On the modified ( -expansion method for finding some analytical solutions of the traveling waves

S.Beher , , N.H.Aljhdly , J.P.S.Virdi

a Department of Physics, Veer Surendra Sai University of Technology, Odisha,India

b Department of Mathematics, Faculty of Sciences and Arts-Rabigh Campus, King Abdulaziz University, Jeddah, SaudiArabia

ABSTRACT This work investigates three nonlinear equations that describe waves on the oceans which are the Kadomtsev Petviashvili-modified equal width (KP-MEW) equation, the coupled Drinfel’d-Sokolov-Wilson(DSW) equation, and the Benjamin-Ono (BO) equation using the modified ( -expansion approach.The solutions of proposed equations by modified ( -expansion approach can be trigonometric, hyperbolic,or rational solutions.As a result, some new exact solutions are obtained and plotted.

Keywords:Nonlinear physical models NLEEs Modified ( ) -expansion approach KP-MEW equation Drinfeld and Soklov and Wilson equation Benjamin-Ono equation

1.Introduction

The Kadomtsev-Petviashvili (KP) equation [21] is one of the contemporary 2-dimensional apparent of the KdV equation which portrays the evolution of slightly nonlinear and slightly dispersive wave, where it shows up in the structure

magnetic waves in ferromagnetic materials and water waves in shallow water with slightly nonlinear restoring forces modeled by KP equation, similarly many important nonlinear physical phenomena can be expressed by KP equation.Apparently the modified equal width (MEW) equation can be written [22,23] :

Wazaz [24] and others presented the above equation in KP sense as follows:

In this work, we desire to examine the 2-dimensional nonlinear KP-MEW equation [25]

Again, we examine the coupled Drinfel’d-Sokolov-Wilson (DSW)equation [26] which is used to model the water wave dynamics have the following form:

Where p, q, r, s are constants.Recently, the DSW equation and the coupled DSW equation, a special case of the classical DSW equation, have been studied by several authors [27] .

In addition, The Benjamin-Ono (BO) equation [28] which is the prototype of the KDV equation used to explain internally waves in fluids with density variations have the following form:

Likewise, it has been utilized to show surface wave propagation on a thin-layered structure, utilizing a surface acoustic wave delay line to dispatch t he waves.

The rest of this paper is arranged as follows.In section 2 ,the mathematical details of the modified ()- expansion approach have been presented.In Section 3 , 4, and 5, by using the modified ()- expansion approach, we constructed the traveling wave solutions for the KP-MEW equation, the DSW equation, and the BO equation respectively.Finally, in section 6, some concluding remarks are given.

2.Glance of the modified ()- expansion approach

where f = f(x, t) is a unknown test function.The-method can be studied as below:

Step 1:In order to calculate the traveling wave solution of Eq.(7) ,by implementing the following wave variable;

than

Similarly:

and can be extended to higher orders.With the help of Eq.(10) ,Eq.(7) can be written as,

Here fφdenotesand ”V”is the speed of wave.

We can integrate the obtained ordinary differential equation(ODE) (11) many times to get a comparatively simpler equation.Then, by assuming f → 0 as φ→ ∞ , the integration constant can be equated to zero.

Step 2:The formal solution of ODE (11) can be written as

where G = G (φ) satisfies

where μ, σand ρare free parameters.To find the general solution,we have to find the constants α0, αn, βn(n = 1 , 2 , 3 , ....N) .

3.Application of the modified () -expansion approach

3.1.The Kadomtsev-Petviashvili-modified equal width (KP-MEW)equation

In order to solve KP-MEW equation via modifiedexpansion approach, by considering the wave transformation method f(x, t) = f(φ) with the wave variable (φ) = (x -V t) ,Eq.(4) can be converted in the form of an ODE,

Now, balancing with the homogeneous balance method, we obtain N = 1 .The formal solution will be in the following form,here α0, α1, β1are arbitrary parameters, while G( φ) satisfies the aforementioned auxiliary Eq.(13) .

Now,

By proper substitution of f, f3, f φφ into Eq.(19) along with Eq.(23) and gathering the terms in terms of exponents of （and then equating these polynomials to zero separately gives a set of algebraic equations for α0, α1, β1, σand ρas follows:

By algebraic simplification of the above Eq.(24) with help of Mathematica, we will get the following sets of solutions for different values of α0, α1, β1, σ, ρ:

We will get five different solutions as below:

Case:1(Trigonometric solution)-By using Eq.(25) and Eq.(14) in expression (12) , we have

Case:2(Hyperbolic solution)- By using Eq.(25) and Eq.(15) in expression (12) , we have

Case:3(Rational solution)-By using Eq.(25) and Eq.(16) in expression (12) , we have

Case:4(Hyperbolic solution)-By using Eq.(25) and Eq.(17) in expression (12) , we have

Case:5(Hyperbolic solution)-By using Eq.(25) and Eq.(18) in expression (12) , we have

3.2.Drinfel’d-Sokolov-Wilson (DSW) equation

Again, in order to solve the DSW equation via modified-expansion approach, by considering the wave transformation method f(x, t) = f(φ) , g(x, t) = g(φ) with the wave variable(φ) = (x -V t) , Eq.(5) can be converted in the form of an ODE.

For simplification integrating Eq.(32) once with respect to (φ) ,and substituting it into Eq.(5) we obtain

Now, balancing with the homogeneous balance method, we obtain N = 1 , where c1and c2are integral constants.The general solution will be the following form,

By algebraic simplification of the above Eq.(35) with help of Mathematica,we will get the following sets of solutions for different values of α0, α1, β1, σ, ρ:

We will get five different solutions as below:

Case:1(Trigonometric solution)- By using Eq.(35) and Eq.(14) in expression (12) , we have

Case:2(Hyperbolic solution)- By using Eq.(35) and Eq.(15) in expression (12) , we have

Case:3(Rational solution)- By using Eq.(35) and Eq.(16) in expression (12) , we have

Case:4(Hyperbolic solution)- By using Eq.(25) and Eq.(17) in expression (12) , we have

Case:5(Hyperbolic solution)- By using Eq.(25) and Eq.(18) in expression (12) , we have

Fig.1.Traveling wave solution corresponding to the KP-MEW equation

3.3.The Benjamin-Ono (BO) equation

Similarly, We use the wave transformation f(x, t) = f(φ) with wave variable φ= (x -V t) the BO equation (6) takes the form of an ODE.

Integrating Eq.(42) with respect to φ, we have

By homogeneous balance method as discussed earlier, we obtain N = 1 , therefore the general solution takes the form,

By proper substitution of f, f2, fφinto Eq.(43) along with Eq.(13) and gathering all the terms ofby considering same power of the terms and then equating these polynomials to zero separately gives a set of algebraic equations for α0, α1, β1, σand ρ as follows:

By algebraic simplification of the above Eq.(45) with help of Mathematica, we will get the following sets of solutions for different values of α0, α1, β1, σ, ρ:

Fig.2.Traveling wave solution corresponding to Drinfel’d-Sokolov-Wilson (DSW) equation

We will get five different solutions as below:

Case:1(Trigonometric solution)- By using

Eq.(46) and Eq.(14) in expression (12) , we have

Case:2(Hyperbolic solution)- By using Eq.(46) and Eq.(15) in expression (12) , we have

Case:3(Rational solution)- By using Eq.(46) and Eq.(16) in expression (12) , we have

Case:4(Hyperbolic solution)- By using Eq.(25) and Eq.(17) in expression (12) , we have

Fig.3.Traveling wave solution corresponding to the Benjamin-Ono (BO) equation

Case:5(Hyperbolic solution)- By using Eq.(25) and Eq.(18) in expression (12) , we have

4.Results and Discussion

The solutions of the KP-MEW equation obtained via modified-expansion approach exhibit different types of soliton profiles such as ( Fig.1 a) and dummyTXdummy-(( Fig.1 b) are periodic soliton solutions analogs to the periodic solution obtained by Adem et al.[29] for the Eq.27.Similarly ( Fig.1 c) is a bright soliton solution analogues to the bright solution obtained by Adem et al.[29]with n = 2 , α= 1 / 2 , β= 1 , γ= 1 , a = 1 , b = 1 , c = 1 , t = 0 , and C =1 of Eq.(14).Kink solution is given in ( Fig.1 d) for Eq.(1), where the left-hand side represents different types of traveling wave solutions and the right-hand side represents their contour plots respectively.The above profiles may help to understand the nature of shallow water waves in presence of weakly nonlinear restoring forces and sometimes traveling waves in ferromagnetic porous media.

The solutions of the DSW equation obtained via the modified-expansion approach exhibit periodic solutions for Eq.(37) and Eq.(38) respectively by ( Fig.2 a) and dummyTXdummy-(( Fig.2 b), which are analogous to the periodic profile of Eq.(86) obtained by Arsed et al.[27] with c2= 4, c4= 1, k1= 3, k2= 2, = 4 and y = 1. The soliton given by( Fig.2 c) is having similar profile when k = 0 .1 , c = 0 .5 , q = 3 and m = 0 .9 for the Jacobi elliptic solution with the initial condition of Eq.(1) by Inc et al.[30] .Kink solution is given by ( Fig.2 d)for Eq.(4).Left hand side represents different types of traveling wave solutions and right hand side represents their contour plots respectively for Fig.2 .Presented profiles of the DSW equation will help to understand water wave dynamics in a greater extent.

The solutions of the BO equation obtained via modifiedexpansion approach exhibit a class of periodic solutions and kink solutions such as ( Fig.3 a) and dummyTXdummy-(( Fig.3 b) are periodic solitons, they are analogous to the doubly periodic wave solution obtained by Z.h.Xu et.al [28] .for BO equation when-6 t, c 1 = 1 , γ= 1 , β= 1 .Whereas ( Fig.3 c),( Fig.3 d) and dummyTXdummy-(( Fig.3 e) are kink solutions for Eq.(6) , the solution profiles are analogues the single solitary wave solution Eq.(7) of the BO equation obtained by Hossen et.al.[31] ,with k1= -1 , H = c0= a1= 1 .Where left hand side represents different types of traveling wave solutions and right hand side represents their contour plots respectively.Presented profiles of the BO equation may leads to understand internal waves in fluid dynamics.

5.Conclusion

In this work, we explored some new solitary solutions, some new periodic solutions, and some of their hybrid solutions.If we take more special cases, we will obtain some more new traveling wave solutions along with traditional (trigonometric, hyperbolic, and rational) wave solutions, but for simplicity, we presented a few of them.The obtained traveling wave solutions are unique and very much useful in applied physics and fluid mechanics.Additionally, these novel solutions and plots are helpful to comprehend the nature of nonlinear, dispersive long gravity waves and their propagation in shallow water.The paper showed the novelty of the modified (G ′ /G2) -expansion approach to obtain a variety of exact solutions for the Kadomtsev Petviashvili-modified equal width equation, coupled Drinfel’d-Sokolov-Wilson equation,and the Benjamin-Ono equation.In the future, the advantages of this method can be utilized for other classes of nonlinear evolution equations and obtain the exact non-differentiable type traveling wave solutions.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.