Research article

Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation

  • Kalim U. Tariq , * ,
  • Raja Nadir Tufail
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  • Department of Mathematics, Mirpur University of Science and Technology, Mirpur, AJK 10250, Pakistan
*E-mail address: (K.U. Tariq).

Received date: 2022-02-27

  Revised date: 2022-04-02

  Accepted date: 2022-04-21

  Online published: 2022-05-06

Abstract

In this paper, the (3+1 )-dimensional nonlinear evolution equation is studied analytically. The bilinear form of given model is achieved by using the Hirota bilinear method. As a result, the lump waves and collisions between lumps and periodic waves, the collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions for the given model are constructed. Furthermore, some new traveling wave solutions are developed by applying the exp(−ϕ(ξ)) expansion method. The 3D, 2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters. As a result, a collection of bright, dark, periodic, rational function and elliptic function solutions are established. The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.

Highlights

● The Hirota bilinear method and the exp(ϕ(ξ)) expansion technique are used to generate some new solitary wave solutions.

● The collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions of the given model are also constructed.

Cite this article

Kalim U. Tariq , Raja Nadir Tufail . Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation[J]. Journal of Ocean Engineering and Science, 2024 , 9(2) : 164 -172 . DOI: 10.1016/j.joes.2022.04.018

1. Introduction

Many phenomena in science and engineering are expressed in the form of nonlinear evolution equations (NLEEs). In the last several decades, the study of nonlinear wave structures is becoming one of the dominant eras of contemporary research. In the presence of solitary waves, the nonlinear evolution models are utilized to simulate the effect of surface for deep water and weakly nonlinear dispersive long waves. Therefore, the exact solutions of such models play a vital role of study of dynamical structures and further properties of physical phenomenon occurring in numerous fields of study, such as electromagnetic theory, fluid motion, nuclear physics, optic fibres, physical chemistry, energy physics, compound physics and fluid mechanics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].
In the soliton theory, the solitary wave interacts with one other without losing amplitude or velocity. Their identities and forms, for example, do not change as a result of their reciprocal interactions. Furthermore, the newly developed solutions and their graphical representations demonstrate various dynamical patterns of solitary waves, which is critical to develop a pre-eminent understanding about NLEEs emerging in diversified disciplines of science and engineering. In literature, a number of travelling wave solutions have been computed with the aid of the latest computing tools such as Mathematica, Maple or Matlab [14], [15], [16], [17], [18], [19], [20], [21].
There are several approaches for finding their exact solutions, including the extended simple equation method [22], the logistic equation method [23], the Kudryashov method [24], the direct perturbation method [25], the extended modified auxiliary equation mapping method [26], [27], the inverse scattering method [28], [29], the ϕ6 -model expansion method [30], [31], the exp(ϕ(ξ)) expansion method [32], the tanh method [33], [34], the Hirota bilinear method [35], [36], [37], the exp-function method [38], [39], the generalized exponential rational function method [40], [41], the Darboux transformation [42], the Bäcklund transformation [43], [44], the residual power series method [45], the homogeneous balance method [46], [47], the modified extended direct algebraic method [48], the Lie group method [49], [50] and the variable separation method [51], [52].
The (3+1 )-dimensional NLEE is a new integrable equation that can be used to explore shallow-water waves and short waves in nonlinear dispersive models. This model was initially used to explore the algebraic geometrical solutions [53] and with the help of the (1 + 1)-dimensional AKNS model, it can be reduced into a systems of solvable ordinary differential equations. In [54], the rogue waves and the rational solutions were obtained. An N-fold Darboux transformation was used to generate the complexiton solution, resonant solution and N-soliton solution [55]. The Hirota bilinear approach was used to create the N-soliton solution and its Wronskian form [56], [57]. The collision of first-order lump and line solitons to the considered model are constructed in Chen et al. [58], Tang et al. [59]. The multiple soliton solutions and a variety of traveling wave solutions of the equation are produced in Wazwaz [60], 61], 62]. The exact solutions, Bäcklund transformation and Bilinear representation were acquired in Liu and Liu [63].
In this work, we study the following (3+1 )-dimensional nonlinear evolution equation [63]
3gxz(2gt+gxxx2ggx)y+2(gxx1gy)x=0,
where x1 represents integral operator of x and is defined as
(x1f)(x)=xf(t)dt.
In recent decades, many powerful approaches for completely integrable evolution equations have been developed and applied. Such NLEEs offer a wide range of applications in ocean engineering, which have been studied in various ways. There are a number of intriguing approaches for obtaining their exact solutions, such as Lie symmetry technique [64], two variable (G/G , 1/G )-expansion method [65], simplified Hirota's method [66], modified simple equation method [67], conservation laws [67], Riccati-Bernoulli sub-ODE method [68], and Bäcklund transformation [68], generalized exponential rational function method [69], generalized Kudryashov method [69], Lie group method [70] and exp-function method [71].

2. The bilinear form of (1)

The following transformation is used to produce the bilinear form of (1)
g=3[lnF]xx.
As a result, we obtain the bilinear form shown below
(Dx3Dy+2DtDy3DxDz)F.F=0,
where the D-operator [72] is defined as
DxrDts(a.b)=(xx)r(tt)s×a(x,t).b(x,t)|x=x,t=t.
Therefore, we attain
FxxxyF3FxxyFx+3FxyFxxFyFxxx+2(FytFFyFt)3(FxzFFxFz)=0,
clearly if F fulfils (1), then g=3[lnF]xx gives the solution of given model (1) directly.

3. The main results

In this part, various symbolic structures (mostly drawn from Ghanbari [73]) are used to construct the analytical solution of the equation.

3.1. Collision among lump wave and strip soliton

In this subsection, we consider the function as follows
F=μ0+μ1ω12+μ2ω22+μ3eω3,
where
ω1=α0+α1x+α2y+α3z+α4t,ω2=β0+β1x+β2y+β3z+β4t,ω3=γ0+γ1x+γ2y+γ3z+γ4t,
here αi,βi,γi 's are the unknown constants. Throughout this paper, these definitions for ω1 , ω2 , and ω3 are still valid.
Inserting (5) into (4), we obtain a set of equations involving various parameters. After solving this system with the help of some computing tool like Mathematica, we acquire the following results:
Case 1:
γ4=γ13γ23γ1γ32γ2,μ1=0,μ2=0,
where γ0 , γ1 , γ2 , γ3 , μ0 , μ3 are free parameters. So using these parameters, Eq. (5) becomes
F=μ0+μ3exp(γ0+γ1x+γ2y+γ3z(γ13γ23γ1γ3)t2γ2).
Using (6) along with (3), we obtain the solution g1(x,y,z,t) of (1).
Case 2:
μ3=0,α4=3(α3β1β2μ2α2β1β3μ2+α1β2β3μ2+α1α2α3μ1)2(α22μ1+β22μ2),β4=3(α2α3β1μ1α1α3β2μ1+α1α2β3μ1+β1β2β3μ2)2(α22μ1+β22μ2),μ0=(α22μ1+β22μ2)(α12β1β2μ1μ2+α2α1β12μ1μ2+α2α13μ12+β13β2μ22)μ1μ2(α2β1α1β2)(α2β3α3β2),
where α0 , α1 , α2 , α3 , β0 , β1 , β2 , β3 , μ1 , μ2 are free parameters. Using all these obtained values along with (5) and then utilizing (3), we get the solution g2(x,y,z,t) of (1).
Case 3:
α2=α3β2β3,α4=3α1β32β2,β4=3β1β32β2,μ2=α12μ1β12,μ3=0,
where α0 , α1 , α3 , β0 , β1 , β2 , β3 , μ0 , μ1 are free parameters. So using these parameters, Eq. (5) becomes
F=μ0+μ1(3α1β3t2β2+α3β2yβ3+α1x+α3z+α0)2α12μ1(3β1β3t2β2+β1x+β2y+β3z+β0)2β12.
Using (7) along with (3), we acquire the solution g3(x,y,z,t) of (1).
Case 4:
α2=α1β2β1,α4=3α3β12+3α1β3β12α1β2β42β1β2,μ2=α12μ1β12,μ0=0,μ3=0,
where α0 , α1 , α3 , β0 , β1 , β2 , β3 , β4 , μ1 are free parameters. So using these parameters, Eq. (5) becomes
F=(t(3α3β12+3α1β3β12α1β2β4)2β1β2+α1β2yβ1+α1x+α3z+α0)2μ1α12μ1(β4t+β1x+β2y+β3z+β0)2β12.
Using (8) along with (3), we attain the solution g4(x,y,z,t) of (1).

3.2. Collision between lump wave and double stripes soliton

In this part, we obtain the following function
F=μ0+μ1ω12+μ2ω22+μ3cosh(ω3).
Putting (9) into (4), we obtain a set of equations involving various parameters. After solving this system with the help of some computing tool like Mathematica, we get the following results:
Case 1:
γ4=4γ13γ23γ1γ32γ2,μ0=0,μ1=0,μ2=0,
where γ0 , γ1 , γ2 , γ3 , μ3 are free parameters. So using these parameters, Eq. (9) becomes
F=μ3cosh((4γ13γ23γ1γ3)t2γ2+γ1x+γ2y+γ3z+γ0).
Using (10) along with (3), we acquire the solution g5(x,y,z,t) of (1).
Case 2:
α2=β1β3μ0μ2α1μ1(α12μ1+β12μ2),α4=3α1(α12μ1+β12μ2)(α1α3μ1β1β3μ2)2β1β3μ0μ2,μ3=0,β2=0,β4=3α1μ1(α3β1+α1β3)(α12μ1+β12μ2)2β1β3μ0μ2,
where α0 , α1 , α3 , β0 , β1 , β2 , β3 , μ0 , μ1 , μ2 are free parameters. Using all these obtained values along with (9) and then utilizing (3), we attain the solution g6(x,y,z,t) of (1).
Case 3:
α2=0,α4=3β1μ2(α3β1+α1β3)(α12μ1+β12μ2)2α1α3μ0μ1,μ3=0,β2=α1α3μ0μ1β1μ2(α12μ1+β12μ2),β4=3β1(α12μ1+β12μ2)(α1α3μ1β1β3μ2)2α1α3μ0μ1,
where α0 , α1 , α3 , β0 , β1 , β3 , μ0 , μ1 , μ2 are free parameters. Using all these obtained values along with (9) and then utilizing (3), we obtain the solution g7(x,y,z,t) of (1).
Case 4:
μ0=0,μ1=0,μ2=0,γ3=2(2γ2γ13+γ2c4)3γ1,
where γ0 , γ1 , γ2 , γ4 , μ3 are free parameters. So using these parameters, Eq. (9) becomes
F=μ3cosh(γ4t+γ1x+γ2y+2(2γ2γ13+γ2γ4)z3γ1+γ0).
Using (11) along with (3), we get the solution g8(x,y,z,t) of (1).

3.3. Collision among lump and periodic waves

In this case, we attain the function F as follows
F=μ0+μ1ω12+μ2ω22+μ3cos(ω3).
Inserting (12) into (4), we obtain a set of equations involving various parameters. After solving this system with the help of some computing tool like Mathematica, we acquire the following results:
Case 1:
γ4=γ1(4γ2γ12+3γ3)2γ2,μ0=0,μ1=0,μ2=0,
where γ0 , γ1 , γ2 , γ3 , μ3 are free parameters. So using these parameters, Eq. (12) becomes
F=μ3cos(γ1(4γ2γ12+3γ3)t2γ2+γ1x+γ2y+γ3z+γ0).
Using (13) along with (3), we acquire the solution g9(x,y,z,t) of (1).

3.4. Collision between lump wave and double stripes soliton

Finally, we obtain the general structure stated as
F=μ0+μ1ω12+μ2cosh(ω2)+μ3cos(ω3).
Inserting (14) into (4), we obtain a set of equations involving various parameters. After solving this system with the help of some computing tool like Mathematica, we attain the following results:
Case 1:
β2=β1β3+γ1γ3β1(β12+γ12),β4=4β13γ1γ3+3β3β12γ12β14β32(β1β3+γ1γ3),μ0=0,γ2=β1β3γ1γ3γ1(β12+γ12),γ4=4β1β3γ133β12γ3γ12+γ3γ142(β1β3+γ1γ3),μ1=0,
where β0 , β1 , β3 , γ0 , γ1 , γ3 , μ2 , μ3 are free parameters.
Using all these obtained values along with (14) and then utilizing (3), we acquire the solution g10(x,y,z,t) of (1).

4. Travelling-wave solutions for Eq. (1)

Now, we will obtain the exact traveling wave solutions for (1) using exp(ϕ(ξ)) expansion technique. Firstly, substituting
g(x,y,z,t)=vx,
in (1) and utilizing the boundary condition gy|x=0 , we get
3vxxz(2vxt+vxxxx2vxvxx)y+2(vxxvxy)x=0
Next, we obtain the transformation stated as
v(x,y,z,t)=Ω(ξ),ξ=αx+βy+γz+λt,
here α,β,γ are constants and λ is the speed of the wave. We use the transformation (17) in (16) to get an ordinary differential equation as
α4βΩ(5)(ξ)+(3α2γ2αβλ)Ω(3)(ξ)+4α3β(Ω(ξ)2+Ω(3)(ξ)Ω(ξ))=0.
Integrating it twice, we get
α4βΩ(3)(ξ)+(3α2γ2αβλ)Ω(ξ)+2α3βΩ(ξ)2.
Using the considered method, we assume the solution of (19) as follows:
Ω(ξ)=i=0MAi(exp(ϕ(ξ)))i,
here Ai(0iM) are parameters, and
ϕ(ξ)=exp(ϕ(ξ))+exp(ϕ(ξ))P+Q.
Group 1: Q24P>0,P0 ,
ϕ(ξ)=ln(Q24Ptanh(12(C+ξ)Q24P)Q2P),
ϕ(ξ)=ln((Q24P)coth(12(C+ξ)Q24P)Q2P),
where C is an integration constant.
Group 2: Q24P<0,P0 ,
ϕ(ξ)=ln(4PQ2tan(12(C+ξ)4PQ2)Q2P),
ϕ(ξ)=ln(4PQ2cot(12(C+ξ)4PQ2)Q2P).
Group 3: Q24P>0 , P=0 , Q0 ,
ϕ(ξ)=ln(QeQ(C+ξ)1).
Group 4: Q24P=0 , P0 , Q0 ,
ϕ(ξ)=ln(2(Q(C+ξ)+2)Q2(C+ξ)).
Group 5: Q24P=0 , P=0 , Q=0 ,
ϕ(ξ)=ln(C+ξ).
Balancing Ω(3)(ξ) and Ω(ξ)2 in (19), we obtain M=1 . Therefore, Eq. (20) can be stated as follows
Ω(ξ)=A0+A1eϕ(ξ).
Putting (29) along with (21) into (19), we obtain an algebraic system of equations given as follows:
6α4A1β+2α3A12β=0,12α4A1βQ+4α3A12βQ=0,α4A1βQ3+8α4A1βPQ+4α3A12βPQ3α2A1γQ+2αA1βλQ=0,2α4A1βP2+α4A1βPQ2+2α3A12βP23α2A1γP+2αA1βλP=0,8α4A1βP+7α4A1βQ2+4α3A12βP+2α3A12βQ23α2A1γ+2αA1βλ=0,
We obtain the following result by solving this system:
A1=3α,λ=4α3βP+α3βQ23αγ2β.
Therefore, using these parameters we attain the following solutions for the given model:
Group 1: Q24P>0,P0 ,
v(x,y,z,t)=A06αPQ24Ptanh(12Q24P(C+ξ))Q,
inserting (31) into (15), we attain the solution as
g11(x,y,z,t)=3α2P(Q24P)sech2(12Q24P(C+ξ))(Q24Ptanh(12Q24P(C+ξ))Q)2,
similarly
v(x,y,z,t)=A06αPQ24Pcoth(12(C+ξ)Q24P)Q,
inserting (33) into (15), we get the solution as
g12(x,y,z,t)=3α2P(Q24P)csch2(12Q24P(C+ξ))(Q24Pcoth(12Q24P(C+ξ))Q)2,
where ξ=t(4α3βP+α3βQ23αγ)2β+αx+βy+γz .
Group 2: Q24P<0,P0 ,
v(x,y,z,t)=A06αP4PQ2tan(12(C+ξ)4PQ2)Q,
inserting (35) into (15), we obtain the solution as
g13(x,y,z,t)=3α2P(4PQ2)sec2(124PQ2(C+ξ))(4PQ2tan(124PQ2(C+ξ))Q)2,
similarly
v(x,y,z,t)=A06αP4PQ2cot(12(C+ξ)4PQ2)Q,
inserting (37) into (15), we acquire the solution as
g14(x,y,z,t)=3α2P(4PQ2)csc2(124PQ2(C+ξ))(4PQ2cot(124PQ2(C+ξ))Q)2,
where ξ=t(4α3βP+α3βQ23αγ)2β+αx+βy+γz .
Group 3: Q24P>0 , P=0 , Q0 ,
v(x,y,z,t)=A03αQexp(Q(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz))1,
inserting (39) into (15), we obtain the solution as
g15(x,y,z,t)=3α2Q2exp(Q(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz))(exp(Q(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz))1)2.
Group 4: Q24P=0 , P0 , Q0 ,
v(x,y,z,t)=A0+3αQ2(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz)2(Q(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz)+2),
inserting (41) into (15), we acquire the solution as
g16(x,y,z,t)=3α2Q3(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz)2(Q(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz)+2)2.
Group 5: Q24P=0 , P=0 , Q=0 ,
v(x,y,z,t)=A03αCt(4α3βP+α3βQ23αγ)2β+αx+βy+γz,
inserting (43) into (15), we attain the solution as
g17(x,y,z,t)=3α2(Ct(4α3βP+α3βQ23αγ)2β+αx+βy+γz)2.

5. Discussion and results

This section displays graphical illustrations of the (3+1 )-dimensional NLEE. For the suitable values of the constants involved, multiple profiles of the resultant solutions have been shown. For a given set of values, a variety of bright, dark, periodic, and singular bell-shaped solitons are shown. Fig. 1 shows the behaviour of solution |g1(x,y,z,t)| for the values of the constants γ0=0.7 , γ1=0.5 , γ2=0.75 , γ3=0.25 , μ0=0.8 , μ3=0.6 and it represents a bright soliton solution. Similarly, Figs. 2, 4, 6, 8, 9 and 12 also represent the bright soliton solutions for the appropriate values of the parameters involving. Fig. 3 represents the singular bell shaped solution for the values of constants α0=0.5 , α1=0.4 , α3=0.6 , β0=0.55 , β1=0.7 , β2=0.56 , β3=0.4 , β4=0.3 , μ1=0.5 . Also, the Figs. 11, 13 and 14 represent the singular bell shaped solutions for the appropriate values of the parameters involving. Singularities can exist anywhere, and they're surprisingly abundant in physicists' mathematics for understanding the universe. Fig. 5 represents the dark soliton solution for the values of the parameters β0=0.15 , β1=0.12 , β3=0.15 , α0=0.1 , α1=0.12 , α3=0.15 , μ0=0.4 , μ1=0.5 , μ2=0.6 . Figs. 7 and 10 illustrate the periodic solitary waves solutions for the various values of the parameters involved. To better understand the behaviour of solutions, 3D, contour and 2D graphs of attained solutions with various parameter values are shown.
Fig.1 Profiles of the solution |g1(x,y,z,t)| of (1) for γ0=0.7 , γ1=0.5 , γ2=0.75 , γ3=0.25 , μ0=0.8 , μ3=0.6 .
Fig.2 Profiles of the solution |g2(x,y,z,t)| of (1) for α0=0.5 , α1=0.55 , α2=0.75 , α3=0.5 , β0=0.6 , β1=0.5 , β2=0.4 , β3=0.25 , μ0=0.5,μ1=0.6 , μ2=0.7 .
Fig.3 Profiles of the solution |g4(x,y,z,t)| of (1) for α0=0.5 , α1=0.4 , α3=0.6 , β0=0.55 , β1=0.7 , β2=0.56 , β3=0.4 , β4=0.3 , μ1=0.5 .
Fig.4 Profiles of the solution |g5(x,y,z,t)| of (1) for γ0=0.25 , γ1=0.55 , γ2=0.3 , γ3=0.025 .
Fig.5 Profiles of the solution |g7(x,y,z,t)| of (1) for β0=0.15 , β1=0.12 , β3=0.15 , α0=0.1 , α1=0.12 , α3=0.15 , μ0=0.4 , μ1=0.5 , μ2=0.6 .
Fig.6 Profiles of the solution |g8(x,y,z,t)| of (1) for γ0=1.55 , γ1=0.75 , γ2=0.8 , γ4=0.25 .
Fig.7 Profiles of the solution |g9(x,y,z,t)| of (1) for γ0=0.1 , γ1=2.25 , γ2=2.55 , γ3=2.25 , μ3=2.6 .
Fig.8 Profiles of the solution |g10(x,y,z,t)| of (1) for β0=0.255 , β1=0.5 , β3=0.255 , γ0=0.25 , γ1=0.225 , γ3=0.125 , μ2=2.6 , μ3=0.25 .
Fig.9 Profiles of the solution |g11(x,y,z,t)| of (1) for α=1.25 , β=1.2 , γ=1.125 , C=1.35 , P=0.125 , Q=1.1 .
Fig.10 Profiles of the solution |g12(x,y,z,t)| of (1) for α=1.7 , β=1.8 , γ=1.9 , C=1.35 , P=1.5 , Q=0.5 .
Fig.11 Profiles of the solution |g13(x,y,z,t)| of (1) for α=0.7 , β=0.85 , γ=0.95 , C=0.8 , P=0.56 , Q=0.55 .
Fig.12 Profiles of the solution |g14(x,y,z,t)| of (1) for α=0.6 , β=0.5 , γ=0.5 , C=0.6 , P=0.4 , Q=0.55 .
Fig.13 Profiles of the solution |g15(x,y,z,t)| of (1) for α=0.7 , β=0.85 , γ=0.95 , C=0.8 , P=1.56 , Q=0.55 .
Fig.14 Profiles of the solution |g17(x,y,z,t)| of (1) for α=0.2 , β=0.1 , γ=0.3 , C=0.9 , P=0.5 , Q=0.75 .

6. Conclusion

In this paper, we use the Hirota bilinear approach and the exp(ϕ(ξ)) expansion technique to investigate the (3+1 )-dimensional NLEE. Using various sorts of functions, we were able to obtain a number of intriguing exact solutions to the given model. We were able to achieve alternative graphical interpretations of the given solutions for different sets of parameter values using Mathematica 11.0. Finally, we can say that the considered approaches can be used to explore a variety of NLEEs in mathematical physics.

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.

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