Imad Jaradat,Marwan Alquran

Department of Mathematics and Statistics,Jordan University of Science & Technology,Irbid 22110,Jordan

Keywords:Exponential-expansion method Generalized equal-width equation Wazwaz-Benjamin-Bona-Mahony equation

ABSTRACT In this paper we are interested in investigating the physical shape-changed propagations to the generalized Equal-Width equation through studying the explicit solutions of Wazwaz-Benjamin-Bona-Mahony model.Both models are of considerable importance in many disciplines of research,including ocean engineering and science,and describe the propagation of equally-width waves.We highlight the effect of the coefficients of both nonlinearity and dispersion terms on changing the physical shape of both models by implementing the new exponential-expansion scheme.2D and 3D graphical plots are provided to validate the findings of the paper.

1.Introduction

In recent years,the study of solitary waves of nonlinear equations gains a lot of attention in the field of ocean dynamics and engineering applications.Investigating travelling solitary waves from a mathematical point of view helps in identifying physical structures and understanding many natural phenomena.Therefore,the construction of explicit solitary wave solutions of nonlinear equations is essential task to describe natural phenomena,such as propagations of long-short waves,equal-width waves,nonlinear lattice-waves,collision of waves and others [1–3].

Recently,several numerical strategies are used in the solitary waves theory to find explicit solutions to nonlinear equations.These approaches provide interesting collection of solutions with different nice physical structures such as kink-wave,solitonwave,cusp-wave,breather-wave,rogue-wave and many others [4–8].In this regard,we mention some of these effective numerical schemes.For example,the(G′/G)-expansion method and its modifications [9–11],Kudryashov method [12–14],Jacobi elliptic function method [15,16],Polynomial function method [17–19],F-expansion and improved simple equation methods [20],Sine-Gordon equation method [21,22],Hirota’s bilinear method combined with the extended ansatze function method [23]and the positive quadratic function [24],and many others,see [25–28].

In this work,we aim to import different types of propagating nonlinear waves to gEW equation from those explicit-solutions obtained for WBBM equation by implementing the new exponentialexpansion technique.

1.1.Formulations of gEW and WBBM equations

The KdV equation is a nonlinear evolutionary model that describes the propagation of unidirectional shallow water waves.The KdV has the following mathematical formulation

wherewwxis the nonlinearity term andwxxxis the spacedispersion term.In (1),ifwxxxis replaced by time-space dispersionwxxt,we get the equal-width (EW) equation

The EW is proposed in [29]as a simulation of the KdV and describes the motion of shallow waves propagation with equally width to all wave amplitudes.In (2),if the nonlinear term is expanded intowmwx,we get gEW equation

Later,Wazawz modified (3) by adding the term-βuxand the resulting equation is known as WBBM [30–32]

The gEW (3) and WBBM (4) are classified as third-order nonlinear equations with time-space dispersion term.

2.Generalized Wazwaz-Benjamin-Bona-Mahony

The first goal of this study is to extract all possible explicit solutions to WBBM for the casesm=1 andm=2 by implementing the new exponential-expansion approach.We consider the new variablez=x-ctto convert (4) into the following differential equation

2.1.WBBM with m=1

WBBM withm=1,(5) reads

The proposed method assumes the solution of (6) in the following form

whereY=Y(z)satisfies the auxiliary differential equation

whereAandBare unknown real constants to be determined later.Upon determining the values ofAandB,the solution of (8) is

Remark 1.The functionw(z)as described in (7) can be regarded as a polynomial of degreenin the variablee-Yand the order of thekth-derivative ofw(z)isn+k.

Now,balancing the order ofw2versus the order ofw′′ appearing in (6),givesn=2.Accordingly,(7) takes the following form

To determineλ0,λ1,λ2,c,A,B,we substitute (10) in (6),and collect the coefficients ofe-iY:i=0,1,2,...Then,we set each coefficient to zero and solve the resulting system.Doing so,we reach at the following output.

From (11),B2-4A=Therefore,the solution of WBBM withm=1 is

2.2.WBBM with m=2

WBBM withm=2,(5) reads

Applying the balance procedure betweenw3andw′′ appearing in(13),givesn=1.Accordingly,the solution of (13) is

Substitution of (14) in (13) leads to the following results:

Note thatB2-4A=Therefore,the solution of WBBM withm=2 is

3.Generalized equal-width equation

The second goal of this work is to derive and analyze graphically the solutions of gEW from the obtained solutions obtained for WBBM in the previous section.We start with the variablez=x-ctto convert (3) into the following differential equation

3.1.gEW with m=1

EW withm=1,(17) reads

It is easy to observe that (18) coincide with (6) whenβ=0.Therefore,considering the result given in (12) and substituteβ=0,we reach at the following explicit solution of gEW withm=1.

Fig.1,presents the 3D-2D plots of (19) forγ＞0,while as Fig.2 is the case forγ＜0.These two figures confirm the importance of the coefficient-parameterγby influencing the dynamics of gEW.It propagates as a bell-shaped wave forγ＞0 and is changed to a breather waves whenγ＜0.

3.2.gEW with m=2

EW withm=2,(17) reads

Now,substitutingβ=0 in (13) will produce (20).Accordingly,by lettingβ=0 in (16),the explicit solution of gEW withm=2 is

Fig.3 shows three different physical shapes,kinky-periodic,kink and singular-kink,for gEW withm=2 upon changing the sign of the parameterγand the value of the free constantλ0.We point here that the solution in (21) forγ＞0 is real-valued function under the condition that the coefficient-parameterα＜0.

Fig.1.3D-2D plots of gEW with m=1 as depicted in (19) for γ＞ 0.

Fig.2.3D-2D plots of gEW with m=1 as depicted in (19) for γ＜ 0.

Fig.3.3D plots of gEW with m=2 as depicted in (21) for {c=1 ,α=-1 ,γ=2 ,λ0=1} ,{c=α=1 ,γ=-2 ,λ0=2} and {c=α=1 ,γ=-2 ,λ0=},respectively.

4.Conclusion

New exponential-expansion algorithm is used to extract different explicit-solitary wave solutions to the gWBBM model.By neglecting the first-order space-dispersive term from gWBBM,we obtained the gEW model.Physical structures of propagating the wave-solutions to gEW equation are investigated by recovery the solutions reported to gWBBM.Bell-shaped,breather,kink,singularkink and kink-periodic waves are different types of solitary-wave solutions extracted to gEW by implementing the proposed method.We believe that the findings of this current work are useful in the field of nonlinear-waves arising in ocean engineering and science.

This study reveal the elastic connections among different physical models,and the simplicity to recover the dynamics of one application by exploring other related applications and models.For future work,we plan to use analytical schemes such as reproducing kernel and residual power series algorithms [33–38]to solve similar proposed models as well as other fractional-classical nonlinear models.

Declaration of Competing Interest

The authors declare that they have no competing interests.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.