solutions of Eq.(24) with the help of Eqs.(17), (26) and (31) are written as
√
√
)
(
( √1 − c2 ) c e( 12 λ2 −4µ)ξ − c e(− 21 λ2 −4µ)ξ
2
1
√ 2
√ 2
,
u(ξ) = ±
1
1
c
c1 e( 2 λ −4µ)ξ + c2 e(− 2 λ −4µ)ξ
√ 2
√ 2
(
)2
1
1
1 c1 e( 2 λ −4µ)ξ − c2 e(− 2 λ −4µ)ξ
√
√
v(ξ) = − 2
.
c c e( 12 λ2 −4µ)ξ + c e(− 12 λ2 −4µ)ξ
1
(31)
(32)
(33)
2
Now, we present some special cases of the general solutions given in Eqs.(32) and (33).
Case 1: When c1 = c2 and λ2 − 4µ > 0. For this particular choice, we have
( √1 − c2 )
(
)
1
u± (ξ) = ±
tanh √
ξ,
c
2(c2 − 1)
(
)
1
1
v(ξ) = − 2 tanh2 √
ξ.
c
2(c2 − 1)
(34)
(35)
Note that solution in Eq.(34) represent the kink shaped soliton solution while solution (35) represents solitary wave
solution.
Case 2: If c1 = c2 and λ2 − 4µ < 0. For this case we get periodic solutions from Eqs.(32) and (33) as
( √c2 − 1 ) (
)
1
u(ξ) = ±
(36)
tan √
ξ,
c
2(1 − c2 )
(
)
1
1
v(ξ) = 2 tan2 √
(37)
ξ.
c
2(1 − c2 )
Similarly, for different choices of c1 and c2 , some other forms of traveling wave solutions of the KGZ equations can also
be obtained.
5
The (2+1)-dimensional BKP equation
A nonlinear dust ion acoustic wave is governed by the KdV-Burger’s equation and such a wave is generated due to the
dissipation caused by the non-adiabatic charge variation of the dust particles. But mostly investigations for the dust charge
fluctuation effects are focused for one-dimensional cases. But, such one-dimensional studies are not sufficient to explain
the observed wave phenomena in the low and higher altitude auroral regions. The wave structure and stability in higher
dimensional systems are subjected to modification because of anisotropy. Therefore, for the study of such phenomena
in higher dimensions the (2+1)-dimensional BKP equation is very helpful [31]. The BKP equation, obtained from the
extension of the KP and the Burger’s equation, is given by [34]
[ut + uux + βuxx ]x + αuyy = 0.
(38)
Following the recipe of the preceding section, we perform a traveling wave reduction of the BKP equation using u(x, y, t) =
u(ξ) with the argument ξ = x + y − ct, generating the nonlinear ODE
(
)′
1
− cu′ + (u2 )′ + βu′′ + αu′′ = 0.
2
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(39)
′
F. Chand,A. K Malik: Exact traveling wave solutions of some nonlinear equations using ( GG )-expansion method
421
On integrating Eq.(39) with respect to ξ twice and setting constants of integration to zero, we obtain
1
(α − c)u + u2 + βu′ = 0.
2
(40)
Again, by balancing procedure, we get n = 1 and form of u(ξ) is same as in Eq.(28). On substituting this ansatz in
′
Eq.(40) and collecting all terms with the same powers of ( GG ) together, the left hand side of Eq.(40) is converted into
′
another polynomial in ( GG ). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for
a0 , a1 , λ, µ and c as
2(α − c)a0 + a20 − 2βµa1 = 0,
(41a)
(α − c)a1 + a1 a0 − βλa1 = 0,
a21 − 2βa1 = 0.
(41b)
(41c)
The set of above algebraic equations yields
(
)
√
a0 = β λ ± λ2 − 4µ ,
a1 = 2β,
c=α±β
√
λ2 − 4µ.
So, using Eqs.(28) and (42), the traveling wave solutions of the BKP equation become
√ 2
√
1
2βc1 (λ2 − 4µ) e 2 (λ −4µ)ξ
√ 2
√ 2
u1 (ξ) =
,
1
1
(c1 e 2 (λ −4µ)ξ + c2 e− 2 (λ −4µ)ξ )
√ 2
√
1
−2βc2 (λ2 − 4µ) e− 2 (λ −4µ)ξ
√ 2
√ 2
u2 (ξ) =
.
1
1
(c1 e 2 (λ −4µ)ξ + c2 e− 2 (λ −4µ)ξ )
(42)
(43)
(44)
Here it is interesting to note that the above two general results reduce in some particular solutions which are obtained by
some other methods. For example, if c1 = c2 and µ = 0, one can obtain
[
{λ
}]
u1 (ξ) = βλ 1 + tanh (x + y − (α + βλ)t) ,
(45)
2
[
{λ
}]
u2 (ξ) = −βλ 1 − tanh (x + y − (α − βλ)t) ,
2
which represent kink shaped soliton solutions. Similarly, if c1 = −c2 and µ = 0, we get
[
{λ
}]
u1 (ξ) = βλ 1 + coth (x + y − (α + βλ)t) ,
2
[
{λ
}]
u2 (ξ) = −βλ 1 − coth (x + y − (α − βλ)t) .
2
(46)
(47)
(48)
which represent traveling wave solutions. The above four solutions are same as derived in [34] using tanh-method.
6
The NLS equation
The NLS equation is one of the most important universal nonlinear model that naturally arises in many physical systems. It
is a generic equation which appears when a quasi-monochromatic wave is propagating in a dispersive and weakly nonlinear
medium, and was used to describe a variety of physical phenomena in hydrodynamics, nonlinear optics, especially in
nonlinear optical fibers, quasi-one-dimensional nonlinear molecular systems, Bose-Einstein condensation etc [3, 4, 32,
35, 36]. The completely integrable NLS equation in one dimension is written as
iwt + wxx + A|w|2 w = 0,
(49)
where w is a complex valued function of the spatial coordinate x and the time t, A is a real parameter. In order to construct
exact solutions of Eq.(49), we transform it into an ODE
u′′ − αu + Au3 = 0,
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(50)
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International Journal of Nonlinear Science, Vol.14(2012), No.4, pp. 416-424
after using the transformation w(x, t) = u(x)eiαt , where α is considered as a real parameter. Suppose the solution of the
′
ODE (50) is expressed by a polynomial in ( GG ) as given in Eq.(11) and G = G(x) satisfies the second order linear ODE
(7).
Considering the homogeneous balance between u′′ and u3 in Eq.(50), we get, n=1. Hence ansatz to Eq.(50) is same as
′
given in Eq.(28). Now substituting Eq.(28) in Eq.(50) and equating the coefficients of various powers of ( GG ), we get the
following algebraic equations
2a1 + Aa31 = 0,
3λa1 + 3Aa21 a0 = 0,
(51a)
(51b)
a1 (2µ + λ2 ) − αa1 + 3Aa1 a20 = 0,
a1 µλ − αa0 + Aa30 = 0.
(51c)
(51d)
These equations are solved for constants a0 , a1 and α in terms of parameters A, µ and λ as
√
λ
−2
(4µ − λ2 )
a1 = ±
, a0 = ∓ √
, α=
.
A
2
−2A
Now using Eqs.(17) and (52) in Eq.(28), we get
√ 2
√ 2
)
(
√
1
1
4µ − λ2 c1 e( 2 λ −4µ)x − c2 e(− 2 λ −4µ)x
√ 2
√ 2
u(x) = ±
.
1
1
2A
c e( 2 λ −4µ)x + c e(− 2 λ −4µ)x
1
(52)
(53)
2
Finally the general solutions of the NLSE can be written as
√ −α )
√ ( √ −α x
α c1 e 2 − c2 e 2 x iαt
√
√ −α
w(x, t) = ±
e .
A c e −α
2 x + c e
2 x
1
(54)
2
Next, we discuss some special cases of Eq.(54).
Case 1: When c1 = c2 and λ2 − 4µ > 0 or α < 0. For this condition, Eq.(54) reduces to
√
(√ −α )
α
w(x, t) = ±
tanh
x eiαt .
A
2
(55)
Note that the real and imaginary parts of Eq.(55) represents periodic wave solutions while modulus part represent kink
shaped soliton solution.
Case 2: If c1 = c2 and λ2 − 4µ < 0 or α > 0. For this case, we obtain
√
(√ α )
α
w(x, t) = ±i
tan
x eiαt ,
(56)
A
2
which again represents the periodic wave solution. Note that the above two results in Eqs.(55) and (56) are same as
obtained by using tanh-method in [35].
Similarly by selecting different choices of c1 and c2 , one can get other forms of exact solutions of the NLS equation with
cubic nonlinearity.
7
Conclusion
In the present work, we successfully obtained the exact solutions of the symmetric regularized long wave equation, the
Klein-Gordon-Zakharov equations, the Burgers-Kadomtsev-Petviashvili equation and the nonlinear Schrödinger wave
′
equation with a cubic nonlinearity within the framework of ( GG )-expansion method. The obtained exact and explicit
analytic solutions are in general forms involving various arbitrary parameters. It is interesting to note that when parameters
are taken as special values in general solutions, the solitary waves such as kink-antikink shaped, bell shaped soliton and
periodic solutions can be derived. It is also interesting to note that from the general results, one can easily recover solutions
′
which are obtained from others methods. We also conclude that the ( GG )-expansion method is a direct, concise, powerful,
effective and convenient technique can be used for all integrable and non-integrable nonlinear models. Performance of
this method is reliable, simple and gives many new solutions. It is also a standard and computerizable method which
allows in to solve complicated nonlinear evolution equations in diverse areas of science. Moreover, this method is capable
of greatly minimizing the size of computational work compared to other existing techniques.
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′
F. Chand,A. K Malik: Exact traveling wave solutions of some nonlinear equations using ( GG )-expansion method
423
Acknowledgements
The authors are very thankful to the reviewer for his useful suggestions to improve the quality of presentation of this paper.
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