Research article

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

  • Oke Davies Adeyemo
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  • International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa

Received date: 2022-03-09

  Revised date: 2022-04-14

  Accepted date: 2022-04-18

  Online published: 2022-04-26

Abstract

The nonlinear evolution equations have a wide range of applications, more precisely in physics, biology, chemistry and engineering fields. This domain serves as a point of interest to a large extent in the world's mathematical community. Thus, this paper purveys an analytical study of a generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model. We further reduce the equation using the subalgebras obtained. Besides, more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique. Consequently, we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright, dark, periodic (cnoidal and snoidal), compact-type as well as singular solitons. The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time, are outlined with suitable diagrams. This can be of interest to oceanographers and ocean engineers for future analysis. Furthermore, to visualize the dynamics of the results found, we present the graphic display of each of the solutions. Conclusively, we construct conservation laws of the understudy equation via the application of Noether's theorem.

Highlights

● We study cnoidal and snoidal wave solutions via optimal system of one-dimensional sub-algebras for a generalized extended (2+1)-D quantum ZK equation with power-law non-linearity in oceanography and ocean engineering.

● We obtain non-topological soliton, cnoidal and snoidal wave and hyperbolic solutions.

● We derive conservation laws using Noether's theorem.

● We highlight the applications of our results (cnoidal and snoidal waves) in oceanography and ocean engineering.

Cite this article

Oke Davies Adeyemo . Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering[J]. Journal of Ocean Engineering and Science, 2024 , 9(2) : 126 -153 . DOI: 10.1016/j.joes.2022.04.012

1. Introduction

The nonlinear evolution equations, as well as their solutions, are highly significant with regards to a good understanding of diverse physical phenomena, for example, in the investigation of the waves observed in plasma, astrophysics, optical fibers, laser, fluids, water waves alongside other areas bothering on engineering. Investigation of autonomous nonlinear evolution equations possesses largely a very rich and long history. For instance, in [1], the generalized variable coefficient Korteweg-de Vries equation with dual power-law nonlinearities possessive of linear damping as well as dispersion terms in quantum field theory were studied. Besides, the significance of the equation in quantum field theory and other theoretic physics areas were highlighted. Moreover, a generalized system of variable-coefficient modified Kadomtsev-Petviashvili-Burgers-type equation in three dimensions was examined in [2]. The authors in [3], investigated the generalized advection-diffusion equation, an important nonlinear evolution equation in fluid mechanics, featuring the transit of buoyancy-projecting-plume existing in a bent-on bibulous medium. A generalized Korteweg-de Vries-Zakharov-Kuznetsov equation in [4], was also examined. The equation recounts the fusion of warm inviscid fluid and hot isochoric alongside a cold static environment which has significance in fluid dynamics. Besides, in [5], a modified-generalized Zakharov-Kuznetsov (ZK) model, delineating ion-acoustic gravitation solitary waves found in a magnetoplasma-electron-positron-ion that exists in the primordial universe was contemplated. This equation was invoked in modelling situations in plasma physics. Moreover, in [6] a study was carried out on interaction characteristics of the vector bright solitons observed via the examination of the coupled Fokas-Lenells system. This system models the femtosecond optical pulses existing in a birefringent optical fibre.
Consequently, nonlinear evolution equations have continued to attract the attention of mathematicians and physicists in these recent years. Further to that, exact or analytic solutions to these equations have been regarded as the key tool for scientists to know the various physical happenstances that govern the real world today. Therefore, searching for exact travelling wave solutions relative to these nonlinear evolution equations plays a key role in the examination of nonlinear physical phenomena that can be associated with many other fields besides the aforementioned areas like meteorology, electromagnetic theory, nonlinear optics as well as other areas of science.
Meanwhile, in recent times scientists have developed effective techniques to obtain viable analytical solutions to nonlinear evolution equations, such as Cole-Hopf transformation approach [8], generalized unified technique [9], [10], exp(Φ(η)) -expansion technique[11], [12], F-expansion technique [1], Painlevé expansion approach [13], mapping technique and extended mapping technique [14], [15], Adomian decomposition approach[16], homotopy perturbation technique[17], Bäcklund transformation [18], rational expansion technique [19], tan-cot approach [20], extended simplest equation method[21], Hirota technique [22], Lie group analysis [23], [24], [25], [26], [27], bifurcation technique [28], the (G/G) expansion method [29], Darboux transformation approach [30], sine-Gordon equation expansion technique [31], Kudryashov method [32], exponential function technique [33], [34], tanh-function technique[35], ansatz technique[36], [37], tanh-coth approach[38], and on and on, the list continues.
Many years back, Zakharov and Kuznetsov[39] introduced an equation modelling nonlinear ion-acoustic waves embedded in a magnetized plasma which comprises cold as well as hot isothermal electrons. The quantum plasmas alongside their new characteristics have captivated the attention of scientists from both theoretical as well as experimental standpoints. Due to the reason that it plays a key role in carrying charge whenever the de Broglie wavelength surpasses the Debye wavelength and as well as approaches the Fermi wavelength[40]. The character of the waves that are weakly nonlinear ion-acoustic which exists in the presence of a magnetic field that is described to be uniform, is administered by the quantum Zakharov-Kuznetsov (QZK) equation. Various authors have investigated the effect of such a magnetic field in diverse quantum plasma models [41].
The two-dimensional Zakharov-Kuznerov equation (ZK)[42]
ut+u2ux+uxxx+uxxy=0,
has been investigated by researchers. In [42], Krishnan and Biswas employed the mapping technique to achieve the solutions of (1.1). The solutions found include shock waves, cnoidal waves, and periodic singular waves alongside solitary waves. Besides, ZK Eq. (1.1) with power-law nonlinearity given as
qt+aqnqx+b(qxx+qxy)x=0,
was also studied by the authors where the ansatz approach was engaged in securing topological solitons and shock wave solutions for some specific values for the parameters of the power law. In [43], Abdou achieved the solutions of ZK (1.1) by utilizing the simplified form of Hirota's bilinear technique. Wazwaz [44], [45] studied equation (1.1) via the use of extended tanh, sine-cosine, with homotopy analysis techniques to gain solutions to the equation. He found solutions in the structure of soliton solutions. In addition, hyperbolic function solutions such as coth and tanh combined, new travelling wave solutions as well as periodic solutions were also gained. Moreover, the author in [46] obtained analytic solutions of the generalized version of (1.1) with time-dependent coefficients and nonlinear dispersion by employing the solitary wave ansatz technique. In [47], the authors found some closed-form solutions of (1.2) via Lie symmetry analysis. They also secure the conserved quantities of the equation by employing Ibragimov's conservation law theorem.
The new extended (2+1)-dimensional quantum Zakharov-Kuznerov equation (QZK) [48], [49]
ut+auux+b(uxxx+uyyy)+c(uxyy+uxxy)=0,
with a , b and c regarded as real-valued constants whereas u(x,y,t) stands for electrostatic wave potential existing in plasmas which is obviously a function depending on spatial variables x , y as well as the temporal variable t . The first term in QZK (1.3) is regarded as the temporal term of evolution, the coefficient of a taken as the nonlinear term whereas the coefficient of b as well as that of c denotes the spatial dispersions which are multi-dimensional. The QZK equation (1.3) recounts ion-acoustic waves that are discovered to be included in a magnetized plasma consisting of hot isothermal alongside cold ions electrons. In [48], Wang et al. engaged group theorem to secure a plethora of analytic solutions of (1.3). Moreover, the authors in [49] found the closed-form solutions of QZK (1.3) with the use of a simplified structure of the bilinear technique, sech approach as well as tanh method. Multiple solitons solutions of the equation were gained. Lately, Jiang et al. in [50] carried out a Lie group analysis of the equation and achieved the conserved quantities of the equation via the utilization of Ibragimov's theorem for the conserved vector.
Remarkable is the fact that symmetry plays a highly significant role in various fields of nature, most especially in integrable systems for the existence of infinitely many symmetries. This technique was introduced towards the end of the nineteenth century, by the Norwegian mathematician, named Sophus Lie. The notion of the Lie group was to study the solutions to differential equations. Roughly speaking, one can say that a Lie point symmetry of a system refers to a local group of transformations that ensures the mapping of every solution of the system to another solution that belongs to the same system. The rigours of achieving an increasing number of solutions of systems of partial differential equations are associated with the group properties of these differential equations. In order to get the Lie point symmetry of a nonlinear equation, diverse effective methods have been proposed, such as the classical Lie group method[26], [51], nonclassical method [52], [53], symmetry reduction[54], [55], and so on. Lie's method is an efficient and the simplest technique among group theoretic techniques and a vast number of equations [56], [57] have been solved with the aid of this technique.
In the same vein, conservation laws have been viewed to serve a leading role in examining differential equations, since they delineate physical conserved quantities, consisting of energy, mass, momentum as well as charge together with other motion constants[4], [58]. Conservation laws have also been utilized to carry out investigations on the existence, uniqueness, as well as stability of solutions regarding nonlinear partial differential equations (NLNPDE) [59]. Moreover, they have as well been applied to numerical techniques [60]. In consequence, it is an optimum necessity to examine the conservation laws of partial differential equations.

1.1. Governing equation

In our study, we contemplate securing various solitary wave solutions of powerlaw nonlinearity [1], [3], [4] of a more generalized structure of (1.3) and it is called a generalized (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation (3D-gextQZKe) given as
ut+aunux+buxxx+cuyyy+duxyy+euxxy=0,abc,be,de,
with parameters a , b , c , d and e taken as real-valued constants with n>0 .
The paper is then catalogued as follows. The first goal of this paper is to carry out the symmetry analysis of the generalized (2+1)-dimensional extended quantum Zakharov-Kuznetsov Eq. (1.4) and this is comprehensively explicated in Section 2. In addition, an optimal system of one-dimensional subalgebras of the symmetries found is computed. Section 3 furnishes the similarity reductions and various group-invariant solutions of (1.4). In Section 4, abundant cnoidal and snoidal waves of the understudy model are achieved via the extended Jacobi elliptic function expansion technique for some particular cases of n . Soliton wave dynamics of the secured solutions through numerical simulation are presented in Section 5, whereas in Section 6, we outline the application of our results in oceanography and ocean engineering. Lastly, conserved currents of (1.4) is constructed in Section 7 via the application of Noether's theorem after which we have the concluding remarks as well as future scope, presented. We notice that the investigation carried out in this paper as catalogued in Sections earlier given, furnishes a plethora of new copious solitons, cnoidal and snoidal waves of (1.4) whose applications in oceanography and ocean engineering fields have never been presented. Thus, the current study bridges this gap, thereby suggesting various possible applications which can be used by oceanographers and ocean engineers in their analysis.

2. Lie group analysis and optimal system of (1.4)

This section furnishes all the pertinent steps of the Lie group technique (whose potency has earlier been given), to enclose this research work within self-bound. We shall calculate Lie point symmetries as well as the 1-D (one-dimensional) optimal system of Lie subalgebras for the 3D-gextQZKe (1.4). Thereafter, we implore the subalgebras to construct classical solutions to the equation.

2.1. Classical Lie point symmetries of (1.4)

Here in this subsection, Let's first contemplate the one-parameter Lie group of infinitesimal transformations;
x*x+εξ1(x,y,t,u)+O(ϵ2),y*y+εξ2(x,y,t,u)+O(ϵ2),t*t+εξ3(x,y,t,u)+O(ϵ2),u*u+εη(x,y,t,u)+O(ϵ2),
with ε standing for the parameter of the group alongside ξ1 , ξ2 , ξ3 , η serving as the infinitesimals of the transformations depending on x , y , t , and u . Thus utilizing ε (one-parameter) Lie group of infinitesimal transformation in compliance with invariant conditions [25], solution space (x,y,t,u) of 3D-gextQZKe (1.4) stays invariant and can as well transform into another space.
In accordance with the technique for deciding the infinitesimal generators of nonlinear differential equations (NLDE), we shall secure infinitesimal generator of (1.4). Symmetry group of 3D-gextQZKe (1.4) will be calculated by exploiting the vector field structured as Υ=ξ1(x,y,t,u)x+ξ2(x,y,t,u)y+ξ3(x,y,t,u)t+η(x,y,t,u)u, with ξ1,ξ2,ξ3 and η regarded as the coefficient functions of the vector field depending on (x,y,t,u) . Vector Υ is a Lie point symmetry of (1.4) if invariance condition
Pr(3)Υ[ut+aunux+buxxx+cuyyy+duxyy+euxxy]=0,
whenever ut+aunux+buxxx+cuyyy+duxyy+euxxy=0 holds. Here Pr(3)Υ denotes the third prolongation of Υ and is defined as
Pr(3)Υ=Υ+ζtut+ζxux+ζxxxuxxx+ζxxyuxxy+ζxyyuxyy+ζyyyuyyy
with the ζt , ζx , ζxxx , ζxxy , ζxyy and ζyyy expressed as[25]
ζiα=Di(Wα)+ξjuijα,ζi1isα=Di1Dis(Wα)+ξjuji1isα,s>1,
whereas the included Lie characteristic function Wα is presented as
Wα=ηαξiujα,
where we also express total derivative Di in this regard as
Di=t+xi+uiαuα+uijαujα+,
which is the operator of total differentiation. Splitting the expanded form of (2.6) over various derivatives of u , we obtain these overdetermined system of linear PDEs
ξu3=0,ξx3=0,ξy3=0,ξtt3=0,ξt1=0,ξu1=0,ξt2=0,ξu2=0,ξx2=0,ξy1=0,3ξx1ξt3=0,3ξy2ξt3=0,3nη+2uξt3=0,
which can be solved without much stress. Thus, we have ξ1=13c1x+c3,ξ2=13c1y+c4,ξ3=c1t+c2,η=23nc1u, where c1,,c4 are arbitrary constants. Hence, we have the following Lie point symmetries:
Υ1=x,Υ2=y,Υ3=t,Υ4=3ntt+nxx+nyy2uu.
Theorem 2.1. The 3D-gextQZKe (1.4) admits a four-dimensional Lie algebra L4 spanned by vectors Υ1,,Υ4 .
Next, we exploit the group of infinitesimal transformations associated with the obtained generators (2.11). First, we state a theorem to that effect.
Theorem 2.2. Given the infinitesimal transformations (2.5) , the associated one-parameter group (G) is computed from the solution of the Lie equations alongside their respective initial conditions expressed as
dx*dε=ξ1(x*,y*,t*,u*),x*|ε=0=x,dy*dε=ξ2(x*,y*,t*,u*),y*|ε=0=y,dt*dε=ξ3(x*,y*,t*,u*),t*|ε=0=t,du*dε=η(x*,y*,t*,u*),u*|ε=0=u.
We secure the group of transformations corresponding to the infinitesimal generators Υ1,,Υ4 by solving the Lie equations given in theorem (2.2) with their respective initial conditions for each of the generators. For instance, in the case of Υ4=3nt/t+nx/x+ny/y2u/u , we have
dx*dε4=nx*,x*|ε4=x,dy*dε4=ny*,y*|ε4=y,dt*dε4=3nt*,t*|ε4=t,du*dε4=2u*,u*|ε4=u,
whose solution is revealed as
x*=enε4x,y*=enε4y,t*=e3nε4t,u*=e2ε4u.
Therefore, taking the same steps in the case of other symmetries, we have one-parameter groups generated by Υ1,,Υ4 as
Gε1:(x*,y*,t*,u*)(x+ε1,y,t,u),Gε2:(x*,y*,t*,u*)(x,y+ε2,t,u),Gε3:(x*,y*,t*,u*)(x,y,t+ε3,u),Gε4:(x*,y*,t*,u*)(enε4x,enε4y,e3nε4t,e2ε4u).
Theorem 2.3. Suppose u(x,y,t) is a solution of 3D-gextQZKe (1.4) , so are the functions Gε1(ε)·u(x,y,t)=u(xε1,y,t),Gε2(ε)·u(x,y,t)=u(x,yε2,t),Gε3(ε)·u(x,y,t)=u(x,y,tε3),Gε4(ε)·u(x,y,t)=e2ε4u(enε4x,enε4y,e3nε4t).
Remark 2.1. The Lie group Gε1×Gε2×Gε3 is a normal Lie subgroup of Gε1Gε2Gε3 . The Lie algebra generated by Υ1 , Υ2 and Υ3 is an ideal of L4 .

2.2. Construction of 1-D optimal system of subalgebras for (2.11)

It is impractical for the full list of all possible group-invariant solutions to be given. Consequently, there is a need for us to engage an effective and systematic approach to classify these solutions; once this is achieved, then we form an optimal system of group-invariant solutions. The more technical issue came up in a bid to do the classification of the subalgebra of Lie algebra occasioned by the obtained Lie point symmetries. Nevertheless, we overcome the barrier by adopting a standard procedure given in [23], [25] to secure all the one-dimensional subalgebras involved. We first present the commutative products for L4 decided by Υ1,Υ2,Υ3,Υ4 according to Lie algebra relation: [Υμ,Υυ]=ΥμΥυΥυΥμ . This is presented in Table 1. Apparently, {Υ1,Υ2,Υ3,Υ4} is closed under the Lie bracket. Next, construction of the adjoint table is done in Table 2, according to the relation
Ad(exp(εv))w0=m=0εmm!(adv)m(w0)=w0ε[v,w0]+ε22[v,[v,w0]]
Table 1 Commutator table of the Lie algebra of 3D-gextQZKe (1.4).
[Υμ,Υυ ]Υ1 Υ2 Υ3 Υ4 Empty Cell
Υ1 000nΥ1
Υ2 000nΥ2
Υ3 0000
Υ4 nΥ1 nΥ2 00
Table 2 Adjoint representation table of the Lie algebra of 3D-gextQZKe (1.4).
AdΥ1 Υ2 Υ3 Υ4 Empty Cell
Υ1 Υ1 Υ2 Υ3 nεΥ1+Υ4
Υ2 Υ1 Υ2 Υ3 nεΥ1+Υ4
Υ3 Υ1 Υ2 Υ3 Υ4
Υ4 enεΥ1 enεΥ2 Υ3 Υ4
From the linear combination
Υ=a1Υ1+a2Υ2+a3Υ3+a4Υ4
Thus, using the procedure introduced in [23], [25] without presenting the detailed calculations, we have the optimal system for the generators as {Υ4,Υ3+βΥ2,Υ3Υ1+βΥ2,Υ3+Υ1+βΥ2} . We now summarize the optimal system in a tabular structure as

3. Similarity reduction, group invariant solutions and direct integration of 3D-gextQZKe (1.4)

Having decided the optimal system of one-dimensional subalgebras, we contemplate the symmetry reductions of (1.4) via the integration of the Lagrangian system associated with each symmetry.
Case 1. Subalgebra Υ4
The differential invariants occasioning the similarity variables for Υ4=3nt/t+nx/x+ny/y2u/u can be determined by finding solutions to the Lagrangian system associated with Υ4 , that is
dt3nt=dxnx=dyny=du2u.
System (3.13) produces the similarity variables
g=x3t,f=xy,u=y2/nG(x3t,xy).
Inserting the values of g , f and u in (1.4), we secure PDE
an3t3y3GfGn+3an3t2x2y4GgGn4cn2t3y3G8ct3y3G12cnt3y3G+2dn3t3y3Gf+6dn2t3y3Gf+4dnt3y3Gf6cn3t3xy2Gf18cn2t3xy2Gf12cnt3xy2Gf2en3t3y3Gff2en2t3y3Gff+4dn3t3xy2Gff+4dn2t3xy2Gff6cn3t3x2yGff6cn2t3x2yGffcn3t3x3Gfff+bn3t3y3Gfffen3t3xy2Gfff+dn3t3x2yGfffn3tx3y6Gg+6bn3t2y6Gg12en2t2xy5Gg+6dn2t2x2y4Gg+12dnt2x2y4Gg+18bn3t2xy5Ggf12en3t2x2y4Ggf12en2t2x2y4Ggf+6dn3t2x3y3Ggf+12dn2t2x3y3Ggf+9bn3t2x2y4Ggff6en3t2x3y3Ggff+3dn3t2x4y2Ggff+54bn3tx3y6Ggg18en2tx4y5Ggg+27bn3tx4y5Gggf9en3tx5y4Gggf+27bn3x6y6Gggg=0,
which gives rise to the translation generators
Γ1=f,Γ2=g,
Imputing the invariants z=fg and group-invariant u=Q(z) secured from the use of Γ1=/f and Γ2=/g , we reduce 3D-gextQZKe (1.4) to third-order nonlinear ordinary differential equation (NODE) written simply as
AQ(z)+BQ(z)Qn(z)+CQ(z)+DQ(z)+EQ(z)=0,
where
A=4c(2+3n+n2)t3y3,B=an3t2y3(t3x2y),C=nty2(y(ny2(6bnty12etxnx3y)2d(n+2)t(nt+t3x2y))+6c(n2+3n+2)t2x),D=2n2ty(y(y(9bnxy2(t3x2y)+e((n+1)t26(n+1)tx2y+9x4y2))+dtx(3(n+2)x2y2(n+1)t))+3c(n+1)t2x2),E=n3(ct3x3y(t3x2y)(y(3x2yt)(by(3x2yt)+etx)+dt2x2)).
The summary of the reduction outcomes from the remaining members of the optimal system is given in Table 3.
Table 3 Summary of optimal system of (1+2)-D gKPle (1.4).
CasesElement selectionsOptimal SystemEmpty Cell
1.a1=0 , a2=0 , a3=0 , a40 Υ4
2.a1=0 , a20 , a30 , a4=0 Υ3+βΥ2
3.a10 , a20 , a30 , a4=0 Υ3Υ1+βΥ2
4.a10 , a20 , a30 , a4=0 Υ3+Υ1+βΥ2
Table 4 Reduction summary for 3D-gextQZKe (1.4) through the dual and triple Lie vectors.
VectorsVariables transformSymmetriesReduced equation
Υ3+βΥ2 f=x , g=yβt Γ1=f , Γ2=g αGg+cGggg+aGnGf+dGfgg
+eGffg+bGfff=0
Υ3Υ1+βΥ2 f=t+x , g=y+βt Γ1=f , Γ2=g (c+β(d+β(bβ+e)))Gggg+dGfgg
+aβGnGg+aGnGf+2eβGfgg+Gf
+3bβ2Gfgg+eGffg+3bβGffg+bGfff=0
Υ3+Υ1+βΥ2 f=yβt , g=yβx Γ1=f , Γ2=g βGfaβGnGg+(cdβ)Gggg
+eβ2Ggggbβ3Gggg+3cGfgg2dβGfgg
+eβ2Gfgg+3cGfggdβGffg+cGfff=0

3.1. Direct integration of the reduced equations

We secure the exact solutions of 3D-gextQZKe (1.4) by direct integration of the NODE earlier obtained via symmetry reductions.
Case a.
Subalgebra Υ3+βΥ2 is reduced to nonlinear ordinary differential equation (NODE)
aQ(z)Q(z)n+bQ(z)cν3Q(z)+ν2dQ(z)νeQ(z)+βνQ(z)=0.
Integration of ODE (3.18) once with regards to z gives
(bcν3+ν2dνe)Q(z)+an+1Q(z)n+1+βνQ(z)+K0=0,
with n1 and K0 regarded as integration constant. Repeating the process of integration one more time after multiplying (3.19) by first derivative of Q , one gets the first-order nonlinear ordinary differential equation (FNODE)
12(bcν3+ν2dνe)Q2(z)+a(n+1)(n+2)Q(z)n+2+12βνQ(z)2+K0Q(z)+K1=0,
where n2 and K1 stands for integration constant. Taking the constants of integration to be zero, integrating (3.20) and reverting to the original variables gives a non-topological soliton solution
u(x,y,t)=(βν(n+1)(n+2)2a)1/nsech2/n[12nβν(n+1)(n+2)(zQ2C*)],
where Q2=(n+1)(n+2)(bν(cν2νd+e)),andz=yβtνx, with C* regarded as an integration constant. In a bid to view the dynamics of solution (3.21), we present the graphic display of the solution with dissimilar values of the involved parameters for n=1 and n=2 in Figs. 1 and 2 accordingly.
Next we get analytic solutions of 3D-gextQZKe (1.4) via direct integration by considering the members of the optimal systems with triple vectors.
Case b.
Subalgebra Υ3+βΥ2+Υ1 transforms (1.4) to third-order NODE
βQ(z)α3bβ3Q(z)+c(α3Q(z)α2Q(z)+2(αQ(z)))+c(α2Q(z)+αQ(z)Q(z))βd(α3Q(z)2α2Q(z)+αQ(z))+β2e(α3Q(z)α2Q(z))aαβQ(z)Q(z)n=0.
Integration of Eq. (3.22) just as we earlier demonstrated, we have a NODE
2aαβ(n+1)(n+2)Q(z)n+2+((α1)3c+αβ(αβ(αbβαe+e)(α1)2d))Q(z)2βQ(z)2=0,
with n1,2 . We contemplate equation (3.23) and set Q=V1/n , therefore transforming to
1n2((α1)3c+αβ(αβ(αbβαe+e)(α1)2d))V2βV2+2aαβ(n+1)(n+2)V3=0.
Obviously, Eq. (3.24) can be presented simply as
V2=AV3+BV2,
with
A=2aαβn2(n+1)(n+2)((α1)3c+αβ(αβ(αbβαe+e)(α1)2d)),B=βn2((α1)3c+αβ(αβ(αbβαe+e)(α1)2d)).
We note here that an ODE of the type
V2=AV3+BV2K,
where K is a constant has solutions
V(ζ)=BAsech2(B2ζ),andV(ζ)=BAcsch2(B2ζ),
with B>0 , when K=0 , see[61]. Therefore, by comparing the form taken by Eqs. (3.25) and (3.27), we see that they are symmetric at K=0 . Hence 3D-gextQZKe (1.4) possesses bright and singular soliton solutons presented respectively as
u=[(n+1)(n+2)2aαsech2(βn24((α1)3c+αβ(αβ(αbβαe+e)Θ))z)]1n,
u=[(n+1)(n+2)2aαcsch2(βn24((α1)3c+αβ(αβ(αbβαe+e)Θ))z)]1n,
with Θ=(α1)2d) and the constraint that aα>0 , β((α1)3c+αβ(αβ(αbβαe+e)Θ)>0 in solution (3.29). Moreover, aα<0 , β((α1)3c+αβ(αβ(αbβαe+e)Θ)>0 in solution (3.30) and z=βtαβx+(α1)y . The streaming pattern of the solutions for n=1 and n=2 are given in Figs. 3, 4, 5 and 6.
Case c.
Subalgebra Υ3+βΥ2Υ1 reduces 3D-gextQZKe(1.4) to NODE
aβσQ(z)Q(z)n+aQ(z)Q(z)nσ3Q(z)(β(β(bβ+e)+d)+c)+3bβ2σ2Q(z)3bβσQ(z)+bQ(z)+dσ2Q(z)+2βeσ2Q(z)eσQ(z)+Q(z)=0,
which on integration a couple of times and allowing the constant of integration to be zero yields
2a(βσ1)(n+1)(n+2)Q(z)n+2+(b(βσ1)3+σ(σ(cσ+βdσd)+e(βσ1)2))Q(z)2Q(z)2=0.
We add that n1,2 in Eq. (3.32). Adopting the technique applied in securing solutions of (1.4) in Case b, we have soliton solutions
u=[(n+1)(n+2)2a(βσ1)sech2(n24(b(βσ1)3+σ(σ(cσ+βdσd)+Θ1)z)]1n,
u=[(n+1)(n+2)2a(βσ1)csch2(n24(b(βσ1)3+σ(σ(cσ+βdσd)+Θ1)z)]1n,
with Θ1=e(βσ1)2 and the constraint that a(βσ1)>0 , b(βσ1)3+σ(σ(cσ+βdσd)+Θ1>0 in solution (3.33). Moreover, a(βσ1)<0 , b(βσ1)3+σ(σ(cσ+βdσd)+Θ1>0 in solution (3.34) and z=t(1+σβ)xσy . We present the pictorial representations of solutions (3.33) and (3.34) in Figs. 7, 8, 9 and 10.

4. Exact solutions of 3D-gextQZKe (1.4)

This subsection presents more general closed-form solutions of (1.4) via some standard techniques. Therefore we aim to secure solutions of the underlying equation by contemplating some special cases of n .

4.1. Solution of (1.4) via extended Jacobi elliptic function expansion technique

This subsection constructs the closed-form solutions of (1.4) with the aid of extended Jacobi elliptic function expansion technique[4]. We assume a formal solution of the third-order NODE (3.22) as
Q(z)=i=mmAiH(z)i,
where we aim to achieve the value of positive integer m by adopting balancing procedure, see[62]. We declare here that function H(z) satisfies the first-order NODE
H(z)=(1H2(z))(1ω+ωH2(z))
or
H(z)=(1H2(z))(1ωH2(z)).
We remind ourselves of the fact that
H(z)=cn(z|ω),
the Jacobi cosine-amplitude function is the solution of (4.36) whereas the solution to NODE (4.37) is the Jacobi sine-amplitude function
H(z)=sn(z|ω),
with parameter ω existing in the interval 0ω1 , see [63], [64].
We quickly recall here that when ω1 , then we have cn(z|ω)sech(z) and when ω0 , then sn(z|ω)tanh(z) . In the same vein, when ω0 , then cn(z|ω)cos(z) and sn(z|ω)sin(z) .

4.1.1. Cnoidal wave solutions of NODE (3.18)

Case. 1. When n=1,
In considering the solution of NODE (3.18), firstly, we investigate the case of n=1. It is observed that balancing procedure produces the value of m as m=2 , therefore (4.35) assumes the form
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
Inserting the value of Q from (4.40) into (3.18) and utilizing (4.36), we get the system
24bν3ω2A224eν2ω2A22aνωA22+24dνω2A224cω2A2=0,6bν3ω2A16eν2ω2A13aνωA1A2+6dνω2A16cω2A1=0,24bν3ω2A2+48bν3ωA2+24eν2ω2A2+2aνωA2224bν3A224dνω2A248eν2ωA22aνA22+24cω2A2+48dνωA2+24eν2A248cωA224dνA2+24cA2=0,64bν3ω2A2+32bν3ωA2+64eν2ω2A22aνωA0A2+4aνωA22aνωA1264dνω2A232eν2ωA22aνA22+64cω2A2+32νdωA22βωA232cωA2=0,6bν3ω2A1+12bν3ωA1+6eν2ω2A1+3aνωA2A16bν3A16dνω2A112eν2ωA13aνA2A1+6cω2A1+12dνωA1+6eν2A112cωA16dνA1+6cA1=0,14bν3ω2A1+7bν3ωA1+14eν2ω2A1aνωA1A2aνωA0A1+6aνωA1A214dνω2A17eων2A13aνA1A2+14cω2A1+7νdωA1βωA17cωA1=0,56bν3ω2A2+56bν3ωA2+56eν2ω2A2+2aνωA224aνωA2A02aνωA128bν3A256dνω2A256eν2ωA2+2νaA2A0+νaA12+56cω2A2+56dνωA2+8eν2A24βωA256cωA28dνA2+2βA2+8cA2=0,56bν3ω2A256bν3ωA256eν2ω2A2+4aνωA0A2+2aνωA122aνωA22+8bν3A2+56dνω2A2+56eν2ωA22νaA0A2νaA12+2aνA2256cω2A256νdωA28eν2A2+4βωA2+56cωA2+8νdA22A2β8cA2=0,64bν3ω2A296bν3ωA264eν2ω2A24aνωA22+2aνωA2A0+aνωA12+32bν3A2+64dνω2A2+96eν2ωA2+2aνA222νaA2A0νaA1264cω2A296dνωA232eν2A2+2βωA2+96cωA2+32dνA22A2β32cA2=0,14bν3ω2A121bν3ωA114eν2ω2A16aνωA2A1+aνωA2A1+aνωA1A0+7bν3A1+14dνω2A1+21eν2ωA1+3aνA2A1aνA2A1aνA1A014cω2A121dνωA17eν2A1+βωA1+21cωA1+7dνA1βA17cA1=0,16bν3ω2A216bν3ω2A28bν3ωA2+24bν3ωA216eν2ω2A2+16eν2ω2A2+2aνωA2A0+aνωA122aνωA0A2aνωA128bν3A2+16dνω2A216dνω2A2+8eν2ωA224eν2ωA2+2νaA0A2+νaA1216cω2A2+16cω2A28dνωA2+24νdωA2+8eν2A2+2βωA22βωA2+8cωA224cωA28νdA2+2βA2+8cA2=0,2bν3ω2A1+10bν3ω2A1bν3ωA110bν3ωA12eν2ω2A110eν2ω2A1+aνωA2A1+aνωA1A0+2aνωA1A2+2aνωA0A13aνωA1A2+bν3A1+2dνω2A1+10dνω2A1+eν2ωA1+10eων2A1aνA1A2aνA0A1+3aνA1A22cω2A110cω2A1dνωA110νdωA1eν2A1+βωA1+2βωA1+cωA1+10cωA1+dνA1βA1cA1=0,10bν3ω2A12bν3ω2A1+10bν3ωA1+3bν3ωA1+10eν2ω2A1+2eν2ω2A1+3aνωA2A12aνωA2A12aνωA1A0aνωA1A2aνωA0A1bν3A1bν3A110dνω2A12dνω2A110eν2ωA13eων2A1+aνA2A1+aνA1A0+aνA1A2+aνA0A1+10cω2A1+2cω2A1+10dνωA1+3νdωA1+eν2A1+eν2A12βωA1βωA110cωA13cωA1dνA1dνA1+βA1+βA1+cA1+cA1=0,
whose solution using Maple software produces the values of Ai,i=2,,2 as
A2=0,A1=0,A1=0,A2=12ω(bν3eν2+dνc)aνA0=(8bω+4b)ν3+(8eω4e)ν2+(8dω+4d)ν+8cω4cβaν,
A2=12(ω1)β0aν,A1=0,A1=0,A2=12ωβ0aνA0=(8bω+4b)ν3+(8eω4e)ν2+(8dω+4d)ν+8cω4cβaν
where β0=bν3eν2+dνc . General solutions associated with (4.41) and (4.42) are respectively given as
u(x,y,t)=A0+A2cn2(z|ω),
u(x,y,t)=A2nc2(z|ω)+A0+A2cn2(z|ω),
with nc=1/cn and z=yβtνx . The behaviour of solutions (4.43) and (4.44) are displayed graphically in Figs. 11 and 12.
Case. 2 When n=2
In this case the balancing term has the value m=1 , so (4.35) becomes
Q(z)=A1H1+A0+A1H.
Substituting the value of Q from (4.45) into (3.18) in association with (4.36), we obtain an algebraic equation which when split gives the system
2aνωA0A12=0,4aνωA12A0+2aνA12A0=0,2aνωA12A02aνA12A0=0,4aνωA12A02aνA12A0=0,2aνωA12A02aνωA0A12+2aνA0A12=0,6bν3ω2A1aνωA136eν2ω2A1+6dνω2A16cω2A1=0,6bν3ω2A1+aνωA13+12bν3ωA1+6eν2ω2A1aνA136bν3A16dνω2A112eν2ωA1+6cω2A1+12dνωA1+6eν2A112cωA16dνA1+6cA1=0,14bν3ω2A1aνωA1A12aνωA02A1+2aνωA13+7bν3ωA1+14eν2ω2A1aνA1314dνω2A17eν2ωA1+14cω2A1+7dνωA1βωA17cωA1=0,14bν3ω2A12aνωA13+aνωA12A1+aνωA1A0221bν3ωA114eν2ω2A1+aνA13aνA12A1aνA1A02+7bν3A1+14dνω2A1+21eν2ωA114cω2A121dνωA17eν2A1+βωA1+21cωA1+7dνA1βA17cA1=0,2bν3ω2A1+10bν3ω2A1+aνωA12A1+aνωA1A02+2aνωA1A12+2aνωA02A1aνωA13bν3ωA110bν3ωA12eν2ω2A110eν2ω2A1aνA1A12aνA02A1+aνA13+bν3A1+2dνω2A1+10dνω2A1+eν2ωA1+10eν2ωA12cω2A110cω2A1dνωA110dνωA1eν2A1+βωA1+2βωA1+cωA1+10cωA1+dνA1βA1cA1=0,10bν3ω2A12bν3ω2A1+aνωA132aνωA12A12aνωA1A02aνωA1A12aνωA02A1+10bν3ωA1+3bν3ωA1+10eν2ω2A1+2eν2ω2A1+aνA12A1+aνA1A02+aνA1A12+aνA02A1bν3A1bν3A110dνω2A12dνω2A110eν2ωA13eν2ωA1+10cω2A1+2cω2A1+10dνωA1+3dνωA1+eν2A1+eν2A12βωA1βωA110cωA13cωA1dνA1dνA1+βA1+βA1+cA1+cA1=0.
The system yields the solution
A1=0,A0=0,A1=6bων36eων2+6dνω6cωaνβ=bν32bων3+2eων22dνωeν2+2cω+dνc,
A1=(3β+β1)4±6aν(8bν38eν2+8dν+3β1+β8c),A0=0A1=±6aν(8bν38eν2+8dν+3β1+β8c)4aνω=(8bν38eν2+8dν+3β1+β8c)28(4bν34eν2+4dν+β1β4c)(2bν32eν2+2dν+β1+β2c)
where β1=8b2ν616beν5+16bdν4+8e2ν416bcν3+β2 , β2=16ceν2+8d2ν216cdν+β2+8c216deν3 . Therefore the related general solutions to (4.46) alongside (4.47) are respectively
u(x,y,t)=A1cn(z|ω),
u(x,y,t)=A1nc(z|ω)+A0+A1cn(z|ω),
with nc=1/cn as well as z=yβtνx . The pictorial representations of the solutions are given accordingly in Figs. 13 and 14.

4.1.2. Snoidal wave solutions of (3.18)

Case. A. When n=1,
We consider the NODE (3.18), for case of n=1 and as earlier shown the balancing term m=2 and then the assumed solution (4.35) is
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
Replacing the value of Q from (4.50) in (3.18) in conjunction with (4.37), and following the procedure earlier demonstrated, we achieve thirteen system of equations whose solution produces the results of A1,A0 ,and A1 as
A2=0,A1=0,A0=(ω+1)(4bν34eν2+4dν)4cω4cβaνA1=0,A2=12ω(bν3eν2+dνc)aν,
A2=12(bν3eν2+dνc)aν,A1=0,A2=12ω(bν3eν2+dνc)aνA1=0,A0=(ω+1)(4bν34eν2+4dν)4cω4cβaν.
Consequently, we have general solutions associated with (4.51) and (4.52) are given respectively as
u(x,y,t)=A0+A2sn2(z|ω),
u(x,y,t)=A2ns2(z|ω)+A0+A2sn2(z|ω),
where ns=1/sn and z=yβtνx . The behavioural pattern of solutions (4.53) and (4.54) with unalike values of the involved parameters are respectively given in Figs. 15 and 16.
Case B. When n=2
In this case as previously shown m=1 and as such (4.35) becomes
Q(z)=A1H1+A0+A1H.
Introducing the value of Q given in (4.55) into (3.18) in consonance with (4.37), we gain an algebraic equation which when split yields a system which gives the solution expressed as
A1=0,A0=0,A1=±6eων26bων36dνω+6cωaν,β=bων3+bν3eων2+dνωeν2cω+dνc,
A1=±6eν26bν36dν+6caν,A0=0,A1=36bν3+6eν26dν+6caν+6aν(8bν38eν2+8dν+β8c)aν,ω=32a(bν3eν2+dνc)ν(6bν3+6eν26dν+6caνaν6β33)2,
where β3=aν(8bν38eν2+8dν+β8c) . Thus, we have the general solution corresponding to (4.56) and (4.57) as
u(x,y,t)=A1sn(z|ω),
u(x,y,t)=A1ns(z|ω)+A1sn(z|ω),
where ns=1/sn and z=yβtνx . We exhibit the dynamics of solutions (4.58) and (4.59) with dissimilar values of the involved parameters in Figs. 17 and 18.

4.2. Solutions of (3.22) using extended Jacobi function expansion technique

Here, we gain cnoidal and snoidal waves solutions of (3.22) using extended Jacobi function expansion techniques for n=1 and n=2 .

4.2.1. Cnoidal wave solutions of NODE (3.22)

Case. 1. When n=1,
Here, contemplating the NODE (3.22), we first consider a case of n=1. Thus, balancing procedure produces m=2 and then (4.35) assumes the structure
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
We follow the procedure highlighted earlier and achieve three possible solutions of the seven system of equations obtained using Mathematica as
A2=A1=0,A0=β(4α3bβ2(12ω)4dα(α1)2(2ω1)+Θ0)+Θ2aαβA1=0,A2=12ω(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ,
A2=12(ω1)(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ,A1=0,A0=β(8α3bβ2ω+4α3bβ24(α1)2αd(2ω1)+Θ0)+Θ2aαβ,A1=0,A2=12ω(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ
A2=A1=0,A0=β(4α3bβ24(α1)2αd+Θ3+1)+4(α1)3caαβ,A1=0,A2=12(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ,ω=1.
with Θ0=4(α1)α2βe(2ω1)+1 , Θ2=4(α1)3c(2ω1) , Θ3=4(α1)α2βe . Therefore, we secure general solution of (1.4) with regards to the solutions itemized in (4.61)-(4.63) respectively as
u(x,y,t)=A0+A2cn2(z|ω),
u(x,y,t)=A2nc2(z|ω)+A0+A2cn2(z|ω),
u(x,y,t)=A0+A2sech2(z|ω),
with nc=1/cn and z=βtαβx+(α1)y . The streaming character of solutions (4.64), (4.65) and (4.66) are accordingly shown in Figs. 19, 20 and 21 with the use of unalike values of the involved parameters.
Case. 2 When n=2
In this case the value of balancing term m=1 and so (4.35) becomes
Q(z)=A1H1+A0+A1H.
Substituting the value of Q from (4.67) into (3.22) in conjunction with (4.36), we obtain an algebraic equation which when split gives six system of equation which when solved, we secure solutions presented as
A1=A0=0,A1=3(β(α3bβ2+(α1)2αdΘ4+1)(α1)3c)aαβb=(α1)3c(2ω1)+β(α(α1)2(d)(2ω1)+Θ4(2ω1)+1)α3β3(2ω1),
A1=A0=0,A1=±6aα,b=(α1)3c+β(α(α1)2(d)+Θ4+1)α3β3,ω=1.
where Θ4=(α1)α2βe . Therefore the general solution of (1.4) with regards to the constant values in (4.68) and (4.69) are respectively expressed as
u(x,y,t)=A0+A1cn(z|ω),
u(x,y,t)=A0+A1sech(z),
where z=βtαβx+(α1)y . Figs. 22 and 23 respectively reveals the streaming pattern of solutions (4.70) and (4.71) using diverse values of the involved parameters.

4.2.2. Snoidal wave solutions

Case. A. When n=1,
We consider the NODE (3.22), for case of n=1 and as earlier shown the balancing term m=2 and then the assumed solution (4.35) is
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
Replacing the value of Q from (4.72) into (3.22) and utilizing (4.37), we get eight system of equations which give the solutions that are
A2=A1=0,A0=β(4α3bβ2ω+4α3bβ2+4(α1)2αd(ω+1)Θ5+1)Θ6aαβ,A1=0,A2=12ω(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ.
A2=12bβ2aA0=4bβ2ω+4bβ2+1a,A1=A1=0,A2=12bωβ2a,α=1.
A2=12(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ,A1=A1=0,A0=(4(α1)3c(ω+1)β(4α3bβ2ω+4α3bβ2+Θ7))aαβ,A2=12ω(αβ(αβ(αbβαe+e)+(α1)2d)(α1)3c)aαβ,
with Θ5=4(α1)α2βe(ω+1) , Θ6=4(α1)3c(ω+1) , Θ7=4(α1)2αd(ω+1)4(α1)α2βe(ω+1)+1 . Thus, we express general solutions of (1.4) in reference to (4.73)-(4.75) as
u(x,y,t)=A0+A2sn2(z|ω),
u(x,y,t)=A2ns2(z|ω)+A0+A2sn2(z|ω),
u(x,y,t)=A2ns2(z|ω)+A0+A2sn2(z|ω),
where ns=1/sn and z=βtαβx+(α1)y . The three solutions (4.76), (4.77) and (4.78) are represented graphically with the use of the dissimilar values of the parameters utilized in Figs. 24, 25 and 26 accordingly.
Case B. When n=2
In this case as previously revealed m=1 and so (4.35) becomes
Q(z)=A1H1+A0+A1H.
Invoking the value of Q stated in (4.79) into (3.22) in consonance with (4.37), we achieve an algebraic equation which when split yields the system whose solution can be presented as
A1=A0=0,A1=6(β(α3bβ2+(α1)2αd(α1)α2βe+1)(α1)3c)aαβb=(α1)3c(ω+1)+β(α(α1)2(d)(ω+1)+α2(α1)βe(ω+1)1)α3β3(ω+1),
A1=35ω12a,A0=0,A1=16(ω+1)35ω12a,b=ω3572β2,α=1,ω=117+122,
A1=±35ω12aα,A0=0,A1=±16(ω+1)35ω12aα,ω=117+122b=72(α1)3c+β(72α(α1)2d+72α2(α1)βe+ω35)72α3β3.
Consequently, we have the corresponding general solution with regards to (4.80)-(4.82) respectively as
u(x,y,t)=A1ns(z|ω),
u(x,y,t)=A1ns(z|ω),+A1sn(z|ω),
u(x,y,t)=A1ns(z|ω),+A1sn(z|ω),
where ns=1/sn and z=βtαβx+(α1)y . We showcase the dynamics of solutions (4.83), (4.84) and (4.85) respectively in Figs. 27, 28 and 29 respectively.

4.3. Solutions of (3.31) using extended Jacobi function expansion technique

Next, we generate closed-form solutions of (3.31) in the form of cnoidal and snoidal waves with the aid of extended Jacobi function expansion approach for n=1 and n=2 .

4.3.1. Cnoidal wave solutions of NODE (3.31)

Considering NODE (3.31), we first contemplate a case of n=1. Thus, m=2 as earlier revealed and then (4.35) assumes the structure
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
We now replace the value of Q from (4.86) into (3.31) and employ (4.36) to secure thirteen system of equations. Engaging Maple software package in gaining the solution of the thirteen long system of equations, we have the outcome as
A2=A1,A0=(8cω4c)σ3+(4d8dω)σ2+(8eω4e)σ+Θ8a,A2=12ω(cσ3+dσ2eσ+b)a,
A2=0,A1=0,A0=8bω4b+1a,A1=0,A2=12bωa,
A2=12b(ω1)a,A1=0,A0=8bω4b+1a,A1=0,A2=12bωa,
A1=0,A0=(8cω4c)σ3+(4d8dω)σ2+(8eω4e)σ+Θ8a,A2=12(ω1)(cσ3+dσ2eσ+b)a,A2=12ω(cσ3+dσ2eσ+b)a.
with Θ8=8bω+4b1 . Hence we have the general solutions related to (4.87)-(4.90) respectively as
u(x,y,t)=A0+A2cn2(z|ω),
u(x,y,t)=A0+A2cn2(z|ω),
u(x,y,t)=A2nc2(z|ω)+A0+A2cn2(z|ω),
u(x,y,t)=A2nc2(z|ω)+A0+A2cn2(z|ω),
with nc=1/cn and z=t+(1σβ)xσy . Engaging the unalike values of the parameters included in the solutions, we display the pictorial representations of solutions (4.91), (4.92), (4.93), and (4.94), in Figs. 30, 31, 32 and 33 accordingly.
Case B. When n=2
In this case as previously revealed m=1 and so (4.35) becomes
Q(z)=A1H1+A0+A1H.
Invoking the value of Q stated in (4.95) into (3.22) in consonance with (4.37), we achieve an algebraic equation which when split yields a system of algebraic equations whose solutions are expressed in terms of constant parameters A1 , A0 , A1 , ω and c as
A1=A0,A1=±6ωa(12ω),c=(2dωd)σ2+(2eω+e)σ+2bωb+1σ3(2ω1),
A1=0,A1=0,A1=3(b1)a,ω=b12b,
A1=(9b38b2+13b3)98b2+1+24b+38a,A0=0,A1=98b2+1+24b+38a,ω=8b+138b2+116b.
The general solutions of 3D-gextQZKe (1.4) related to (4.96)-(4.98) are
u(x,y,t)=A1cn(z|ω),
u(x,y,t)=A1cn(z|ω),
u(x,y,t)=A1nc(z|ω)+A0+A1cn(z|ω),
where nc=1/cn and z=t+(1σβ)xσy . We exhibit the graphs of solutions (4.99), (4.100) and (4.101) respectively in Figs. 34, 35 and 36.

4.3.2. Snoidal wave solutions

Case. A. When n=1,
Now, we contemplate the NODE (3.31) and secure its snoidal solution for the usual two cases of n . Case n=1 as earlier shown possesses the balancing number m=2 and then an assumed solution (4.35) becomes
Q(z)=A2H2+A1H1+A0+A1H+A2H2.
Inserting the value of Q from (4.102) into (3.31) and utilizing (4.37), we get thirteen system of equations which when solved furnish the solutions
A2=0A1=0A0=0A2=12ω(cσ3+dσ2eσ+b)a,A0=4c(ω+1)σ3+4d(ω+1)σ24e(ω+1)σ+4bω+4b1a,
A2=0A1=0A0=4bω+4b1aA1=0,A2=12bωa,
A2=12baA1=0A0=4bω+4b1aA1=0,A2=12bωa,
A2=12cσ312dσ2+12eσ12ba,A2=ω(12cσ312dσ2+12eσ12b)a,A1=A1=0,A0=4c(ω+1)σ3+4d(ω+1)σ24e(ω+1)σ+4bω+4b1a.
Thus, corresponding general solutions to (4.103)-(4.106) are presented respectively as
u(x,y,t)=A0+A2sn2(z|ω),
u(x,y,t)=A0+A2sn2(z|ω),
u(x,y,t)=A2ns2(z|ω)+A0+A2sn2(z|ω),
u(x,y,t)=A2ns2(z|ω)+A0+A2sn2(z|ω)
with ns=1/sn alongside z=t+(1σβ)xσy . We give the graphic display of solutions (4.107), (4.108), (4.109) and (4.110) in Figs. 37, 38, 39 and 40 accordingly using the unalike values of the included parameters.
Case B. When n=2
In this case as previously demonstrated m=1 and so (4.35) becomes
Q(z)=A1H1+A0+A1H.
Invoking the value of Q given in (4.111) into (3.31) in conjunction with (4.37), we achieve an algebraic equation which when split yields eleven system of equation. In consequence, solving the system furnishes the values of the involved parameters as
A1=0,A0=0,A1=±6ωa(ω+1)c=d(ω+1)σ2e(ω+1)σ+bω+b1σ3(ω+1),
A1=0,A0=0,A1=±6(b1)a,σ=0,ω=b1b,
A1=(108b+368b2+b)636a(68b2+b17b1),A0=0,A1=6a(68b2+b17b1)a,σ=0,ω=(68b2+b17b1)2(3b+8b2+b)2.
The associated general solution to (4.112)-(4.114) are respectively
u(x,y,t)=A1ns(z|ω),
u(x,y,t)=A1sn(z|ω),
u(x,y,t)=A1ns(z|ω),+A1sn(z|ω),
where ns=1/sn as well as z=t+(1σβ)xσy . The streaming patterns of solutions (4.115), (4.116), and (4.117) are revealed in Figs. 41, 42, and 43 respectively.

5. Soliton wave dynamics and analysis of solutions

Physical interpretations of various solitary wave solutions secured in this study are presented in this section in order to reveal their physical meaning. Thus, we depict these solutions via 3-D plot, 2-D plot as well as density plots. In order to exhibit the most advantageous representation of the graph, we make appropriate arbitrary choice of constants. The dynamical character of soliton solution (3.21) is portrayed in Fig. 1, using the suitable constant parameters ν=0.5 , β=10 , a=1 , b=1.000001 , c=2 , d=7 , e=1.00000001 , C1=0 when n=1 at t=0 with 7x,y7 . In addition, Fig. 2 is plotted with the aid of assigned values ν=0.1 , β=10 , a=1 , b=1.000001 , c=2 , d=7 , e=1.0000001 , C1=0 when n=2 with the same value of t and (x,y) intervals. Soliton (3.29) is represented in Fig. 3 by invoking the parametric values ν=0.5 , β=2 , α=2 , a=10 , b=1 , c=3 , d=5 , e=1 with n=1 , variables t=1 and 10x,y10 . Fig. 4 further reveals the streaming behaviour of (3.29) through dissimilar constant values ν=0.5 , β=2 , α=3 , a=3 , b=1 , c=4 , d=5 , e=1.00000001 , when n=2 , t=1 and 7x,y7 . Next, multi-soliton structure in Fig. 5 portrays the soliton solution (3.30) via unalike parameters ν=0.1 , β=1 , α=2 , a=0.4 , b=1 , c=3 , d=5 , e=5.0000000001 , where n=1 , t=2 with (x,y) in the interval [7,7] . Moreover, we plot Fig. 6 to exhibit the solution by assigning suitable constant values ν=0.5 , β=2 , α=3 , a=3 , b=1.0000000001 , c=4 , d=5 , e=1 , when n=2 , t=2 and 7x,y7 . The Figure reveals another multi-soliton wave of the solution. The compacton-type soliton wave structure in Fig. 7 is plotted through the numerical simulation of bright soliton (3.33) with parameters σ=0.9 , β=2 , α=2 , a=10 , b=2 , c=2.000000001 , d=5 , e=1 , when n=1 , t=0.3 as well as (x,y)[10,10] . In the same vein, bell-shaped Fig. 8 further depicts the solution with constant values σ=0.7 , β=1.5 , α=2 , a=10 , b=2.00000000001 , c=3 , d=2 , e=1 , when n=2 , t=0.3 and 10x,y10 . The combo-type solitons which is an interesting wave structure in physical sciences exhibited in Fig. 9 portrays the singular soliton solution (3.34). This is achieved via the dissimilar constant values σ=0.1 , β=3 , α=2 , a=0.4 , b=1 , c=3 , d=5 , e=5.000000001 , when n=1 with t=0.8 and variables (x,y) existing in the interval [7,7] . Besides, the motion of the solution is further simulated numerically yielding Fig. 10 with parameters σ=2 , β=3 , α=3 , a=3 , b=1 , c=4 , d=5 , e=1.000000000001 , where n=2 with variable t assuming the same value and (x,y) , the same range of interval. Next, we examine the dynamics of soliton waves of periodic soliton solution (4.43) in Fig. 11 with values ν=0.4 , β=0.2 , ω=0.2 , a=1 , b=1.0000001 , c=1.000000001 , d=2 , e=0 , where t=1 and 10x,y10 . Solitary wave solution (4.44) in Fig. 12 with values ν=0.4 , β=0.2 , ω=1.01 , a=1.03 , b=1 , c=1.000000000001 , d=2 , e=0 , where t=3 and variables (x,y)[10,10] . In the same vein numerical simulation of soliton (4.48) in Fig. 13 with values ν=0.4 , ω=0.4 , a=10 , b=1 , c=0 , d=2 , e=0 , where we have t=2 and variables (x,y)[13,13] . Moreover, Fig. 14 depict the motion of soliton (4.49) with values ν=0.6 , β=0 , a=1 , b=1 , c=0 , d=2 , e=0.000000000000001 , where variables t=0 and (x,y)[10,10] . The snoidal wave solution (4.53) is represented with Fig. 15 using unalike parameters ν=0.4 , β=0.4 , ω=0.4 , a=5 , b=1 , c=0.1 , d=2 , e=1.000000000001 , where t=5 and (x,y)[12,12] . Numerical simulation of solitary wave solution (4.54) is depicted with Fig. 16 by invoking parameters ν=0.4 , β=0.9 , ω=0.2 , a=5 , b=1 , c=0.1 , d=2 , e=1.000000000001 , where t=5 and (x,y)[12,12] . Snoidal wave solution (4.58) is portrayed via Fig. 17 using parametric values ν=0.4 , ω=0.4 , a=5 , b=1 , c=0.1 , d=2 , e=1 , where t=4 and (x,y)[12,12] . Further to that, Fig. 18 represents the dynamics of solution (4.59) through dissimilar parameters ν=0.4 , ω=0.4 , β=0.5 , a=5 , b=1 , c=0.1 , d=2 , e=1 , where t=2.3 and (x,y)[15,15] . Fig. 19 portrays the wave motion of cnoidal wave solution (4.64) where the graphs in the Figure are plotted using the constant parameters β=0.2 , ω=0.4 , a=1 , b=1.0000000000001 , c=1.00000000001 , d=2 , e=0 , with t=4.2 as well as 10x,y10 . In the same format, we have solution (4.65) depicted in Fig. 20
Fig.1 Bright soliton wave profile of (3.21), 10x,y10 at t=0 for n=1 .
Fig.2 Bright soliton wave profile of (3.21), 10x,y10 at t=0 for n=2 .
Fig.3 Dark soliton wave profile of (3.29), 10x,y10 at t=1 for n=1 .
Fig.4 Bright soliton wave profile of (3.29), 7x,y7 at t=1 for n=2 .
Fig.5 Multi-soliton wave structure of (3.30), 7x,y7 at t=2 for n=1 .
Fig.6 Multi-soliton wave structure of (3.30), 7x,y7 at t=2 for n=2 .
Fig.7 Bright soliton wave profile of (3.33), 10x,y10 at t=0.3 for n=1 .
Fig.8 Bright soliton wave profile of (3.33), 10x,y10 at t=0.3 for n=2 .
Fig.9 Multi-soliton wave profile of (3.34), 7x,y7 at t=0.8 for n=1 .
Fig.10 Multi-soliton wave profile of (3.34), 7x,y7 at t=0.8 for n=2 .
Fig.11 Smooth periodic wave profile of (4.43) with 10x,y10 at t=1 .
Fig.12 Smooth periodic wave profile of (4.44) with 10x,y10 at t=3 .
Fig.13 Smooth periodic wave profile of solution (4.48), 13x,y13 at t=2 .
Fig.14 Smooth periodic wave profile of solution (4.49), 10x,y10 at t=0 .
Fig.15 Smooth periodic wave profile of (4.53), 12x,y12 at t=5 .
Fig.16 Singular periodic wave profile of (4.54), 15x,y15 at t=10 .
Fig.17 Smooth periodic wave profile of (4.58), 12x,y12 at t=4 .
Fig.18 Singular periodic wave profile of (4.59), 15x,y15 at t=2.3 .
Fig.19 Smooth periodic wave profile of (4.64) with 10x,y10 at t=4.2 .
Fig.20 Smooth periodic wave profile of (4.65) with 13x,y13 at t=3.7 .
via the selection of parametric values β=0.2 , ω=1.01 , a=1.03 , b=1 , c=1.00000000001 , d=2 , e=0 , with t=3.7 as well as 13x,y13 . The numerical simulation of periodic soliton (4.66) occasions the diagrammatic representations in Fig. 21 with unalike constant values β=0.01 , α=0.2 , a=1.03 , b=1 , c=1.00000000001 , d=2 , e=0 , with t=6.2
Fig.21 Bell shape wave profile of (4.66) with 10x,y10 at t=6.2 .
whereas on the (x,y) -plane, (x,y)[10,10] . Moreover, the wave portrayal of cnoidal wave solution (4.70) is achieved in Fig. 22 through the use of variant parametric values designated as β=10 , α=13 , ω=10 , a=10 , c=10.00000000001 , d=2 , e=1 , where variables t=20 and 10x,y10 . Fig. 23 reveals the wave motion of bright soliton solution (4.71) by invoking constant values β=0.1 , α=0.3 , a=1 , c=10 , d=2 , e=1.00000000001 , with variables t=20
Fig.22 Periodic wave profile of (4.70) with 10x,y10 at t=20 .
Fig.23 Bell shape wave profile of (4.71) with 10x,y10 at t=20 .
whereas on the (x,y) -plane, (x,y)[10,10] . Next, we examine the wave dynamics of snoidal wave solutions. Thus, solution (4.76) is represented via Fig. 24 with parametric values α=0.4 , ω=0.4 , a=5 , b=1 , c=0.1 , d=2 , e=1.00000000001 , with t=8.3 as well as 12x,y12 . Besides, we plot solution (4.77) in Fig. 25 by assigning α=1 , ω=0.6 , a=5 , b=1.00000000001 , c=0.1 , d=2 , e=1 , with t=0.6 alongside (x,y)[15,15] . Solution (4.78) is depicted in Fig. 26 through the dissimilar constant values α=0.3 , β=0.9 ω=0.3 , a=5 , b=1 , c=0.1 , d=2 , e=1.00000000001 , whereas t assumes the same value and (x,y) , the same range. Periodic soliton (4.83) is plotted in Fig. 27 with α=0.4 , ω=0.4 , a=5 , c=0.1 , d=2 , e=1 , with t=0.2
Fig.24 Smooth periodic wave profile of (4.76) with 12x,y12 at t=8.3 .
Fig.25 Singular periodic wave profile of (4.77) with 15x,y15 at t=0.6 .
Fig.26 Singular periodic wave profile of (4.78) with 15x,y15 at t=0.6 .
Fig.27 Smooth periodic wave profile of (4.83) with 12x,y12 at t=0.2 .
where (x,y)[12,12]
where solution (4.84) is portrayed in Fig. 28 using β=0.08 , a=5 , c=0.1 , d=2 , e=1 , variable t=1.8 and (x,y)[15,15] . We reveal the dynamics of snoidal wave solution (4.85) in Fig. 29 through α=0.7 , β=0.001 , a=0.1 , c=0 , d=0.02 , e=0.00000000001 , whereas t=0.5 together with (x,y)[10,10] . Furthermore, cnoidal wave solution (4.91) is depicted in Fig. 30 via constants σ=0.4 , β=0.2 , ω=0.7 , a=1 , b=1.00000000001 , c=1.000000001 , d=2 , e=0 , with t=10 alongside 10x,y10 . In the same vein, periodic solution (4.91) is plotted in Fig. 31 through variant parametric values σ=0.5 , β=0.3 , ω=1.01 , a=1.03 , b=1.00000000001 , c=1 , d=2 , e=0 , where we have t=9 as well as 13x,y13 . In Fig. 32, we represent cnoidal wave (4.91) via constant values σ=0.6 , β=0.2 , ω=1.01 , a=0.1 , b=10 , c=1 , d=2 , e=0 , where t=6 and 10x,y10 . Besides, solitary wave solution (4.92) is furnished in Fig. 33 using σ=1 , β=0.6 , ω=1.05 , a=0.1 , b=1 , c=0 , d=2 , e=0.00000000001 , with t=0.7 alongside 10x,y10 . Solitary wave solution (4.99) is represented in Fig. 34 by utilizing parameters σ=0.3 , β=0.2 , a=1 , d=2 , e=1 , whereas t=1 together with (x,y)[10,10] . In addition to that cnoidal wave (4.100) is exhibited in Fig. 35 by invoking parametric values σ=0.3 , β=0.1 , a=1 , b=10 , d=2 , e=1.00000000001 , where t=2 alongside (x,y)[8,8] . We depict periodic soliton (4.101) in Fig. 36 via the use of σ=1 , β=0.1 , a=1 , b=100 , d=2 , e=1.00000000001 , whereas t=12 and (x,y)[10,10] . Furthermore, solitary wave solution (4.107) is purveyed in Fig. 37 with dissimilar constants σ=0.4 , β=0.3 , ω=0.6 , a=5 , b=1 , c=0.1 , d=2 , e=1.00000000001 , where we have t=0.3 along with 10x,y10 . Periodic solution (4.108) is exhibited in Fig. 38 with unalike constants σ=0.41 , β=0.3 , ω=0.8 , a=5 , b=1.00000000001 , c=0.1 , d=2 , e=1 , with variables t=0.22 together with 10x,y10 on the (x,y) -plane. The snoidal wave (4.109) plotted in Fig. 39 is achieved via choice of constants σ=0.41 , β=0.5 , ω=0.8 , a=5 , b=1 , c=0.1 , d=2 , e=1.00000000001 , whereas variable t=5.6 alongside 15x,y15 .
Fig.28 Singular periodic wave profile of (4.84) with 15x,y15 at t=1.8 .
Fig.29 Singular periodic wave profile of (4.85) with 10x,y10 at t=0.5 .
Fig.30 Periodic wave profile of (4.91) with 10x,y10 at t=10 .
Fig.31 Periodic wave profile of (4.92) with 13x,y13 at t=9 .
Fig.32 Periodic wave profile of (4.93) with 10x,y10 at t=6 .
Fig.33 Periodic wave profile of (4.94) with 10x,y10 at t=0.7 .
Fig.34 Periodic wave profile of (4.99) with 10x,y10 at t=1 .
Fig.35 Periodic wave profile of (4.100) with 8x,y8 at t=2 .
Fig.36 Singular periodic wave profile of (4.101) with 10x,y10 at t=12 .
Fig.37 Smooth periodic wave profile of (4.107) with 10x,y10 at t=0.3 .
Fig.38 Smooth periodic wave profile of (4.108) with 10x,y10 at t=0.22 .
Fig.39 Singular periodic wave profile of (4.109) with 15x,y15 at t=5.6 .
Moreover, dynamics of snoidal wave (4.110) is numerically simulated thereby purveying Fig. 40 by engaging most beneficial parametric constant values σ=0.41 , β=0.5 , ω=0.1 , a=5 , b=1.00000000001 , c=0.1 , d=2 , e=1 , whereas we have variable t=2.4 and also (x,y)[15,15] . Fig. 41 depicts smooth periodic solution (4.115) using σ=0.4 , β=0.2 , ω=0.2 , a=2 , c=0.1 , d=2 , e=1 , with variable t=20 and (x,y)[12,12] on the (x,y) -plane. In the same vein, smooth snoidal wave (4.116) is portrayed in Fig. 42 with dissimilar constant values β=0.08 , a=5 , b=3.3 , d=2 , e=1 , where t=30 and 15x,y15 . Finally, we simulate periodic soliton (4.117) numerically via β=1 , a=1 , b=1 , c=0.2 , d=0.02 , e=0 , where we assign variable t=0 and (x,y)[1,1] on the (x,y) -plane.
Fig.40 Singular periodic wave profile of (4.110) with 15x,y15 at t=2.4 .
Fig.41 Smooth periodic wave profile of (4.115) with 12x,y12 at t=20 .
Fig.42 Smooth periodic wave profile of (4.116) with 15x,y15 at t=30 .
Fig.43 Singular periodic wave profile of (4.117) with 1x,y1 at t=0 .

6. Applications of cnoidal and snoidal waves in ocean engineering and oceanography

In this study, we are aware that various solutions attained comprise copious cnoidal and snoidal waves which are solitary waves or periodic solitons. Cnoidal waves can be presented as an infinite sum of periodically repeated solitary waves. These waves have been comprehensively demonstrated through numerical simulations of the solutions using computer software by making adequate choices of parameters.
A solitary wave [7], [65], [66], [67] is the kind of wave propagating in the absence of any temporal evolution with regards to shape or size when it is viewed concerning the reference frame travelling with the group velocity existing in the wave. Furthermore, the envelope of the wave is possessive of one global peak whose decays occur far away from the peak. It is observed that solitary waves surface in various contexts that are inclusive of the elevation of the surface of the water as well as the intensity of light which is resident in optical fibres. In addition, a soliton is a nonlinear type of solitary wave which is possessive of an additional characteristic that the wave, after interacting with another soliton still conserves its permanent structure. For instance, in two solitons generating in the directions opposite to each other, successfully move through each other as it were without any breakage [68]. (See Figs. 44-45).
Fig.44 Ocean's surface solitary wave triggered by atmosphere lower boundary's decreasing pressures [69].
Fig.45 Schematic diagram of the life cycle of nonlinear internal waves in the region southeast of Hainan Island [70].
Solitons constitute a remarkable group of solutions related to some model equations. These equations comprise the Korteweg de-Vries as well as the Nonlinear Schrödinger equations. These equations are approximations, that are true under a confining set of criteria. The soliton results achieved from the model equation give us some understanding of the dynamical behaviour of solitary waves. Nevertheless, they are restrained by the criteria under which the model equations operate. A surrogate approach, relating directly to the precise equations where the model equations are constructed, gives an awareness of a larger class of solitary waves than those which are achieved via the model equations. Information with regards to the possible existence of a certain class of solitary waves can be achieved via a phase-plane formalism which is a common technique engaged in dynamical systems. Thus, a solitary wave, in this framework, corresponds to a homoclinic orbit that is existing in a spatial dynamical system. In addition, the occurrence of solitary waves is discovered in many diverse cases, which are determined, by their level of decay concerning the distance from their global peak. In consequence, a keen investigation of the tail regions, relative to their distance from the global peak, reveals that four possible cases of solitary waves can be achieved. Three of these cases include; the one secured via the steady-state solution of the Korteweg de-Vries equation. Secondly, a generalized solitary wave decays to non-zero oscillations with uniform amplitude as well as wave number. Thirdly, an envelope solitary wave, which fulfils the Nonlinear Schrödinger equation[68].
Moreover, solitary waves can be contemplated as a class of nonlinear, nonsinusoidal, more-or-less isolated waves which have complex shapes frequently occurring in nature. Internal waves are waves existing and also travelling within the interior of a given fluid. Such waves are very common as oscillations and they are visible in a two-layer fluid contained in a clear plastic box. The coherence of these waves is sustained. In the same vein, the visibility of the waves is conserved via nonlinear hydrodynamics that surface in imagery, as long quasilinear stripes. Besides, the signatures of internal waves are made conspicuous through current or wave interactions. In such a case, the related near-surface current to the internal wave modulates locally with regard to the height of the surface wave spectrum[71]. (See Fig. 45).
Ocean engineering is comprehensively defined as an area of technological investigation relating to the design alongside operations of man-made systems that is existing in the ocean as well as other marine bodies. In fact, it makes provision for essential interconnections that is existing among other disciplines of oceanography which consist of chemical together with physical oceanography, marine biology as well as marine geology and geophysics. The innovations in the instrumentation along with equipment design effectuated by ocean engineers have restructured the field of oceanography. In the same vein, the oceanographers' central point has been to drive the demand for design skills in conjunction with the technical expertise of ocean engineers. Moreover, unswervingly, ocean engineering incorporates diverse other areas of engineering which comprise a merge of mechanical, electrical, acoustical, chemical alongside civil engineering skills as well as techniques in consonance with a basic understanding of the way oceans operate (see the schematic diagram in Fig. 47). In addition to designing the building instruments that must outlive the wear and tear of persistent use, ocean engineers must also design instruments that can come through the harsh realities surrounding the ocean environment.
Hazardous events, which include rock falls or avalanches, rock slides, and landslides, frequently generate great, impulsive waves when they enter or are submerged in water bodies [73]. Therefore, these waves are then approximated through the use of solitary waves whose overwhelming potential damage with regard to the built environment has been investigated by researchers. In order to further deepen the understanding of solitary waves which run up a constant beach slope and also propagate over a subsequent horizontal plane can assist in minimizing as well as attenuating damage and the number of casualties that can be caused as a result of the damning effect of such a hazardous event [74]. The interaction exiting between structures alongside waves is a typical phenomenon in naval architecture as well as ocean engineering. Consequently, we present some relevance of solitary waves in oceanography and ocean engineering .
Oceanographers and acousticians have realized that shallow water internal solitary waves are a major topic of interest . The investigation into internal solitary waves found in ocean acoustic was motivated by the discovery of an anomalous frequency response of sound propagation measurements which was noticed in shallow water (see Fig. 46). The spatial, as well as temporal scales of internal bores and solitary waves that are induced by tides, are the type that can significantly affect the acoustic field via the structure of sound speed. The interaction between the acoustical field and the solitary wave train at certain frequencies can be quite significant. Various interactions that existed between the acoustical field, as well as the solitary wave train, can also emerge between the receiver and source. In [75], the scientist conducted theoretical research into the coupling of acoustical normal modes where the presence of solitary-wave-type disturbances was observed. They discovered that strong lateral gradients of sound speed in the waves occasioned the transfer of energy between various acoustical modes. The phase mode within the solitary wave packet differs with time scales of minutes. This eventually causes coupling and signal fluctuations at comparable time scales. Furthermore, right from the studies conducted by Zhou et al. [76], oceanographic alongside littoral acoustical experiments have been conducted on the synoptic as well as local scales, where the experiment fronts reveal that there was a prevalence of eddies, tidal effects, internal waves, mesoscale variability together with solitary waves. Surveys have also been conducted on the New England and New Jersey continental shelves[77].
Fig.46 Russell's solitary-wave solitons occur in shallow water channels, (a) as shown in a laboratory reconstruction of Russell's original observation. The inset is a typical localized solution of the Korteweg-de Vries equation for extremely shallow water. (b) In deep water, waves propagate (from left to right), whereas water particles orbit in circles around their average position. The diameter of the circular orbits shrinks gradually with increasing distance from the surface down to half the wavelength L . In shallow waters with depth h<L/2 , the orbits become elliptical and essentially flat at the bottom of the tank or seabed [72].
Fig.47 A simplified concept of the main wave transformation and attenuation processes which must be considered by coastal engineers in designing coastal defence schemes [78].

7. Conservation laws of 3D-gextQZKe (1.4)

In this section we construct the conservation laws for the 3D-gextQZKe (1.4). We achieve that by using Noether's theorem.

7.1. Construction of conservation laws using the Noether theorem

This subsection begins with the employment of the well-celebrated Noether theorem [79] to derive the conservation laws of the 3D-gextQZKe (1.4). Here we discovered that (1.4) does not admit any Lagrangian in its current state and as such possesses no variational principle. Notwithstanding, to circumvent that we engage the transformation u=Ux and interestingly, we secure a fourth-order form of Eq. (1.4) structured as
E*=Utx+aUxnUxx+bUxxxx+cUxyyy+dUxxyy+eUxxxy=0,
which readily has a Lagrangian. Hence, we present a Lemma to that effect.
Lemma 7.1. The 3D-gextQZKe (1.4) manifests the Euler-Lagrange equation having the functional
J(U)=000L(x,y,t,Ux,Ut,Uxx,Uxy,Uyy)dxdydt,
where the conforming function of Lagrange L is expressed as
L=12UtUxa(n+1)(n+2)Uxn+2+b2Uxx2+c2UxyUyy+d2UxxUyy+e2UxxUxy.
One can readily proof that the second-order Lagrangian for the fourth-order PDE (7.118) which is given in (7.120) satisfies the Euler-Lagrangian equation δL/δU=0=E* as envisaged, by utilizing the standard Euler operator δ/δU expressed as
δδU=UDtUtDxUx+Dt2Utt+Dx2Uxx+DtDxUtx,
where total derivatives Dt , Dx and Dy can be calculated from (2.10). In a bid to secure the variational symmetries Σ=ξ1/x+ξ2/y+ξ3/t+η/U which is correspondent to the stated Lagrangian (7.120), we shall utilize the invariance condition
Pr(2)ΣL+L[Dx(ξ1)+Dy(ξ2)+Dt(ξ3)]=Dx(Bx)+Dy(By)+Dt(Bt),
where second prolongation Pr(2)Σ is defined as
Pr(2)Σ=ζtUt+ζxUx+ζyUy+ζxxUxx+ζxyUxy+ζyyUyy,
We notice that coefficient functions ξ1 , ξ2 , ξ3 , η as well as gauge functions Bx , By and Bt are depending on (x,y,t,U) . To determine the Noether symmetry generators Σ , we expand invariance condition (7.121) and separate the monomials and that furnishes the sixty-four system of linear PDEs, which are
ξU1=0,ξx3=0,ξUU3=0,ηx=0,ξUx3=0,ξt3=0,ξU3=0,ξUU1=0,ξU2=0,ξy1=0,ξUU2=0,ξUx3=0,ξUx1=0,ξUy1=0,ξx1=0,ξt2=0,ξU3=0,BUy=04bξUx1+eξUy1=0,ξU1+BUt=0,eηxx+cηyy=0,ξU3+BUt=0,cηxy+dηxx=0,ηUU2ξUy3=0,cξUy1+2dξUx1=0,cξxy1+dξxx1=0,cξU2+dξU3=0,cξU2+3dξU3=0,cξUU2+dξUU3=0,2dξx1+cξy1=0,ηUU2ξUx2=0,eξx1+2dξy1=0,eξy1+4bξx1=0,eξUx1+2dξUy1=0,eξxx1+cξyy1=0,eξU2+bξU3=0,eξUU2+2bξUU3=0,eξU3+dξU2=0,eξU3+3dξU2=0,eξx3+cξy2=0,eξUU3+dξUU2=0,eξUx3+cξUy2=0,ηt+2BUx=0,Byy+Bxx=0,dηUUcξUy22dξUx2=0,cηUxcξxy3dξxx3=0,2eηUxeξxx2cξyy2=0,2bηxx+eηxy+dηyy=0,eξxy1+dξyy1+2bξxx1=0,2bηUUeξUy24bξUx2=0,dηUUeξUx32dξUy3=0,2cηUyeξxx3cξyy3=0,(n+1)ξU1+BUt=0,Bttξy32ηU=0,cηUUcξUx2cξUy32dξUx3=0,cηUycξxy2+2dηUxdξxx2=0,cξt1cBtt2dξx3+2cηU2cξy3=0,eηUUeξUx2eξUy32dξUy24bξUx3=0,eηUxeξxy3+2dηUydξyy32bξxx3=0,eηUyeξxy2dξyy2+4bηUx2bξxx2=0,bξt1+bξy3bBtteξy2+2bηU3bξx2=0,eξt1eBtt2dξy2+2eηU2eξx24bξx3=0,(n+2)(ξx2ηU)ξt1ξx2ξy3+Btt=0,dξt1dBtteξx3+2dηUdξx2dξy3cξy2=0.
From the system of PDEs, we secure the values of the coefficient functions as
ξ1=13C1x+C2,ξ2=13C1y+C3,ξ3=C1t+C4,η=n23nC1U+G(t)y+E(t),Bx=12G(t)yU12E(t)U+F1(y,t)Fy2dx,By=F3(x,y,t),Bt=(3n43n)C1t+F4(x,y).
Functions F1,F2 and F3 are set to zero due to the reason that they contribute to the trivial part of the conserved vectors. Thus, the coefficient functions yields the six Noether symmetries alongside their respective gauge functions, viz.,
Σ1=x,Bt=0,Bx=0,By=0,Σ2=y,Bt=0,Bx=0,By=0,Σ3=t,Bt=0,Bx=0,By=0,Σ4=tt+13xx+13yy+(n23n)UU,Bt=(3n43n)t,BΣ=0,By=0.ΣE=E(t)U,Bt=0,Bx=12E(t)U,By=0,ΣG=G(t)yU,Bt=0,Bx=12G(t)yU,By=0.
The respective conserved vectors [26], [80] for the Noether symmetry operators
Σ1,,ΣG associated with Lagrangian (7.120) can be secured from [80]
Ci=Ni(L)Bi,i=1,,q,
in such a way that condition DtCt+DxCx+DyCy=0 , holds. The Noether operator Ni given in (7.123) is expressed as
Ni=ξi+WαδδUiα+s1Di1Di1(Wα)δδUi1i2isα,α=1,,m,
with q and m representing the total number of independent and dependent variables respectively. The Euler-Lagrange operators with respect to derivatives of Uα is achieved from
δδUα=Uα+s1(1)sDi1DisδδUi1i2isα,i=1,,q,α=1,,m,
by substituting Uα with the corresponding derivatives. For instance,
δδUiα=Uiα+s1(1)sDj1DjsδδUij1j2jsα,j=1,,q,α=1,,m.
Utilizing (7.120), (7.122) and (7.123), we generate local and nonlocal conserved vectors for respective generators Σ1 , Σ2 , Σ3 , Σ4 , ΣE and ΣG as
C1t=12u2,C1x=an+2un+2+12(bux2+duuyy+e(uuxyuxuy))+buuxx,C1y=cuuyy+12(euuxxcuy2+d(uuxyuxuy));C2t=12uuydxC2x=(12utdx+an+1un+1+buxx+12euxy+12duyy)uydx12(euy2+[eux+duy+cuyydx]uyydx)buxuy,C2y=a(n+1)(n+2)un+2+12(uutdx+bux2+euxuy)+(cuyy+12euxx+12duxy)uydx;C3t=a(n+1)(n+2)un+2+12(bux2+euxuy+cuyuyydx),+12duxuyydx,C3x=(12utdx+an+1un+1+buxx+12euxy+12duyy)utdx12[euyut+(eux+cuyydx)uytdx]butux12dutuyydx,C3y=(12euxx+cuyy+12duxy)utdx12(cuy+dux)uytdx;C4t=12((ctuy+dtux)uyydx+btux2+etuxuyn23nudx)a(n+1)(n+2)un+2t+13(12yuuydx+12xu23n4nt),C4x=13(an+2xun+212bxux2+bxuuxx+12dxuuyy+12exuuxy)+12(n23nudx+tutdx+13yuydx+an+1un+1t)utdx13(a(n2)n(n+1)un+1+(n2)nbuxx+(n2)2nduyy+(n2)2neuxy)udx+13(an+1yun+1+byuxx+12dyuyy+12eyuxy+cnuyy)uydx+12t(2buxx+duyy+euxy)utdx23nbuuxbtutux13byuxuy12(23ndu+dtut+13dyuy+ctuytdx+13cyuyydx)uyydx13(enuuy+32etutuy+12eyuy2+enuxuydx+32etuxuytdx)16e(xuxuy+yuxuyydx),C4y=13(3ctuyy12yu+32etuxx+32dtuxy)utdx+16(byux2+eyuxuy)13(a(n+1)(n+2)yun+2+n2ncuyyudxcxuuyy12exuuxx)+13(cyuyy+12eyuxx+12dyuxycnuydnux)uydx+16dxuuxy(n2)6n(euxx+duxy)udx12t(cuy+dux)uytdx16cxuy216dxuxuy;CEt=12E(t)u,CEx=(12utdx+an+1un+1+buxx+12(duyy+euxy))E(t)+12E(t)udx,CEy=12(euxx+2cuyy+duxy)E(t);CGt=12G(t)yu,CGx=(12utdx+an+1un+1+buxx+12(duyy+euxy))G(t)y+12(eG(t)ux+cG(t)uyydx+G(t)yudx),CGy=12(euxx+2cuyy+duxy)G(t)y+12(cuy+dux)G(t).

8. Conclusions

In this paper, we carried out a comprehensive investigation on a generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity (1.4) in oceanography and ocean engineering. The concept of the Lie group theory was engaged in achieving the task. In consequence, four-dimensional Lie algebra associated with (1.4) was gained. Besides, we constructed a 1-D optimal system of Lie subalgebras corresponding to the secured symmetries which were invoked to achieve various classical solutions for Eq. (1.4) through symmetry reductions as well as direct integration approach. Besides, the Jacobi functions expansion approach was engaged in securing general solutions to the underlying equation for some particular cases of n . The solutions contain bright, compact-type, dark, singular, non-topological and periodic solitons which represent both the bounded and unbounded solution-type related to the underlying equation. Moreover, to complete the solutions, we depicted the dynamics of the results with suitable graphical representations through numerical simulation using a computer software package. Sequel to that, applications of cnoidal and snoidal waves which were abundantly obtained in the course of the study were outlined in oceanography and ocean engineering fields. Furthermore, we constructed the conservation laws of (1.4) by imploring Noether's theorem of conserved quantities. These conservation laws are expressed in terms of both local and nonlocal conserved vectors. The nonlocal conserved vectors are the first integrals associated with the variational principle secured. In addition, conservations of energy and momentum which are highly applicable in theoretical physics are obtained in the study. In consequence of the aforementioned, the various results gained in this research work could be of interest to scientists in oceanography and theoretical physics as well as ocean and coastal engineers.

8.1. Future scope

Some of the achieved soliton solutions and various other results which include the conserved currents have diverse applications in physical sciences and some other engineering fields including plasma physics, mathematical physics, coastal science and engineering, solitons dynamics, and a wide range of other nonlinear sciences as well as engineering physics. In consequence, in the future, the dynamics of various soliton solutions could contribute to a broad-gauge theory to delineate the experience of the intricated diversity in diverse nonlinear physical systems. In the same vein, due to the fact that the evolutionary character of soliton solutions has invigorated outstanding research alongside development in a vast range of fields, it will be of great interest to examine the potential advantages of interactions that could exist between cnoidal-snoidal waves and that of the dynamical behaviour of mixed solitons (singular and non-singular).

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.
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
No financial interests/personal relationships which may be considered as potential competing interests.

Acknowledgements

The author would like to thank the editor of the journal as well as the anonymous reviewers for their generous time in providing detailed comments along with suggestions that assisted in improving the paper.
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Outlines

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