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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 810729, 11 pageshttp://dx.doi.org/10.1155/2013/810729

Research ArticleNew Exact Solutions of Ion-Acoustic Wave Equations by(𝐺/𝐺)-Expansion Method

Wafaa M. Taha, M. S. M. Noorani, and I. Hashim

School of Mathematical Sciences, Universiti Kebangsaan Malaysia UKM, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Wafaa M. Taha; wafaa [email protected]

Received 24 May 2013; Revised 19 September 2013; Accepted 20 September 2013

Academic Editor: Saeid Abbasbandy

Copyright © 2013 Wafaa M. Taha et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The (𝐺/𝐺)-expansion method is used to study ion-acoustic waves equations in plasma physic for the first time. Many new exacttraveling wave solutions of the Schamel equation, Schamel-KdV (S-KdV), and the two-dimensional modified KP (Kadomtsev-Petviashvili) equation with square root nonlinearity are constructed. The traveling wave solutions obtained via this method areexpressed by hyperbolic functions, the trigonometric functions, and the rational functions. In addition to solitary waves solutions,a variety of special solutions like kink shaped, antikink shaped, and bell type solitary solutions are obtained when the choice ofparameters is taken at special values. Two- and three-dimensional plots are drawn to illustrate the nature of solutions. Moreover,the solution obtained via this method is in good agreement with previously obtained solutions of other researchers.

1. Introduction

The ion-acoustic solitary wave is one of the fundamentalnonlinear waves phenomena appearing in fluid dynamics [1]and plasma physics [2]. To allowing for the trapping of someof the electrons on ion-acoustic waves, Schamel proposed amodified equation for ion-acoustic waves [3] given by

𝑢𝑡+ 𝑢1/2𝑢𝑥+ 𝛿𝑢𝑥𝑥𝑥

= 0, (1)

where 𝑢 is thewave potential and 𝛿 is a constant, this equationdescribing the ion-acoustic wave, where the electrons do notbehave isothermally during their passage of the wave in acold-ion plasma. Then, combining the equations of Schameland the KdV equation, one obtains the so-called one-dimensional form of the Schamel-KdV (S-KdV) equationequation [4, 5]:

𝑢𝑡+ (𝛼𝑢

1/2+ 𝛽𝑢) 𝑢

𝑥+ 𝛿𝑢𝑥𝑥𝑥

= 0, 𝛿𝛽 ̸= 0, (2)

where 𝛽, 𝛼, and 𝛿 are constants. This equation is establishedin plasma physics in the study of ion acoustic solitonswhen electron trapping is present, and also it governs theelectrostatic potential for a certain electron distribution in

velocity space. Note that we obtain the KdV equation when𝛼 = 0 and the Schamel equation when 𝛽 = 0 for 𝛿 = 1.Due to the wide range of applications of (2), it is importantto find new exact wave solutions of the Schamel-KdV (S-KdV) equation. Another equation arising in the study ofion-acoustic waves is the so-called modified Kadomtsev-Petviashvili (KP) equation given by [6]

(𝑢𝑡+ 𝛼𝑢1/2𝑢𝑥+ 𝛽𝑢𝑥𝑥𝑥

)𝑥+ 𝛿𝑢𝑦𝑦

= 0. (3)

Equation (3) was firstly derived by Chakraborty andDas [7]; the modified KP equation containing a square rootnonlinearity is a very attractive model for the study of ion-acoustic waves in a multispecies plasma made up of non-isothermal electrons in plasma physics.

In the literature, theKP equation is also known as the two-dimensional KdV equation [8].

It has lately become more interesting to obtain exactanalytical solutions to nonlinear partial differential equationssuch as the one arising from the ion-acoustic wave phe-nomena, by using appropriate techniques. Several importanttechniques have been developed such as the tanh-method [9,10], sine-cosine method [11, 12], tanh-coth method [13], exp-functionmethod [14], homogeneous-balancemethod [15, 16],

2 Journal of Applied Mathematics

Jacobi-elliptic function method [17, 18], and first-integralmethod [19, 20] to solve analytically nonlinear equations suchas the above ion-acoustic wave equations.

Moreover, in the standard tanh method developed byMalfliet in 1992 [21], the tanh is used as a new variable. Sinceall derivatives of a tanh are represented by tanh itself, thesolution obtained by this method may be solitons in termsof sech2 or may be kinks in terms of tanh. We believe thatthe (𝐺/𝐺)-expansion method is more efficient than the tanhmethod. Moreover, the tanh method may yield more thanone soliton solution, a capability which the tanh methoddoes not have. The sine-cosine method yields a solutionin trigonometric form. The Exp-function method leads toboth generalized solitary solution and periodic solutions.Thehomogeneous-balancemethod is a generalized tanh functionmethod for many nonlinear PDEs. The first integral method,which is based on the ring theory of commutative algebra,was first proposed by Feng. There is no general theory tellingus how to find its first integrals in a systematic way; so, akey idea of this approach to find the first integral is to utilizethe division theorem.The traveling wave solutions expressedby the (𝐺/𝐺)-expansion method, which was first proposedby Wang et al. [22], transform the given difficult probleminto a set of simple problems which can be solved easily toget solutions in the forms of hyperbolic, trigonometric, andrational functions. The main merits of the (𝐺/𝐺)-expansionmethod over the other methods are as follows.

(i) Higher-order nonlinear equations can be reduced toODEs of order greater than 3.

(ii) There is no need to apply the initial and boundaryconditions at the outset. The method yields a generalsolution with free parameters which can be identifiedby the above conditions.

(iii) The general solution obtained by the (𝐺/𝐺)-expan-sion method is without approximation.

(iv) The solution procedure can be easily implemented inMathematica or Maple.

In fact, the (𝐺/𝐺)-expansion method has been success-fully applied to obtain exact solution for a variety of NLPDE[23–34].

In this paper, the (𝐺/𝐺)-expansion method is used tostudy ion-acoustic waves equations in plasma physic for thefirst time.We obtainmany new exact traveling wave solutionsfor the Schamel equation, S-KdV, and the two-dimensionalmodified KP equation.The traveling wave solutions obtainedvia this method are expressed by hyperbolic functions,the trigonometric functions, and the rational functions. Inaddition to solitary waves solutions, a variety of specialsolutions like kink shaped, antikink shaped, and bell typesolitary solutions are obtained when the choice of parametersis taken at special values. Two- and three-dimensional plotsare drawn to illustrate the nature of solutions. Moreover, thesolution obtained via this method is in good agreement withpreviously obtained solutions of other researchers.

Our paper is organized as follows: in Section 2, we presentthe summary of the (𝐺/𝐺)-expansionmethod, and Section 3describes the applications of the (𝐺/𝐺)-expansion methodfor Schamel equation, S-KdV equation, and modified KPequation, and lastly, conclusions are given in Section 4.

2. Summary of the (𝐺/𝐺)-Expansion Method

In this section, we describe the (𝐺/𝐺)-expansion methodfor finding traveling wave solutions of NLPDE. Suppose thata nonlinear partial differential equation in two independentvariables, 𝑥 and 𝑡, is given by

𝑝 (𝑢, 𝑢𝑡, 𝑢𝑥, 𝑢𝑥𝑡, 𝑢𝑡𝑡, 𝑢𝑥𝑥, . . .) = 0, (4)

where 𝑢 = 𝑢 (𝑥, 𝑡) is an unknown function, 𝑃 is a polynomialin 𝑢 = 𝑢 (𝑥, 𝑡) and its various partial derivatives, in whichhighest order derivatives and nonlinear terms are involved.

The summary of the (𝐺/𝐺)-expansion method can bepresented in the following six steps.

Step 1. To find the traveling wave solutions of (4), weintroduce the wave variable:

𝑢 (𝑥, 𝑡) = 𝑢 (𝜁) , 𝜁 = (𝑥 − 𝑐𝑡) , (5)

where the constant 𝑐 is generally termed the wave velocity.Substituting (5) into (4), we obtain the following ordinary dif-ferential equations (ODE) in 𝜁 (which illustrates a principaladvantage of a traveling wave solution; i.e., a PDE is reducedto an ODE):

𝑝 (𝑢, 𝑐𝑢, 𝑢, 𝑐𝑢, 𝑐2𝑢, 𝑢, . . .) = 0. (6)

Step 2. If necessary, we integrate (6) asmany times as possibleand set the constants of integration to be zero for simplicity.

Step 3. Suppose that the solution of nonlinear partial differ-ential equation can be expressed by a polynomial in (𝐺/𝐺)as

𝑢 (𝜁) =

𝑚

∑

𝑖=0

𝑎𝑖(𝐺

𝐺

)

𝑖

, (7)

where 𝐺 = 𝐺(𝜁) satisfies the second-order linear ordinarydifferential equation

𝐺

(𝜁) + 𝜆𝐺

(𝜁) + 𝜇𝐺 (𝜁) = 0, (8)

where 𝐺 = 𝑑𝐺/𝑑𝜁, 𝐺 = 𝑑2𝐺/𝑑𝜁2, and 𝑎𝑖, 𝜆, and 𝜇 are real

constants with 𝑎𝑚

̸= 0. Here, the prime denotes the derivativewith respect to 𝜁. Using the general solutions of (8),

Journal of Applied Mathematics 3

we have

(𝐺

𝐺

)

=

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

−𝜆

2

+

√𝜆2− 4𝜇

2

×(

𝑐1sinh {(√𝜆2 − 4𝜇/2) 𝜁}+ 𝑐

2cosh {(√𝜆2 − 4𝜇/2) 𝜁}

𝑐1cosh {(√𝜆2 − 4𝜇/2) 𝜁}+ 𝑐

2sinh {(√𝜆2 − 4𝜇/2) 𝜁}

) ,

𝜆2− 4𝜇 > 0,

−𝜆

2

+

√4𝜇 − 𝜆2

2

×(

−𝑐1sin {(√4𝜇 − 𝜆2/2) 𝜁}+ 𝑐

2cos {(√4𝜇 − 𝜆2/2) 𝜁}

𝑐1cos {(√4𝜇 − 𝜆2/2) 𝜁}+ 𝑐

2sin {(√4𝜇 − 𝜆2/2) 𝜁}

) ,

𝜆2− 4𝜇 < 0,

(𝑐2

𝑐1+ 𝑐2𝜁

) −𝜆

2

, 𝜆2− 4𝜇 = 0.

(9)The above results can be written in simplified forms as

(𝐺

𝐺

)

=

{{{{{{{{{{{{{

{{{{{{{{{{{{{

{

−𝜆

2

+

√𝜆2− 4𝜇

2

tanh{{

{{

{

√𝜆2− 4𝜇

2

𝜁

}}

}}

}

, 𝜆2− 4𝜇 > 0,

−𝜆

2

+

√4𝜇 − 𝜆2

2

tan{{

{{

{

√4𝜇 − 𝜆2

2

𝜁

}}

}}

}

, 𝜆2− 4𝜇 < 0,

(𝑐2

𝑐1+ 𝑐2𝜁

) −𝜆

2

, 𝜆2− 4𝜇 = 0.

(10)

Step 4. The positive integer 𝑚 can be accomplished byconsidering the homogeneous balance between the highestorder derivatives and nonlinear terms appearing in (6) asfollows: if we define the degree of 𝑢(𝜁) as 𝐷[𝑢(𝜁)] = 𝑚, thenthe degree of other expressions is defined by

𝐷[𝑑𝑞𝑢

𝑑𝜁𝑞] = 𝑚 + 𝑞,

𝐷 [𝑢𝑟(𝑑𝑞𝑢

𝑑𝜁𝑞)

𝑠

] = 𝑚𝑟 + 𝑠 (𝑞 + 𝑚) .

(11)

Therefore, we can get the value of𝑚 in (7).

Step 5. Substituting (7) into (6), useing general solutions of(8), and collecting all terms with the same order of (𝐺/𝐺)together, then setting each coefficient of this polynomial tozero yields a set of algebraic equations for 𝑎

𝑖, 𝑐, 𝜆, and 𝜇.

Step 6. Substitute 𝑎𝑖, 𝑐, 𝜆, and 𝜇 obtained in Step 5 and the

general solutions of (8) into (7). Next, depending on the signof the discriminant 𝐴 = 𝜆2 − 4𝜇, we can obtain the explicitsolutions of (4) immediately.

3. Applications of the(𝐺/𝐺)-Expansion Method

3.1. Schamel Equation. In order to find the solitary wave solu-tion of (1), we use the transformations

𝑢 (𝑥, 𝑡) = V2 (𝑥, 𝑡) , V (𝑥, 𝑡) = V (𝜁) , 𝜁 = 𝑘𝑥 − 𝑐𝑡.(12)

Then, (1) becomes

−𝑐VV + 𝑘V2V + 𝛿𝑘3 (VV + 3VV) = 0. (13)

Integrating (13) with respect to 𝜁 and setting the integrationconstant equal to zero, we have

−𝑐

2

V2 +𝑘

3

V3 + 𝑘3𝛿(V)2

+ 𝑘3𝛿VV = 0. (14)

According to the previous steps, using the balancing proce-dure between V3 with VV in (14), we get 3𝑚 = 2𝑚 + 2 so that𝑚 = 2. Now, assume that (14) has the following solution:

V (𝜁) = 𝑎0+ 𝑎1(𝐺

𝐺

) + 𝑎2(𝐺

𝐺

)

2

, 𝑎2

̸= 0, (15)

where 𝑎0, 𝑎1, and 𝑎

2are unknown constants to be determined

later. Substituting (15) along with (8) into (14) and collectingall terms with the same order of (𝐺/𝐺), the left hand side of(14) is converted into a polynomial in (𝐺/𝐺). Equating eachcoefficient of the resulting polynomials to zero yields a set ofalgebraic equations for 𝑎

0, 𝑎1, 𝑎2, 𝛿, 𝜆, 𝑐, 𝑘, and 𝜇 as follows:

(𝐺

𝐺

)

6

:1

3

𝑘𝑎3

2+ 10𝑘

3𝛿𝑎2

2= 0,

(𝐺

𝐺

)

5

: 18𝑘3𝛿𝑎2

2𝜆 + 12𝑘

3𝛿𝑎1𝑎2+ 𝑘𝑎1𝑎2

2= 0,

(𝐺

𝐺

)

4

: 6𝑘3𝛿𝑎0𝑎2+ 16𝑘

3𝛿𝑎2

2𝜇 + 8𝑘

3𝛿𝑎2

2𝜆2−1

2

𝑐𝑎2

2

+ 21𝑘3𝛿𝑎2

1+ 𝑘𝑎0𝑎2

2+ 𝑘𝑎2

1𝑎2= 0,

(𝐺

𝐺

)

3

: 14𝑘3𝛿𝑎2

2𝜆𝜇 + 5𝑘

3𝛿𝑎2

1𝜆 + 18𝑘

3𝛿𝑎1𝑎2𝜇

+ 9𝑘3𝛿𝑎1𝜆2𝑎2+ 2𝑘𝑎

0𝑎1𝑎2− 𝑐𝑎1𝑎2+1

3

𝑘𝑎3

1

+ 2𝑘3𝛿𝑎0𝑎1+ 10𝑘

3𝛿𝑎0𝑎2𝜆 = 0,


(𝐺

𝐺

)

2

: 2𝑘3𝛿𝑎2

1𝜆2+ 4𝑘3𝛿𝑎2

1𝜇 + 8𝑘

3𝛿𝑎0𝑎2𝜇 −

1

2

𝑐𝑎2

1− 𝑐𝑎0𝑎2

+ 15𝑘3𝛿𝑎1𝑎2𝜆𝜇 + 𝑘𝑎

0𝑎2

1+ 4𝑘3𝛿𝑎0𝑎2𝜆2

+ 6𝑘3𝛿𝑎2

2𝜇2+ 3𝑘𝛿𝑎

0𝑎1𝜆 + 𝑘𝑎

2

0= 0,

(𝐺

𝐺

)

1

: 2𝑘3𝑎0𝑎1𝜇 + 𝑘3𝛿𝑎0𝑎1𝜆2+ 6𝑘3𝛿𝑎0𝑎2𝜆𝜇 + 𝑘𝑎

2

0

− 𝑐𝑎0𝑎1+ 6𝑘3𝑎1𝑎2𝛿𝜇2+ 3𝑘3𝛿𝜆𝜇𝑎2

1= 0,

(𝐺

𝐺

)

0

: 𝑘3𝛿𝑎2

1𝜇2−1

2

𝑐𝑎2

0+1

3

𝑘𝑎3

0+ 𝑘3𝛿𝜆𝜇𝑎0𝑎1

+ 2𝑘3𝛿𝜇2𝑎0𝑎2= 0.

(16)

On solving the above set of algebraic equations by Maple, wehave

𝑎0= −30𝜇𝛿𝑘

2, 𝑎

1= −30𝑘

2𝛿𝜆,

𝑎2= −30𝑘

2𝛿, 𝑐 = 𝛿𝑘

3(4𝜆2− 16𝜇) .

(17)

Now, (15) becomes

V (𝜁) = −30𝜇𝛿𝑘2 − 30𝑘2𝛿𝜆(𝐺

𝐺

) − 30𝑘2𝛿(

𝐺

𝐺

)

2

. (18)

Substituting the general solution of (8) into (18), we obtainthe three types of traveling wave solutions depending on thesign of 𝐴 = 𝜆2 − 4𝜇.

If 𝐴 > 0, we have the following general hyperbolictraveling wave solutions of (1):

V (𝑥, 𝑡)

= −30𝜇𝛿𝑘2− 30𝑘

2𝛿𝜆

× [−𝜆

2

+

√𝐴

2

× (

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

− 30𝑘2𝛿 [

−𝜆

2

+

√𝐴

2

×(

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

2

,

(19)

where 𝑐1and 𝑐2are arbitrary constants.

If 𝐴 < 0, we have the following general trigonometricfunction solutions of (1):

V (𝑥, 𝑡)

= −30𝜇𝛿𝑘2− 30𝑘

2𝛿𝜆

× [−𝜆

2

+

√𝐴

2

×(

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {√𝐴/2}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

− 30𝑘2𝛿 [

−𝜆

2

+

√𝐴

2

× (

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {(√𝐴/2) 𝜁}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

2

.

(20)

If 𝐴 = 0, we have the following general rational functionsolutions of (1):

V (𝑥, 𝑡) = − 30𝜇𝛿𝑘2 − 30𝑘2𝛿𝜆 [−𝜆

2

+ (𝑐2

𝑐1+ 𝑐2𝜁

)]

− 30𝑘2𝛿[−

𝜆

2

+ (𝑐2

𝑐1+ 𝑐2𝜁

)]

2

,

(21)

where 𝜁 = 𝑘𝑥 − 𝛿𝑘3(4𝜆2 − 16𝜇)𝑡.Writing 𝑢(𝑥, 𝑡) = V2(𝑥, 𝑡) and setting 𝑐

2= 𝜇 = 0 and 𝜆 = 2

in (19), we reproduce the result of Khater andHassan [35] (seetheir Equation (4.7)),

𝑢 (𝑥, 𝑡) = 4 (900𝑘4𝛿2sech4 {𝜁}) , (22)

where 𝜁 = 𝑘𝑥 − 16𝛿𝑘3𝑡.Note that Khater and Hassan [35] obtained only hyper-

bolic solutions, but in this work, we found two additionaltypes of solutions, that is, trigonometric and rational solu-tions.

3.2. S-KdV Equation. To find the general exact solutions of(2), we first write 𝑢(𝑥, 𝑡) = V2(𝑥, 𝑡) to transform (2) into

VV𝑡+ (𝛼V + 𝛽V2) V

𝑥+ 𝛿VV𝑥𝑥𝑥

= 0. (23)

Assume the traveling wave solution of (23) in the formV (𝑥, 𝑡) = 𝑉 (𝜁) , 𝜁 = 𝑘 (𝑥 − 𝑐𝑡) . (24)

Hence, (23) becomes−𝑐𝑉𝑉

+ (𝛼𝑉

2+ 𝛽𝑉3)𝑉+ 𝑘2𝛿 (𝑉𝑉

+ 3𝑉𝑉) = 0.

(25)

Suppose that the solution of (25) can be expressed by apolynomial in (𝐺/𝐺) as

𝑉 (𝜁) =

𝑚

∑

𝑖=0

𝑎𝑖(𝐺

𝐺

)

𝑖

, 𝑎𝑖̸= 0 (26)


and𝐺(𝜁) satisfies (8).The homogeneous balance between thehighest order derivative 𝑉𝑉 and the nonlinear term 𝑉3𝑉appearing in (25) yields 𝑚 = 1, and hence, we take thefollowing formal solution:

𝑉 (𝜁) = 𝑎0+ 𝑎1(𝐺

𝐺

) , (27)

where the positive integers 𝑎0and 𝑎

1are to be determined

later. Substituting (27) along with (8) into (25), collectingall the terms with the same power of (𝐺/𝐺), and equatingeach coefficient to zero yield a set of simultaneous algebraicequations for 𝑎

0, 𝑎1, 𝑐, 𝑘, 𝛼, 𝛽, and 𝛿 as follows:

(𝐺

𝐺

)

5

: −𝛽𝑎4

1− 12𝛿𝑘

2𝑎2

1= 0,

(𝐺

𝐺

)

4

: − 3𝛽𝑎3

1𝑎0− 𝛽𝑎4

1𝜆 − 27𝛿𝑘

2𝑎2

1𝜆 − 𝛼𝑎

3

1

− 6𝛿𝑘2𝑎0𝑎1= 0,

(𝐺

𝐺

)

3

: − 3𝛽𝑎3

1𝜆𝑎0− 19𝛿𝑘

2𝑎2

1𝜆2− 12𝛿𝑘

2𝑎0𝑎1𝜆

− 20𝛿𝑘2𝑎2

1𝜇 − 𝛼𝑎

3

1𝜆 − 𝛽𝑎

4

1𝜇 + 𝑐𝑎

2

1− 2𝛼𝑎

2

1𝑎0

− 3𝛽𝑎2

1𝑎2

0= 0,

(𝐺

𝐺

)

2

: 𝑐𝑎0𝑎1− 3𝛽𝑎

2

1𝜆𝑎2

0− 2𝛼𝑎

2

1𝜆𝑎0+ 𝑐𝑎2

1𝜆 − 26𝛿𝑘

2𝑎2

1𝜆𝜇

− 8𝛿𝑘2𝑎0𝑎1𝜇 − 7𝛿𝑘

2𝑎0𝑎1𝜆2− 3𝛽𝑎

3

1𝜇𝑎0− 𝛽𝑎1𝑎3

0

− 𝛼𝑎1𝑎2

0− 𝛼𝑎3

1𝜇 = 0,

(𝐺

𝐺

)

1

: − 𝛼𝜆𝑎1𝑎2

0+ 𝑐𝑎0𝑎1𝜆 − 8𝛿𝑘

2𝑎2

1𝜇2− 7𝛿𝜇𝑘

2𝑎2

1𝜆2

− 3𝛽𝜇𝑎2

1𝑎2

0+ 𝑐𝑎2

1𝜇 − 𝛽𝜆𝑎

1𝑎3

0− 8𝛿𝜆𝜇𝑘

2𝑎0𝑎1

− 2𝛼𝜇𝑎2

1𝑎0= 0,

(𝐺

𝐺

)

0

: 𝑐𝜇𝑎0𝑎1− 𝛼𝜇𝑎

1𝑎2

0− 𝛽𝜇𝑎

1𝑎3

0− 3𝛿𝜆𝑘

2𝑎2

1𝜇2

− 𝛿𝜇𝜆2𝑘2𝑎0𝑎1− 2𝛿𝜇

2𝑘2𝑎0𝑎1= 0.

(28)

The above system admits the following sets of solutions:

𝑎0= 0, 𝑎

1=

4𝛼

5𝛽𝜆

, 𝜇 = 0,

𝑐 = −16𝛼2

75𝛽

, 𝑘 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

,

(29)

𝑎0= −

4𝛼

5𝛽

, 𝑎1= −

4𝛼

5𝛽𝜆

, 𝜇 = 0,

𝑐 = −16𝛼2

75𝛽

, 𝑘 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

,

(30)

𝑎0=

5𝛽𝑎1𝜆 − 4𝛼

10𝛽

, 𝜇 =

25𝛽2𝑎2

1𝜆2− 16𝛼

2

100𝛽2𝑎2

1

,

𝑐 = −16𝛼2

75𝛽

, 𝑘 = ±√𝛽

−12𝛿

𝑎1.

(31)

Now, substituting (29)-(30) into (27) gives, respectively,

𝑉1(𝜁) =

4𝛼

5𝛽𝜆

(𝐺

𝐺

) ,

𝑉2(𝜁) = −

4𝛼

5𝛽

−4𝛼

5𝛽𝜆

(𝐺

𝐺

) ,

𝑉3(𝜁) =

5𝛽𝑎1𝜆 − 4𝛼

10𝛽

+ 𝑎1(𝐺

𝐺

) .

(32)

When substituting the general solutions (9) into (32), weobtain the following three types of traveling wave solutions.

Case 𝐴 > 0: (hyperbolic type)

V1(𝑥, 𝑡) = −

2𝛼

5𝛽

+2𝛼

5𝛽

(𝑐1sinh {(𝜆/2) 𝜁} + 𝑐

2cosh {(𝜆/2) 𝜁}

𝑐1cosh {(𝜆/2) 𝜁} + 𝑐

2sinh {(𝜆/2) 𝜁}

) ,

𝜁 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

𝑥 +16𝛼2

75𝛽

𝑡,

V2(𝑥, 𝑡) = −

2𝛼

5𝛽

−2𝛼

5𝛽

(𝑐1sinh {(𝜆/2) 𝜁} + 𝑐

2cosh {(𝜆/2) 𝜁}

𝑐1cosh {(𝜆/2) 𝜁} + 𝑐

2sinh {(𝜆/2) 𝜁}

) ,

𝜁 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

𝑥 +16𝛼2

75𝛽

𝑡,

V3(𝑥, 𝑡)

= −2𝛼

5𝛽

+2𝛼

5𝛽

× (

𝑐1sinh {(2𝛼/5𝛽𝑎

1) 𝜁} + 𝑐

2cosh {(2𝛼/5𝛽𝑎

1) 𝜁}

𝑐1cosh {(2𝛼/5𝛽𝑎

1) 𝜁} + 𝑐

2sinh {(2𝛼/5𝛽𝑎

1) 𝜁}

) ,

𝜁 = ±√𝛽

−12𝛿

𝑎1𝑥 +

16𝛼2

75𝛽

𝑡.

(33)


If we set 𝑐2= 0 and write 𝑢(𝑥, 𝑡) = V2(𝑥, 𝑡), then the above

solutions can be written as

𝑢1(𝑥, 𝑡) =

4𝛼2

25𝛽2[−1 ± tanh( 𝛼

5√−3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

,

(34)

𝑢2(𝑥, 𝑡) =

4𝛼2

25𝛽2[1 ± tanh( 𝛼

5√−3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

,

(35)

𝑢3(𝑥, 𝑡) =

4𝛼2

25𝛽2[−1 ± 2 tanh( 𝛼

5√−3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

.

(36)

Note that (35) is exactly the same solution of Khater andHassan [35] as given in their first equation of (3.9) with 𝜁

0= 0.

Similarly we can obtain the second solution of (36) in Hassan[5] if we set 𝑐

1= 0 in our solution (35). The solution (35)

represents kink shaped solitary and antikink shaped solitarysolutions (depending upon the choice of sign) which areshown graphically in Figure 1 for the case 𝑐

1= 1.

Case 𝐴 < 0: (trigonometric type)

𝑉1(𝑥, 𝑡)=−

2𝛼

5𝛽

+2𝛼𝑖

5𝛽

(−𝑐1sin {(𝑖𝜆/2) 𝜁} + 𝑐

2cos {(𝑖𝜆/2) 𝜁}

𝑐1cos {(𝑖𝜆/2) 𝜁} + 𝑐

2sin {(𝑖𝜆/2) 𝜁}

) ,

𝜁 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

𝑥 +16𝛼2

75𝛽

𝑡,

𝑉2(𝑥, 𝑡) = −

2𝛼

5𝛽

−2𝑖𝛼

5𝛽

× (−𝑐1sin {(𝑖𝜆/2) 𝜁} + 𝑐

2cos {(𝑖𝜆/2) 𝜁}

𝑐1cos {(𝑖𝜆/2) 𝜁} + 𝑐

2sin {(𝑖𝜆/2) 𝜁}

) ,

𝜁 = ±

2√1/ − 75𝛿𝛽𝛼

𝜆

𝑥 +16𝛼2

75𝛽

𝑡,

𝑉3(𝑥, 𝑡)

= −2𝛼

5𝛽

+2𝑖𝛼

5𝛽

× (

−𝑐1sin {(2𝑖𝛼/5𝛽𝑎

1) 𝜁} + 𝑐

2cos {(2𝑖𝛼/5𝛽𝑎

1) 𝜁}

𝑐1cos {(2𝑖𝛼/5𝛽𝑎

1) 𝜁} + 𝑐

2sin {(2𝑖𝛼/5𝛽𝑎

1) 𝜁}

) ,

𝜁 = ±√𝛽

−12𝛿

𝑎1𝑥 +

16𝛼2

75𝛽

𝑡.

(37)

But if 𝑐2= 0 and 𝑢(𝑥, 𝑡) = V2(𝑥, 𝑡), then trigonometric

type solution becomes

𝑢1(𝑥, 𝑡) =

4𝛼2

25𝛽2[−1 ± 𝑖 tan( 𝛼

5√3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

,

𝑢2(𝑥, 𝑡) =

4𝛼2

25𝛽2[1 ± 𝑖 tan( 𝛼

5√3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

,

𝑢3(𝑥, 𝑡) =

4𝛼2

25𝛽2[−1 ± 2𝑖 tan( 𝛼

5√3𝛽𝛿

[𝑥 +16𝛼2

75𝛽

𝑡])]

2

.

(38)

Case 𝐴 = 0: (rational type)

𝑢1(𝑥, 𝑡) =

4𝛼2

25𝛽2[−1 + 2(

𝑐2

𝑐1+ 𝑐2𝜁

)]

2

,

𝑢2(𝑥, 𝑡) =

4𝛼2

25𝛽2[1 + 2(

𝑐2

𝑐1+ 𝑐2𝜁

)]

2

,

𝑢3(𝑥, 𝑡) = [

2𝛼

5𝛽

+ 𝑎1(

𝑐2

𝑐1+ 𝑐2𝜁

)]

2

.

(39)

As mentioned before, the (𝐺/𝐺)-expansion methodgives more general types of solutions than that found byKhater and Hassan [35] and Hassan [5].

3.3.TheModified Two-Dimensional KP (Kadomtsev-Petviash-vili) Equation. The modified KP equation (3) containing asquare root nonlinearity is a very attractive model for thestudy of ion-acoustic waves in plasma physic [8]. We willobtain more general exact solutions of the modified KPequation. In order to find the traveling wave solution of (3),we let

V (𝑥, 𝑦, 𝑡) = V (𝜁) , 𝜁 = (𝑥 + 𝑘𝑦 − 𝑐𝑡) . (40)

Now, taking 𝑢(𝑥, 𝑦, 𝑡) = V2(𝑥, 𝑦, 𝑡), (3) becomes

(−𝑐 + 𝛿𝑘2) VV + (−𝑐 + 𝛿𝑘2) V2 + 𝛼V2V + 2𝛼VV2

+ 𝛽VV + 4𝛽VV + 3𝛽(V)2

= 0,

(41)

where 𝑘, 𝑐, 𝛽, 𝛿, and 𝛼 are constants and the prime denotesdifferentiation with respect to 𝜁. Integrating (41) with respectto 𝜁 and setting the integration constant equal to zero, weobtain

(−𝑐 + 𝛿𝑘2) VV + 𝛼V2V + 3𝛽VV + 𝛽VV = 0. (42)

Balancing V2V with VV gives𝑚 = 2. Therefore, we can writethe solution of (42) in the form

V (𝜁) = 𝑎0+ 𝑎1(𝐺

𝐺

) + 𝑎2(𝐺

𝐺

)

2

, (43)

where 𝑎0, 𝑎1, and 𝑎

2are constants to be determined later.

Substituting (43) along with (8) into (42) and collecting allterms with the same order of (𝐺/𝐺), the left hand sides of(42) are converted into a polynomial in (𝐺/𝐺). Setting eachcoefficient of each polynomial to zero, we derive a set of


0.6

0.5

0.4

0.3

0.2

0.1

0−40 −20 0 20 40

x

u(x)

(a)

0246810

t

0.6

0.5

0.4

0.3

0.2

0.1

0

−40−2002040x

u(x,t)

(b)

Figure 1: (a) 2D profile of (35): kink shaped solitary (𝑢+, blue line), anti-kink shaped solitary (𝑢−, black line). (b) Corresponding 3D plotswhen + sign is taken and when −ve sign is taken, with 𝛼 = 1, 𝛽 = −1, and 𝛿 = 1.

algebraic equations for 𝑎0, 𝑎1, 𝑎2, 𝛿, 𝜆, 𝛼, 𝛽, 𝑐, 𝑘, and 𝜇 as

follows:

(𝐺

𝐺

)

7

: −60𝛽𝑎2

2− 2𝛼𝑎

3

2= 0,

(𝐺

𝐺

)

6

: −2𝛼𝑎3

2𝜆 − 5𝛼𝑎

1𝑎2

2− 60𝛽𝑎

1𝑎2− 150𝑎

2

2𝜆 = 0,

(𝐺

𝐺

)

5

: − 12𝛽𝑎2

1− 5𝛼𝑎

1𝑎2

2𝜆 − 124𝛽𝑎

2

2𝜆2− 2𝑘2𝛿𝑎2

2

− 24𝛽𝑎0𝑎2− 4𝛼𝑎

0𝑎2− 144𝛽𝑎

1𝑎2𝜆 + 2𝑐𝑎

2

2

− 4𝛼𝑎2

1𝑎2− 2𝛼𝑎

3

2𝜇 = 0,

(𝐺

𝐺

)

4

: − 196𝛽𝑎2

2𝜆𝜇 − 6𝛼𝑎

0𝑎1𝑎2− 111𝛽𝑎

1𝑎2𝜆2

− 4𝛼𝑎0𝑎2

2𝜆 − 32𝛽𝑎

2

2𝜆3− 𝛼𝑎3

1− 2𝑘2𝛿𝑎2

2𝜆 + 2𝑐𝑎

2

2𝜆

− 54𝛽𝑎0𝑎2𝜆 − 5𝛼𝑎

1𝑎2

2𝜇 − 4𝛼𝑎

2

1𝑎2𝜆 + 3𝑐𝑎

1𝑎2

− 6𝛽𝑎0𝑎1− 3𝑘2𝛿𝑎1𝑎2= 0,

(𝐺

𝐺

)

3

: 2𝑐𝑎0𝑎2− 76𝛽𝑎

2

2𝜇2− 4𝛼𝑎

0𝑎2

2𝜇 − 𝑘2𝛿𝑎2

1

− 27𝛽𝑎1𝜆3𝑎2− 74𝛽𝑎

2

2𝜆2𝜇 + 𝑐𝑎

2

1− 6𝛼𝑎

0𝑎1𝑎2𝜆

+ 3𝑐𝑎1𝑎2𝜆 − 40𝛽𝑎

0𝑎2𝜇 − 38𝛽𝑎

0𝑎2𝜆2

− 19𝛽𝑎2

1𝜆2− 2𝛼𝑎

0𝑎2

1− 4𝛼𝑎

2

1𝑎2𝜇

− 168𝛽𝑎1𝑎2𝜆𝜇 − 3𝑘

2𝛿𝑎1𝑎2𝜆

+ 2𝑐𝑎2

2𝜇 − 2𝛼𝑎

2

0𝑎2− 20𝛽𝑎

2

1𝜇 − 𝛼𝑎

3

1𝜆

− 12𝛽𝑎0𝑎1𝜆 − 2𝑘

2𝛿𝑎0𝑎2= 0,

(𝐺

𝐺

)

2

: 2𝑐𝑎0𝑎2𝜆 − 4𝛽𝑎

2

1𝜆3− 8𝛽𝑎

0𝑎1𝜇

− 3𝑘2𝛿𝑎1𝑎2𝜇 − 𝛼𝑎

3

1𝜇 − 52𝛽𝑎

0𝑎2𝜆𝜇 + 𝑐𝑎

0𝑎1

+ 3𝑐𝑎1𝑎2𝜇 − 6𝛼𝑎

0𝑎1𝑎2𝜇 − 𝑘2𝛿𝑎2

1𝜆

− 60𝛽𝑎1𝑎2𝜇2− 𝛼𝑎2

0𝑎1− 2𝛼𝑎

0𝑎2

1𝜆

− 2𝑘3𝛿𝑎0𝑎2𝜆 − 2𝛼𝑎

2

0𝑎2𝜆 − 7𝛽𝑎

0𝑎1𝜆2

− 26𝛽𝑎2

1𝜆𝜇 + 𝑐𝑎

2

1𝜆 − 57𝛽𝑎

1𝜆2𝑎2𝜇

− 8𝛽𝑎0𝑎2𝜆3− 𝑘2𝛿𝑎0𝑎1− 54𝛽𝑎

2

2𝜆𝜇2= 0,

(𝐺

𝐺

)

1

: − 2𝛼𝑎2

0𝜇 + 2𝑐𝑎

0𝑎2𝜇 − 𝛼𝑎

2

0𝑎1𝜆 − 2𝑘

2𝛿𝑎0𝑎2𝜇

− 8𝛽𝑎2

1𝜇2+ 𝑐𝑎0𝑎1𝜆 − 16𝛽𝑎

0𝑎2𝜇2

− 8𝛽𝑎0𝑎1𝜆𝜇 − 𝛽𝑎

0𝑎1𝜆3− 14𝛽𝑎

0𝑎2𝜆2𝜇

− 36𝛽𝜆𝜇2𝑎1𝑎2− 12𝛽𝑎

2

2𝜇3− 𝑘2𝛿𝑎0𝑎1𝜆

− 𝑘2𝛿𝑎2

1𝜇 + 𝑐𝑎

2

1𝜇 − 2𝛼𝑎

0𝑎2

1𝜇 − 7𝛽𝜇𝜆

2𝑎2

1= 0,


(𝐺

𝐺

)

0

: − 6𝛽𝜇3𝑎1𝑎2+ 𝑐𝑎0𝑎1𝜇 − 𝛼𝜇𝑎

2

0𝑎1− 3𝛽𝜆𝜇

2𝑎2

1

− 𝛽𝜇𝜆2𝑎0𝑎1− 6𝛽𝜆𝜇

2𝑎0𝑎2− 2𝛽𝑎

0𝑎1𝜇2

− 𝑘2𝛿𝜇𝑎0𝑎1= 0.

(44)

Solving this system by Maple gives two sets of solutions.

Case 1. We have

𝑎0=

−30𝛽𝜇

𝛼

, 𝑎1=

−30𝛽𝜆

𝛼

, 𝑎2=

−30𝛽

𝛼

,

𝑐 = −16𝛽𝜇 + 4𝛽𝜆2+ 𝑘2𝛿.

(45)

Substituting the above case and the general solution (8) into(43) and according to (42), we obtain three types of travelingwave solutions of (3) as follows.

If 𝐴 > 0, we have the hyperbolic type

V (𝑥, 𝑦, 𝑡)

=

−30𝛽𝜇

𝛼

−

30𝛽𝜆

𝛼

× [−𝜆

2

+

√𝐴

2

× (

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

−

30𝛽

𝛼

[−𝜆

2

+

√𝐴

2

× (

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

2

.

(46)

In particular, if 𝑐1

̸= 0, 𝑐2= 0, 𝜆 > 0, and 𝜇 = 0, then 𝑢(𝑥, 𝑦, 𝑡)

becomes

𝑢 (𝑥, 𝑦, 𝑡) =

225𝛽2𝜆4

4𝛼2

sech4 {𝜆2

𝜁} ,

𝜁 = 𝑥 + 𝑘𝑦 − (4𝛽𝜆2+ 𝑘2𝛿) 𝑡.

(47)

If 𝐴 < 0, we have the trigonometric type

V (𝑥, 𝑦, 𝑡)

=

−30𝛽𝜇

𝛼

−

30𝛽𝜆

𝛼

× [−𝜆

2

+

√𝐴

2

× (

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {(√𝐴/2) 𝜁}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

−

30𝛽

𝛼

[−𝜆

2

+

√𝐴

2

×(

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {(√𝐴/2) 𝜁}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

2

.

(48)

So, the traveling wave solutions of (3) in this case are

𝑢 (𝑥, 𝑦, 𝑡) =

225𝛽2𝜆4

4𝛼2

sec4 {√−𝜆2

2

𝜁} ,

𝜁 = 𝑥 + 𝑘𝑦 − (4𝛽𝜆2+ 𝑘2𝛿) 𝑡.

(49)

Case 2. We have

𝑎0=

−5𝛽 (𝜆2+ 2𝜇)

𝛼

, 𝑎1=

−30𝛽𝜆

𝛼

, 𝑎2=

−30𝛽

𝛼

,

𝑐 = 16𝛽𝜇 − 4𝛽𝜆2+ 𝑘2𝛿.

(50)

If 𝐴 > 0, we have the hyperbolic type

V (𝑥, 𝑦, 𝑡)

= −

5𝛽 (𝜆2+ 2𝜇)

𝛼

−

30𝛽𝜆

𝛼

× [−𝜆

2

+

√𝐴

2

× (

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

−

30𝛽

𝛼

[−𝜆

2

+

√𝐴

2

×(

𝑐1sinh {(√𝐴/2) 𝜁} + 𝑐

2cosh {(√𝐴/2) 𝜁}

𝑐1cosh {(√𝐴/2) 𝜁} + 𝑐

2sinh {(√𝐴/2) 𝜁}

)]

2

.

(51)

However, if 𝑐1

̸= 0, 𝑐2= 0, 𝜆 > 0, and 𝜇 = 0, then 𝑢(𝑥, 𝑦, 𝑡)

becomes

𝑢 (𝑥, 𝑦, 𝑡) =

25𝛽2𝜆4

4𝛼2

[2 − 3sech2 {𝜆2

𝜁}]

2

,

𝜁 = 𝑥 + 𝑘𝑦 − (−4𝛽𝜆2+ 𝑘2𝛿) 𝑡.

(52)

If 𝐴 < 0, we have the trigonometric type

V (𝑥, 𝑦, 𝑡)

= −

5𝛽 (𝜆2+ 2𝜇)

𝛼

−

30𝛽𝜆

𝛼


0 5 100

1

1.5

0.5

2

−10 −5

x

u(x)

(a)

u(𝜁)

0510

−10−5x

05

−10−5

y

0

1

1.5

0.5

2

(b)

Figure 2: Bell type solitary (a) 2D profile and (b) corresponding 3D plot of (47) for parameters 𝛼 = 2, 𝛽 = 0.4 and 𝛿 = 0.1, 𝜆 = 1, 𝑘 = 1, and𝑡 = 0.5.

× [−𝜆

2

+

√𝐴

2

× (

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {(√𝐴/2) 𝜁}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

−

30𝛽

𝛼

[−𝜆

2

+

√𝐴

2

×(

−𝑐1sin {(√𝐴/2) 𝜁} + 𝑐

2cos {(√𝐴/2) 𝜁}

𝑐1cos {(√𝐴/2) 𝜁} + 𝑐

2sin {(√𝐴/2) 𝜁}

)]

2

.

(53)

In the particular case when 𝑐1

̸= 0, 𝑐2= 0, 𝜆 > 0, and 𝜇 = 0,

𝑢(𝑥, 𝑦, 𝑡) becomes

𝑢 (𝑥, 𝑦, 𝑡) =

25𝛽2𝜆4

4𝛼2

[2 − 3sec2 {√−𝜆2

2

𝜁}]

2

,

𝜁 = 𝑥 + 𝑘𝑦 − (−4𝛽𝜆2+ 𝑘2𝛿) 𝑡.

(54)

If 𝐴 = 0, we have the rational type

V (𝑥, 𝑦, 𝑡) = −5𝛽 (𝜆2+ 2𝜇)

𝛼

− 30𝑘2𝛿𝜆 [−

𝜆

2

+ (𝑐2

𝑐1+ 𝑐2𝜁

)]

− 30𝑘2𝛿[−

𝜆

2

+ (𝑐2

𝑐1+ 𝑐2𝜁

)]

2

,

(55)

where 𝜁 = 𝑥 + 𝑘𝑦 − (16𝛽𝜇 − 4𝛽𝜆2 + 𝑘2𝛿)𝑡.

We remark that our results in (47) and (52), when 𝑐1

̸= 0,𝑐2= 0, 𝜆 > 0, and 𝜇 = 0, match those of Khater et al. [6] (2.19)

when 𝑎 = 1. In Figure 2, we plot the bell type solitary for 2Dprofile and the corresponding 3D plot of (47) for parameters𝛼 = 2, 𝛽 = 0.4 and 𝛿 = 0.1, 𝜆 = 1, 𝑘 = 1, and 𝑡 = 0.5.

4. Conclusion

The (𝐺/𝐺)-expansion was applied to solve the model of ion-acoustic waves in plasma physics where these equations eachcontain a square root nonlinearity. The (𝐺/𝐺)-expansionhas been successfully used to obtain some exact travelingwave solutions of the Schamel equation, Schamel-KdV (S-KdV) equation, and modified KP (Kadomtsev-Petviashvili)equation. Moreover, the reliability of the method and thereduction in the size of computational domain give thismethod a wider applicability. This fact shows that ouralgorithm is effective and more powerful for NLPDE. Inall the general solutions (22), (35), (47), and (52), we havethe additional arbitrary constants 𝑐

1, 𝑐2, 𝜆, and 𝜇. We note

that the special case 𝑐1

̸= 0, 𝑐2

= 0, 𝜆 > 0, and 𝜇 = 0reproduced the results of Khater and Hassan [35], Hassan [5]and Khater et al. [6]. Many different new forms of travelingwave solutions such as the kink shaped, antikink shaped, andbell type solitary solutions were obtained. Finally, numericalsimulations are given to complete the study.

Moreover, all the methods have some limitations in theirapplications. In fact, there is no unified method that canbe used to handle all types of nonlinear partial differentialequations (NLPDE). Certainly, each investigator in the fieldof differential equations has his own experience to choosethe method depending on form of the nonlinear differentialequation and the pole of its solution. So, the limitations of


the (𝐺/𝐺)-expansion method used a rise only when theequation has the traveling wave and becomes powerful infinding traveling wave solutions of NLPDE only.

In our future works, we can extend our method byintroducing a more generalized ansätz 𝐺2 = 𝑑

2𝐺2+ 𝑑3𝐺3+

𝑑4𝐺4, where 𝐺 = 𝐺(𝜁), to solve Schamel equation, Schamel-

KdV (S-KdV) equation, and modified Kadomtsev-Petviash-vili (KP) equation.

Acknowledgment

This work is financially supported by Universiti KebangsaanMalaysia Grant: UKM-DIP-2012-31.

References

[1] G. B. Whitham, Linear and Nonlinear Waves, Pure and AppliedMathematics, John Wiley & Sons, New York, NY, USA, 1974.

[2] R. C.Davidson,Methods inNonlinear PlasmaTheory, AcademicPress, New York, NY, USA, 1972.

[3] H. Schamel, “A modified Korteweg de Vries equation for ionacoustic waves due to resonant electrons,” Journal of PlasmaPhysics, vol. 9, pp. 377–387, 1973.

[4] J. Lee and R. Sakthivel, “Exact travelling wave solutions of theSchamel–Korteweg–de Vries equation,” Reports on Mathemati-cal Physics, vol. 68, no. 2, pp. 153–161, 2011.

[5] M. M. Hassan, “New exact solutions of two nonlinear physicalmodels,” Communications in Theoretical Physics, vol. 53, no. 4,pp. 596–604, 2010.

[6] A. H. Khater, M. M. Hassan, and D. K. Callebaut, “Travellingwave solutions to some important equations of mathematicalphysics,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 1–19, 2010.

[7] D. Chakraborty and K. P. Das, “Stability of ion-acoustic solitonsin a multispecies plasma consisting of non-isothermal elec-trons,” Journal of Plasma Physics, vol. 60, no. 1, pp. 151–158, 1998.

[8] B. B. Kadomtsev and V. I. Petviashvili, “On the stability ofsolitary in weakly dispersivemedia,” Soviet Physics Doklady, vol.15, pp. 539–541, 1970.

[9] M. Krisnangkura, S. Chinviriyasit, and W. Chinviriyasit, “Ana-lytic study of the generalized Burger’s-Huxley equation byhyperbolic tangent method,” Applied Mathematics and Compu-tation, vol. 218, no. 22, pp. 10843–10847, 2012.

[10] A.-M. Wazwaz, “The tanh method for traveling wave solutionsof nonlinear equations,”AppliedMathematics andComputation,vol. 154, no. 3, pp. 713–723, 2004.

[11] Q. Shi, Q. Xiao, and X. Liu, “Extended wave solutions for a non-linear Klein-Gordon-Zakharov system,” Applied Mathematicsand Computation, vol. 218, no. 19, pp. 9922–9929, 2012.

[12] A.-M. Wazwaz, “The tanh and the sine-cosine methods for areliable treatment of the modified equal width equation and itsvariants,” Communications in Nonlinear Science and NumericalSimulation, vol. 11, no. 2, pp. 148–160, 2006.

[13] A. Jabbari and H. Kheiri, “New exact traveling wave solutionsfor the Kawahara and modified Kawahara equations by usingmodified tanh-coth method,” Acta Universitatis Apulensis, no.23, pp. 21–38, 2010.

[14] K. Parand and J. A. Rad, “Exp-function method for somenonlinear PDE’s and a nonlinear ODE’s,” Journal of King SaudUniversity-Science, vol. 24, no. 1, pp. 1–10, 2012.

[15] M. K. Elboree, “New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equa-tion,”AppliedMathematics andComputation, vol. 218, no. 10, pp.5966–5973, 2012.

[16] A. S. Abdel Rady, E. S. Osman, and M. Khalfallah, “On solitonsolutions of the (2 + 1) dimensional Boussinesq equation,”AppliedMathematics and Computation, vol. 219, no. 8, pp. 3414–3419, 2012.

[17] B. Hong and D. Lu, “New Jacobi elliptic function-like solutionsfor the general KdV equation with variable coefficients,”Mathe-matical and ComputerModelling, vol. 55, no. 3-4, pp. 1594–1600,2012.

[18] J. Lee, R. Sakthivel, and L. Wazzan, “Exact traveling wave solu-tions of a higher-dimensional nonlinear evolution equation,”Modern Physics Letters B, vol. 24, no. 10, pp. 1011–1021, 2010.

[19] S. Abbasbandy and A. Shirzadi, “The first integral method formodified Benjamin-Bona-Mahony equation,” Communicationsin Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp.1759–1764, 2010.

[20] N. Taghizadeh, M. Mirzazadeh, and F. Tascan, “The first-integral method applied to the Eckhaus equation,” AppliedMathematics Letters, vol. 25, no. 5, pp. 798–802, 2012.

[21] W. Malfliet, “Solitary wave solutions of nonlinear wave equa-tions,” American Journal of Physics, vol. 60, no. 7, pp. 650–654,1992.

[22] M. Wang, X. Li, and J. Zhang, “The (𝐺/𝐺)-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physics,” Physics Letters A, vol. 372, no. 4, pp.417–423, 2008.

[23] W. M. Taha and M. S. M. Noorani, “Exact solutions of equa-tion generated by the Jaulent-Miodek hierarchy by (𝐺/𝐺)-expansion method,”Mathematical Problems in Engineering, vol.2013, Article ID 392830, 7 pages, 2013.

[24] W.M. Taha,M. S. M. Noorani, and I. Hashim, “New applicationof the (𝐺/𝐺)-expansion method for thin film equations,”Abstract and Applied Analysis, vol. 2013, Article ID 535138, 6pages, 2013.

[25] B. Ayhan and A. Bekir, “The (𝐺/𝐺)-expansion method forthe nonlinear lattice equations,” Communications in NonlinearScience and Numerical Simulation, vol. 17, no. 9, pp. 3490–3498,2012.

[26] J. Feng, W. Li, and Q. Wan, “Using (𝐺/𝐺)-expansion methodto seek the traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation,” Applied Mathematics and Computation,vol. 217, no. 12, pp. 5860–5865, 2011.

[27] M. M. Kabir, A. Borhanifar, and R. Abazari, “Application of(𝐺/𝐺)-expansion method to regularized long wave (RLW)equation,” Computers & Mathematics with Applications, vol. 61,no. 8, pp. 2044–2047, 2011.

[28] A. Malik, F. Chand, H. Kumar, and S. C. Mishra, “Exactsolutions of the Bogoyavlenskii equation using the multiple(𝐺/𝐺)-expansion method,” Computers & Mathematics withApplications, vol. 64, no. 9, pp. 2850–2859, 2012.

[29] A. Malik, F. Chand, and S. C. Mishra, “Exact travelling wavesolutions of some nonlinear equations by (𝐺/𝐺)-expansionmethod,” Applied Mathematics and Computation, vol. 216, no.9, pp. 2596–2612, 2010.

[30] A. Jabbari, H. Kheiri, and A. Bekir, “Exact solutions of the cou-pled Higgs equation and the Maccari system using He’s semi-inverse method and (𝐺/𝐺)-expansion method,” Computers &Mathematics with Applications, vol. 62, no. 5, pp. 2177–2186,2011.


[31] H. Naher and F. A. Abdullah, “Some new traveling wave solu-tions of the nonlinear reaction diffusion equation by using theimproved (𝐺/𝐺)-expansion method,” Mathematical Problemsin Engineering, vol. 2012, Article ID 871724, 17 pages, 2012.

[32] E. M. E. Zayed and M. A. M. Abdelaziz, “The two-variable(𝐺/𝐺, 1/𝐺)-expansion method for solving the nonlinear KdV-

mKdV equation,” Mathematical Problems in Engineering, vol.2012, Article ID 725061, 14 pages, 2012.

[33] H. Naher and F. A. Abdullah, “The improved (𝐺/𝐺)-expansionmethod for the (2 + 1)-dimensional modified Zakharov-Kuznetsov equation,” Journal of Applied Mathematics, vol. 2012,Article ID 438928, 20 pages, 2012.

[34] H. Naher and F. A. Abdullah, “New traveling wave solutions bythe extended generalized Riccati equation mapping method ofthe (2 + 1)-dimensional evolution equation,” Journal of AppliedMathematics, vol. 2012, Article ID 486458, 18 pages, 2012.

[35] A. H. Khater and M. M. Hassan, “Exact solutions expressiblein hyperbolic and jacobi elliptic functions of some importantequations of ion-acoustic waves,” in Acoustic Waves—FromMicrodevices to Helioseismology, 2011.

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Research Article New Exact Solutions of Ion-Acoustic Wave ...downloads.hindawi.com/journals/jam/2013/810729.pdf · of sech 2 or may be kinks in terms of tanh. We believe that the