# The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations

Authors: Kamruzzaman Khan; M. Ali Akbar; Norhashidah Hj. Mohd. Ali
Journal: ISRN Mathematical Physics
 (2013)

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## Abstract

The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.


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## Body

## 1. Introduction

The importance of nonlinear evolution equations (NLEEs) is now well established, since these equations arise in various areas of science and engineering, especially in fluid mechanics, biology, plasma physics, solid-state physics, optical fibers, biophysics and so on. As a key problem, finding their analytical solutions is of great importance and is actually executed through various efficient and powerful methods such as the Exp-function method [1–4], the tanh-function method [5, 6], the homogeneous balance method [7, 8], the (G′/G)-expansion method [9–16], the Hirota’s bilinear transformation method [17, 18], the Backlund transformation method [19], the inverse scattering transformation [20], the Jacobi elliptic function method [21], the modified simple equation method [22–24] and so on.The objective of this paper is to look for new study relating to the MSE method via the well-recognized GZK-BBM equation and right-handed nc-Burgers’ equation and establish the originality and effectiveness of the method.The paper is organized as follows: in Section2, we give the description of the MSE method. In Section 3, we use this method to the nonlinear evolution equations pointed out above, and in Section 4 conclusions are given.


## 2. Description of the MSE Method

Suppose the nonlinear evolution equation is in the following form:(1)F(u,ut,ux,uxx,utt,…)=0,
where F  is a polynomial of u(x,t) and its partial derivatives wherein the highest order derivatives and nonlinear terms are concerned. The main steps of the MSE method [22–24] are as follows.Step 1.
The traveling wave transformation(2)u(x,t)=u(ξ),ξ=x±νt
permits us to reduce (1) into the following ordinary differential equation (ODE):
(3)P(u,u′,u′′,…)=0,
where P is a polynomial in u(ξ) and its total derivatives, wherein u′(ξ)=du/dξ.Step 2.
We suppose the solution of (3) is of the form
(4)u(ξ)=∑k=0NCk[Φ′(ξ)Φ(ξ)]k,
where Ck  (k=0,1,2,3,…) are arbitrary constants to be determined, such that CN≠0, and Φ(ξ) is an unidentified function to be determined afterwards. In Exp-function method, (G′/G)-expansion method, tanh-function method, Jacobi elliptic function method, and so forth, the solution is offered in terms of some predefined functions, but in the MSE method, Φ is not predefined or not a solution of any predefined differential equation. Therefore, some fresh solutions might be found by this method. This is the merit of the MSE method.Step 3.
We determine the positive integerN come out in (4) by considering the homogeneous balance between the highest order derivatives and the highest order nonlinear terms occurring in (3).Step 4.
We compute all the required derivativesu′,u′′,…, and substitute (4) and the derivatives into (3) and then we account for the function Φ(ξ). As a result of this substitution, we get a polynomial of (Φ′(ξ)/Φ(ξ)) and its derivatives. In this polynomial, we equate all the coefficients to zero. This procedure yields a system of equations whichever can be solved to find Ck and Φ(ξ).


## 3. Applications

### 3.1. The GZK-BBM Equation

In this subsection, we will use the MSE method to look for the exact solutions and then the solitary wave solutions to the GZK-BBM equation:(5)ut+ux+α(u3)x+β(uxt+uyy)x=0,
where α and β are nonzero constants.Using traveling wave transformation,(6)u=u(x,y,t),ξ=x+y-νt,u(x,y,t)=u(ξ),
(5) reduces to the following ODE:
(7)-νu′+u′+α(u3)′+β(-νu′′+u′′)′=0.
Integrating (7) with respect to ξ, we obtain
(8)(1-ν)u+αu3+β(1-ν)u′′=0.
Balancing the highest order derivative u′′ and nonlinear term u3, we obtain N=1.Therefore, the solution (4) turns into the following form:
(9)u(ξ)=C0+C1(Φ′Φ),
where C0 and C1 are constants such that  C1≠0, and Φ(ξ) is an unidentified function to be determined. It is easy to find that
(10)u′=C1[Φ′′Φ-(Φ′Φ)2],(11)u′′=C1Φ′′′Φ-3C1Φ′′Φ′Φ2+2A1(Φ′Φ)3,(12)u3=C13(Φ′Φ)3+3C12C0(Φ′Φ)2+3C1C02(Φ′Φ)+C03.
Substituting the values of u, u′′, and u3 from (9)–(12) into (8) and then equating the coefficients of Φ0,Φ-1,Φ-2, and Φ-3 to zero, we obtain
(13)αC03+C0-νC0=0,(14)β(ν-1)Φ′′′-(3αC02-ν+1)Φ′=0,(15)β(ν-1)Φ′′+αC0C1Φ′=0,(16)(αC13+2βC1-2βνC1)(Φ′)3=0.
Equations (13) and (16), respectively, yield
(17)C0=0,±ν-1α,C1=±2β(ν-1)αsinceC1≠0,
where ν≠1.From (14) and (15), we obtain
(18)Φ′′′Φ′′=-l,
where l=(3αC02-ν+1)/αC0C1.Integrating (18), we obtain
(19)Φ′′(ξ)=c1e-lξ,
where c1 is a constant of integration.And from (15) and (19), we obtain
(20)Φ′=-me-lξ,
where m=β(ν-1)c1/αC0C1Integrating (20) with respect to ξ, we obtain
(21)Φ(ξ)=c2+mle-lξ,
where c2 is a constant of integration.Substituting the value ofΦ and Φ′ into solution (9) yields
(22)u(ξ)=C0+C1(-lme-lξc2l+me-lξ).Case 1.
WhenC0=0, solution (22) collapses, and hence this case is rejected.Case 2.
WhenC0=±(ν-1)/α and C1=±2β(ν-1)/α, substituting the values of C0 and C1 into (22) and simplifying, we obtain the exact solutions:
(23)u(x,y,t)=±(ν-1α)×{1-2βc1exp(±2/β(x+y-νt))βc1exp(±2/β(x+y-νt))+2c2}.
Since c1 and c2 are arbitrary constants, therefore, if we set c2=c1/2, the exact solution (23) turns out to the following solitary wave solutions
(24)u(x,y,t)=±(ν-1α)tanh(12(x+y-νt)),
when β=1. On the other hand, if β=-1, solution (23) turns into
(25)u(x,y,t)=±(ν-1α)coth(12(x+y-νt)).
In particular, when α=1 and ν=4, solutions (24) and (25) convert to
(26)u(x,y,t)=±3tanh(12(x+y-4t)),(27)u(x,y,t)=±3coth(12(x+y-4t)),
respectively.Fory=0, the solution u(x,y,t) presented in (26) is sketched in Figure 1.Figure 1Again fory=0, the solution u(x,y,t) presented in (27) is sketched in Figure 2.Figure 2The MSE method is applied to investigate solitary wave solutions to the GZK-BBM equation and obtained solutions with free parameters involving the known solutions in the open literature. Obviously we might choose the values of the arbitrary constantsc1 and c2 equal to other values, resulting in diverse solitary shapes. The free parameters imply some physical meaningful results in gravity water waves in the long-wave regime.


### 3.2. The Right-Handed nc-Burgers’ Equation

In this subsection, we will bring to bear the MSE method to find the traveling wave solutions and then the solitary wave solutions to the right-handed nc-Burgers’ equation:(28)ut=uxx+2uux.
Using traveling wave transformation (2), (28) is reduced to the following ODE:
(29)u′′+2uu′+νu′=0.
Integrating (29) with respect to  ξ and setting the constant of integration to zero, we obtain
(30)u′+u2+νu=0.
Balancing the highest order derivative and nonlinear term, we obtain N=1.Therefore, solution (4) becomes
(31)u(ξ)=C0+C1(Φ′Φ).
Executing the parallel course of action which is described in Section 3.1, we obtain
(32)νC0+C02=0,(33)Φ′′+(2C0+ν)Φ′=0,(34)C12-C1=0.
Solving (32) and (34), we obtain C0=0,-ν and  C1=1, since  C1≠0, respectively.Case 1.
WhenC0=0 and C1=1 and solving (33), we receive the value of Φ, and substituting the value of Φ into (31), we obtain the following exact solution:
(35)u(x,t)=νc1exp(-ν(x-νt))νc2-c1exp(-ν(x-νt)),
where c1 and c2 are constants of integration. Therefore, we can make choices at random the parameters c1 and c2; if we choose c1=νand c2=1, the exact solution (35) turns into the under determined solitary wave solution
(36)u(x,t)=-ν2{1-coth(ν(x-νt)2)}.
And if c1=-νand c2=1, the solution (35) turn into,
(37)u(x,t)=-ν2{1-tanh(ν(x-νt)2)}.Case 2.
WhenC0=-ν and solving (33), we get the value of Φ, and substituting this value into (31), we obtain the subsequent exact solution:
(38)u(x,t)=-ν+νc1exp(ν(x-νt))c1exp(ν(x-νt))+νc2.
We can arbitrarily pick the parameters c1 and c2. Therefore, exact solution (38) turns into the following solitary wave solutions:
(39)u(x,t)=-ν+ν2{1+tanh(ν(x-νt)2)},
when c1=ν and c2=1, and
(40)u(x,t)=-ν+ν2{1+coth(ν(x-νt)2)},
when c1=-ν and c2=1.The solutionu(x,t) given in (39) is presented in Figure 3.Figure 3The solutionu(x,t) given in (40) is presented in Figure 4.Figure 4For specific values of the parameters in the generalized exact solutions (35) and (38), we obtain the solitary wave shape solutions to the right-handed nc-Burgers’ equation which are shown in Figures 3 and 4. Of course we might choose other values of the arbitrary constants c1 and c2, resulting in diverse solitary wave shapes. The free parameters may imply some physical meaningful results in fluid mechanics, gas dynamics, and traffic flow.


## 3.1. The GZK-BBM Equation

In this subsection, we will use the MSE method to look for the exact solutions and then the solitary wave solutions to the GZK-BBM equation:(5)ut+ux+α(u3)x+β(uxt+uyy)x=0,
where α and β are nonzero constants.Using traveling wave transformation,(6)u=u(x,y,t),ξ=x+y-νt,u(x,y,t)=u(ξ),
(5) reduces to the following ODE:
(7)-νu′+u′+α(u3)′+β(-νu′′+u′′)′=0.
Integrating (7) with respect to ξ, we obtain
(8)(1-ν)u+αu3+β(1-ν)u′′=0.
Balancing the highest order derivative u′′ and nonlinear term u3, we obtain N=1.Therefore, the solution (4) turns into the following form:
(9)u(ξ)=C0+C1(Φ′Φ),
where C0 and C1 are constants such that  C1≠0, and Φ(ξ) is an unidentified function to be determined. It is easy to find that
(10)u′=C1[Φ′′Φ-(Φ′Φ)2],(11)u′′=C1Φ′′′Φ-3C1Φ′′Φ′Φ2+2A1(Φ′Φ)3,(12)u3=C13(Φ′Φ)3+3C12C0(Φ′Φ)2+3C1C02(Φ′Φ)+C03.
Substituting the values of u, u′′, and u3 from (9)–(12) into (8) and then equating the coefficients of Φ0,Φ-1,Φ-2, and Φ-3 to zero, we obtain
(13)αC03+C0-νC0=0,(14)β(ν-1)Φ′′′-(3αC02-ν+1)Φ′=0,(15)β(ν-1)Φ′′+αC0C1Φ′=0,(16)(αC13+2βC1-2βνC1)(Φ′)3=0.
Equations (13) and (16), respectively, yield
(17)C0=0,±ν-1α,C1=±2β(ν-1)αsinceC1≠0,
where ν≠1.From (14) and (15), we obtain
(18)Φ′′′Φ′′=-l,
where l=(3αC02-ν+1)/αC0C1.Integrating (18), we obtain
(19)Φ′′(ξ)=c1e-lξ,
where c1 is a constant of integration.And from (15) and (19), we obtain
(20)Φ′=-me-lξ,
where m=β(ν-1)c1/αC0C1Integrating (20) with respect to ξ, we obtain
(21)Φ(ξ)=c2+mle-lξ,
where c2 is a constant of integration.Substituting the value ofΦ and Φ′ into solution (9) yields
(22)u(ξ)=C0+C1(-lme-lξc2l+me-lξ).Case 1.
WhenC0=0, solution (22) collapses, and hence this case is rejected.Case 2.
WhenC0=±(ν-1)/α and C1=±2β(ν-1)/α, substituting the values of C0 and C1 into (22) and simplifying, we obtain the exact solutions:
(23)u(x,y,t)=±(ν-1α)×{1-2βc1exp(±2/β(x+y-νt))βc1exp(±2/β(x+y-νt))+2c2}.
Since c1 and c2 are arbitrary constants, therefore, if we set c2=c1/2, the exact solution (23) turns out to the following solitary wave solutions
(24)u(x,y,t)=±(ν-1α)tanh(12(x+y-νt)),
when β=1. On the other hand, if β=-1, solution (23) turns into
(25)u(x,y,t)=±(ν-1α)coth(12(x+y-νt)).
In particular, when α=1 and ν=4, solutions (24) and (25) convert to
(26)u(x,y,t)=±3tanh(12(x+y-4t)),(27)u(x,y,t)=±3coth(12(x+y-4t)),
respectively.Fory=0, the solution u(x,y,t) presented in (26) is sketched in Figure 1.Figure 1Again fory=0, the solution u(x,y,t) presented in (27) is sketched in Figure 2.Figure 2The MSE method is applied to investigate solitary wave solutions to the GZK-BBM equation and obtained solutions with free parameters involving the known solutions in the open literature. Obviously we might choose the values of the arbitrary constantsc1 and c2 equal to other values, resulting in diverse solitary shapes. The free parameters imply some physical meaningful results in gravity water waves in the long-wave regime.


## 3.2. The Right-Handed nc-Burgers’ Equation

In this subsection, we will bring to bear the MSE method to find the traveling wave solutions and then the solitary wave solutions to the right-handed nc-Burgers’ equation:(28)ut=uxx+2uux.
Using traveling wave transformation (2), (28) is reduced to the following ODE:
(29)u′′+2uu′+νu′=0.
Integrating (29) with respect to  ξ and setting the constant of integration to zero, we obtain
(30)u′+u2+νu=0.
Balancing the highest order derivative and nonlinear term, we obtain N=1.Therefore, solution (4) becomes
(31)u(ξ)=C0+C1(Φ′Φ).
Executing the parallel course of action which is described in Section 3.1, we obtain
(32)νC0+C02=0,(33)Φ′′+(2C0+ν)Φ′=0,(34)C12-C1=0.
Solving (32) and (34), we obtain C0=0,-ν and  C1=1, since  C1≠0, respectively.Case 1.
WhenC0=0 and C1=1 and solving (33), we receive the value of Φ, and substituting the value of Φ into (31), we obtain the following exact solution:
(35)u(x,t)=νc1exp(-ν(x-νt))νc2-c1exp(-ν(x-νt)),
where c1 and c2 are constants of integration. Therefore, we can make choices at random the parameters c1 and c2; if we choose c1=νand c2=1, the exact solution (35) turns into the under determined solitary wave solution
(36)u(x,t)=-ν2{1-coth(ν(x-νt)2)}.
And if c1=-νand c2=1, the solution (35) turn into,
(37)u(x,t)=-ν2{1-tanh(ν(x-νt)2)}.Case 2.
WhenC0=-ν and solving (33), we get the value of Φ, and substituting this value into (31), we obtain the subsequent exact solution:
(38)u(x,t)=-ν+νc1exp(ν(x-νt))c1exp(ν(x-νt))+νc2.
We can arbitrarily pick the parameters c1 and c2. Therefore, exact solution (38) turns into the following solitary wave solutions:
(39)u(x,t)=-ν+ν2{1+tanh(ν(x-νt)2)},
when c1=ν and c2=1, and
(40)u(x,t)=-ν+ν2{1+coth(ν(x-νt)2)},
when c1=-ν and c2=1.The solutionu(x,t) given in (39) is presented in Figure 3.Figure 3The solutionu(x,t) given in (40) is presented in Figure 4.Figure 4For specific values of the parameters in the generalized exact solutions (35) and (38), we obtain the solitary wave shape solutions to the right-handed nc-Burgers’ equation which are shown in Figures 3 and 4. Of course we might choose other values of the arbitrary constants c1 and c2, resulting in diverse solitary wave shapes. The free parameters may imply some physical meaningful results in fluid mechanics, gas dynamics, and traffic flow.


## 4. Conclusions

The modified simple equation method presented in this paper has been successfully implemented to find the exact and the solitary wave solutions for NLEEs via the GZK-BBM and right-handed nc-Burgers’ equation. The method offers solutions with free parameters that might be important to explain some intricate physical phenomena. Some special solutions including the known solitary wave solution are originated by setting appropriate values for the parameters. Compared to the currently proposed method with other methods, such as the(G′/G)-expansion method, the Exp-function method and the tanh-function method, we might conclude that the exact solutions to (5) and (28) can be investigated using these methods with the help of the symbolic computational software such as Mathematica and Maple to facilitate the complex algebraic computations. On the other hand, via the proposed method, the exact and solitary wave solutions to these equations have been achieved without using any symbolic computation software because the method is very simple and has easy computations.



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