Md.Rezwn Ahmed Fhim ,Puroi Rni Kundu ,Md.Ekrmul Islm ,M.Ali Akr ,M.S.Osmn

a Department of Mathematics,Pabna University of Science and Technology,Bangladesh

b Department of Applied Mathematics,University of Rajshahi,Bangladesh

c Department of Mathematics,Faculty of Science,Cairo University,Giza 12613,Egypt

d Department of Mathematics,Faculty of Applied Science,Umm Alqura University,Makkah 21955,Saudi Arabia

Keywords:Sine-Gordon expansion approach Kadomtsev-Petviashvili equation modified KdV-Zakharov-Kuznetsov equation soliton solutions

ABSTRACT The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation,dynamics of shallow water wave,plasma fluid model,chemical kinematics,chemical engineering,geochemistry,and many other topics.In this article,we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it.As a result,wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations,containing kink,breather soliton,bell-shaped soliton,and periodic waves.To extract the soliton solutions of the high-dimensional nonlinear evolution equations,a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly.The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations.This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs).Closed-form solutions are carefully illustrated and discussed through diagrams.

1.Introduction

Exact solutions of nonlinear equations have a significant role in nonlinear physical science and extensively used in applied science,ocean engineering and in other fields of technology,since majority of physical phenomena appearing in hydrodynamics,plasma,fibers optics,chemistry etc.are normally have intricate nonlinear features [1–10].They can provide much applied data and more insight into the physical characteristics of the problem and deliver further understanding and applications.But,extraction of solutions of nonlinear equations is difficult and cumbersome,and generally requires special schemes for certain models.Due to the complicated structure of the nonlinear wave equations,the investigation of travelling wave solutions is turning out to be increasingly appealing.Successively,with immediate advancement of computation systems,a variety of novel approaches has been effectively discussed and improvement by assorted researchers,for instance: the improved Bernoulli sub-equation function method[11,12 ],the F-expansion method [13],the improved F-expansion method [14],the traveling-wave method [15],the sine-cosine method [16],the modified auxiliary equation method [17],Jacobi elliptic function method [18],extended direct algebraic method [19],the homogeneous balance method [20],Hirota bilinear method [21–24],the(G′/G)-expansion method [25],modified extended tanh method [26],the reductive perturbation method [27],the sine-Gordon expansion method [28–33],B?cklund transformation [34],the test function method [35]and others[36–49].We adapt sine-Gordon expansion approach: a relatively advanced mathematical method.The sine-Gordon expansion method stands on a strong basis of the known sine-Gordon equation.In this method,the solutions of nonlinear equations are estimated as polynomials of hyperbolic functions and to simplify the calculation involved in the determination of these polynomials a transformation based on the use of the sine-Gordon equation is functional.The sine-Gordon approach is a latterly developed theory to bring out advanced and broad-ranging solutions to the HNEEs.The underlying motivation of the present study is to make use of the sine-Gordon expansion approach to unravel the(3+1)-dimensional nonlinear physical models.The consequence is a new analogous system with two additional dimensions which may considered more general than the classic original system.Subsequently,we utilize the method to obtain solutions for a couple of nonlinear evolution equations,namely,Kadomtsev-Petviashvili equation [50–52],which describes many nonlinear processes in plasma physics,fluid dynamics,and other nonlinear sciences when the non-uniformities of borders and the inhomogeneities of media are considerd,and modified KdV Zakharov-Kuznetsov equation[53–57]that is modeled to investigate the waves in magnetized plasma.

The grouping of this investigation goes as follows: Section 2 is dedicated for presentation and additional evaluation of the sine-Gordon expansion method.Section 3 contains execution of the proposed method to extract solutions.Physical explanations with graphs of solution behavior are given in Section 4.The last section is a brief conclusion.

2.Methodology

We will provide a brief explanation of the sine-Gordon expansion method in this section.Let us consider a nonlinear evolution equation including four variablesx,y,z,tas follows:

HereU=U(x,y,z,t)is the wave function andmis an arbitrary real constant.Now,we assume that,

whereα,βandγare wave direction ratios andωis the velocity of the traveling wave.

After applying (2.2),the equation (2.1) can be modified to get a nonlinear equation of the subsequent form

Equation (2.3) can be reestablish as follows

whereinkis the constant of integration.

If,k=0,φ(ξ)=andinto

equation (2.4),resultant will be

Forl=1,equation (2.5) transmutes to

By considering the separation variable technique,we reach the next relations,

wherefis the integration constant.

Now,we consider a (3+1)-dimensional nonlinear evolution equation as follows:

in whichU=U(x,y,z,t)is an unknown wave function,ψis a polynomial of the variableUand its derivatives.

In line with the mentioned method we suppose the solution of equation (2.9) as

From (2.7) and (2.8) in (2.10),we get

From the obtained nonlinear equation with the help of balancing principle,it can be determined the value ofR.Balancing the coefficients of [sinq(φ(ξ))cosq(φ(ξ))]to zero,provides many algebraic equations which can be solved to find the values ofMr,Nr,α,β,γandω.Finally,plugging the values ofMr,Nr,α,β,γandωinto solution (2.10),the required solution to the nonlinear evolution equation (2.9) can be found.

3.Mathematical analysis of solutions

In this paragraph,we formulate the stable solitary wave solutions for the Kadomtsev-Petviashvili equation and the modified KdV-Zakharov-Kuznetsov equation through the implementation of the extended sine-Gordon expansion approach.

3.1.The Kadomtsev-Petviashvili equation

The (3+1)-dimensional Kadomtsev-Petviashvili equation is of the form [51]:

Now,we consider the succeeding wave transformation

whereincis the soliton’s speed.After treating (3.1.2),the Kadomtsev-Petviashvili equation (3.1.1) alters into the following new shape

Integrating (3.1.3) two times and disregarding the integral constants,we achieve

From the previous section,we attainR=2.Thus,the general solution (3.1.4) has the type,

The second derivative of the solution (3.1.5) is given below

Embedding (3.1.5) and (3.1.6) into the equation (3.1.4),we get:

Fig.1.Different geometrical structures for the solution given by Eq.(3.1.21).

From above equation by setting the coefficients of same power of sin(φ)and cos(φ)to zero:

Fig.2.Different geometrical structures for the solution given by Eq.(3.1.23).

Unraveling the above documented equations with the help of any software package,we attain:

Case I: By means of the values in (3.1.17) through equation (3.1.2) into (3.1.5),we attain

Case II:Substituting the values in (3.1.18) along with (3.1.2) in(3.1.5),we find the solution

Fig.3.Different geometrical structures for the solution given by Eq.(3.2.18).

Case III: As in previous two cases,from (3.1.19) and (3.1.2) into(3.1.5),we accomplish

Case IV: Putting the values of parameter in (3.1.20) with equation (3.1.2) in (3.1.5),we determine

3.2.The modified KdV-Zakharov-Kuznetsov equation

We consider the (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation of the form [53]:

whereβis a constant.We may accept the following transformation:

whereλis the wave velocity.After treating the transformation (3.2.2),the modified KdV-Zakharov-Kuznetsov equation (3.2.1) transforms into the resulting new form:

Integrating (3.2.3) once,we develop

Using the homogeneous balancing principle,we getR=1 for the equation (3.2.4).Therefore,as stated in the methodology section,we begin the solution of equation (3.2.4) as follows:

The successive derivatives ofUcan be determined using (3.2.5),and the results are shown below

We find the resulting double variable polynomial by introducing(3.2.5) and (3.2.6) into (3.2.4):

Similarly,as we did in the previous section,from (3.2.7) we establish the following system of equations:

The following values of the unknown parameters are determined from the above equations using Maple software:

Family 1

Family 2

Family 3

Category I:Using the values of parameter arranged in (3.2.15)in place of the unknown constants into solution (3.2.5) together with (3.2.2),we ascertain

Category II: Use (3.2.16) into solution (3.2.5),we bring out the next solution

Category III:Introducing the values of parameter prescribed in(3.2.17) into solution (3.2.5),we accomplish

The attained wave solutions of the modified KdV-Zakharov-Kuznetsov equation are significant findings and might be useful in the analysis of fluid flow,chemical engineering,plasma physics,geochemistry,and other fields.

4.Results and discussion about the findings

This section is divided in two sub-sections.First part is concerned about the Kadomtsev-Petviashvili equation and in second part we pertained to the modified KdV-Zakharov-Kuznetsov equation.Herein,we have discussed the essence of the completed solutions carefully with graphical demonstrations for different parameter values.

4.1.To the Kadomtsev-Petviashvili equation

With the support of Matlab,the dynamical behaviors of the solutions to the Kadomtsev-Petviashvili equation are demonstrated.

It is significant to understand the of the wave profile’s characteristic subject to the associated free factors.To analyze in detail,we drew several portrayals of (3.1.21) for diverse values of the wave velocitycwhich depends on free parametersk,landm.For solution function (3.1.21),we have used the valuekandcarranged in equation (3.1.17) impliesEquation (3.1.21) offers different portraits for real and imaginary parts.For the valuesN2=1,l=±1,m=±1,(3.1.21) introduces spike type irregular soliton for both real part imaginary part depicted in Fig.1 (a) within the interval-2＜x,t＜2 andy,z=1.ForN2=-1,l=±1,m=±1,that is for negative values ofN2,we attain dark-bright soliton given in Fig.1 (b) for both parts.Changes made with the values ofl,mdo not alter the nature of the solutions as we found multi-soliton for real part and special soliton asserted in Fig.1(c) forN2=1,l=±0.7,m=±0.7,as we got for Fig.1 (a).Now,forN2=-with the valuesl=±1,m=±1,it depicts dark and dark-bright soliton for real and imaginary part,respectively illustrated in Fig.1 (d).For growing the values ofl,mup to 100,both real and imaginary part changes their nature and real part becomes kink wave,while imaginary part depicts bell wave as shown in Fig.1 (e).Then again,shrinking the values ofl,mto zero,each part represents special type soliton portrayed in Fig.1 (f).The outlines are portrayed in-2＜x,t＜2 andy,z=1.

It is clear that solution (3.1.23) allows the values the parameters ask∈ R-{0} andl,m∈ R.Equation (3.1.23) represents breather soliton shown in Fig.2 (a) fork=±1,l=±1,m=±1 within the interval-10＜x,t＜10 andy,z=1.Increasing the value ofkmeans increase the wave velocitycwhich rises the distance between the breather spikes and reducing the value ofkincrease the length of the waves portrayed in Fig.2 (b) and Fig.2 (c),respectively.As in Fig.2 (b) fork=0.5,l=±1,m=±1,the spikes have wave-length larger than that of Fig.2 (a).On the other hand,fork=-2,l=±1,m=±1 the distance between the spikes increased which marked in Fig.2 (c).Therefore,changing the values ofl,monly change the number of spikes but the nature of the equation remains same as we can see in Fig.2 (d) fork=1,l=±1,m=±0.5.

For the other solutions,we attain almost similar figures,and thus to avoid recurrence,have not been drafted here.

From the previous analysis,it can be concluded that we have found different forms of waves to the Kadomtsev-Petviashvili equation whose characteristics are affected by wave velocity and other parameters.

Fig.4.Different geometrical structures for the solution given by Eq.(3.2.19).

Fig.5.Different geometrical structures for the solution given by Eq.(3.2.20).

4.2.To the modified KdV-Zakharov-Kuznetsov equation

The characteristic diagrams and evolution of the solutions obtained for the modified KdV-Zakharov-Kuznetsov equation are discussed in this section.(3.2.18) depicts periodic wave for both real and imaginary parts in Fig.3 (a) form=1,n=1,M0=1,β=1 within the interval 0＜x,t＜4 andy=1,z=1.It is clear thatM0∈ R-{0} andm,n,β∈ R but they can not be all zero together which yields trivial solution.If the partmy+nz=0 then the solution (3.2.18) becomes purely imaginary and the modulus embodies spike singular irregular type of periodic waves,shown in Fig.3 (b)form=-1,n=1,M0=1,β=1 within the interval-10＜x,t＜10 andy=1,z=1.Again,for higher value ofM0=15 in Fig.3 (c),we get parabolic wave and kink shaped soliton within the interval-7＜x,t＜7 andy=1,z=1.For-36n2-36m2-6βM21≥0 the solution becomes all real and displays ideal kink soliton solution as given in Fig.3 (d) for precise valuesm=0.1,n=0.1,M0=

The solutionU2provided in (3.2.19) portrays approximately similar structure of the solution (3.2.18) except for some special cases.One of the cases is whenmy+nz=0,the solution(3.2.19) becomes real and embodies spike type irregular singular periodic waves which is just opposite of the previous solution of the equation (3.2.18).The figure is shown in the Fig.4 (a) for the valuesm=-1,n=1,N1=1,β=1 with the interval-3＜x,t＜3 andy=1,z=1.Again in Fig.4 (b),we use following valuesm=0.1,n=0.1,β=1 to make-36n2-36m2+≥0 for which we get bell-shaped wave with no imaginary part,within the interval-5＜x,t＜5 andy=1,z=1.

Finally,for the solution (3.2.20),we encounter the similar cases with different profile.The solutionU3illustrates exclusively real solution with the portrayal of spike type irregular singular periodic waves in the figure Fig.5 (a),if we choosem=1,n=-1,N1=1,β=1 and vanish the partmy+nz.Then we try to create a situation where-9n2-9m2＞0 and get Fig.5(b) with the valuesm=0.1,n=0.1,N1=1.5,β=1.This time Fig.5 (b) displays bell-shape and kink wave for real and imaginary part,respectively.

All the experiments above indicate that,we get kink,bell-shape soliton,parabolic soliton,spike type irregular singular periodic waves,kink type waves,singular periodic soliton for the modified KdV-Zakharov-Kuznetsov equation.However,it is significant that we acquire both bright and dark solitons.We typically get different related periodic waves,with the exception of three special cases.But solutions attained in this study changed their nature in those particular cases.We also found that the nature of the obtained solutions changed with the velocity.

5.Conclusion

We have effectually put into use the sine-Gordon expansion approach to study nonlinear evolution equations with higher dimensions in this analysis.This method with computerized symbolic computation systems,namely Maple and MATLAB have been used to achieve solitary wave solutions,videlicet,kink,bell-shape soliton,anti-bell shape soliton,spike type irregular singular periodic waves,breather soliton,kink type waves,periodic solutions,singular periodic to the Kadomtsev-Petviashvili equation and the modified KdV-Zakharov-Kuznetsov equation.Wave behaviors for various cases have been carefully analyzed and wave propagation has been estimated in the results and discussion section and the physical description has also been conferred.The newly found hyperbolic function solutions may be useful for understanding physical phenomena especially in long-wave propagation,dynamics of shallow water wave,and plasma fluid.The reliability and the scope of computation give this method an eclectic applicability for future research to higher-dimensional nonlinear evolution equations.

Declaration of Competing Interest

The authors have declared no conflict of interest

Acknowledgments

The authors thank to Faculty of Science,University of Rajshahi,Bangladesh for supporting project number (TURSP-2019/16).