1. Introduction
Nowadays, the recognition of nonlinear evolution equations (NLEEs) that delineate physical significance of nonlinear phenomena of science alongside engineering can not be overemphasized [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Besides, mathematicians, physicists and engineers have also found out that generalized higher-dimensional NLEEs are highly important and key in modelling some of the daunting and nagging phenomena that are encountered on a day to day basis. Thus, NLEEs are the most fundamental models crucial to inevitably study nonlinear phenomena of life. In general the study of nonlinear partial differential equations (NPDEs) remains an active area of research with regards to applied mathematics, signal processing, hydrodynamics alongside thermodynamics, solid-state physics, theoretical physics, plasma physics, nonlinear control, optical fibres, fluid mechanics, chemical physics, chemical kinematics, elastic media, atmospheric, biology, image processing, as well as oceanic phenomena to mention but few. Consequently, there has been largely considerable interest vested on securing exact travelling wave solutions of the NLEEs that are discovered to recount some important procedures and processes that are dynamical and as well as physical. Moreover, the study of travelling wave solutions relating to some of these observed NLEEs from the fields earlier highlighted played numerous significant roles in carrying out analysis on these phenomena.
One of such NLEEs is the generalized Zakharov-Kuznetsov equation (gZKe) in three dimensions given as Gu and Qi [9]
${{u}_{t}}+{{r}_{1}}{{u}^{2}}{{u}_{x}}+{{r}_{2}}{{u}_{xxx}}+{{r}_{3}}{{u}_{xyy}}+{{r}_{4}}{{u}_{xzz}}+{{r}_{5}}u{{u}_{x}}+{{r}_{6}}{{u}_{txx}}=0$
with ${{r}_{i}},i,\ldots,6$ as real constants. Zakharov–Kuznetsov (ZK) Eq. (1) is of a generalized setting. Kuznetsov together with Zakharov [10] presented the equation
$u_{t}+u u_{x}+\nabla^{2} u_{x}=0$,
which recounts the progression of the infirmly nonlinear ion with acoustic waves within a plasma comprising hot plutonic electrons alongside cold ions in the attendance of an unvarying magnetic field pushed in the direction of x-axis. Eq. (2) also emerged in sundry other fields in science counting, optical fibre, geochemistry, as well as physics in solid states [11], [12], [13], [14]. This type of equation is found out to govern diverse as well as varieties of physical phenomena, in the purely dispersive limit, thus including the Rossby waves in a rotating atmosphere [15], long waves subsisting on a thin liquid film [16], as well as the isolated vortex of drift waves in a plasma that is three-dimensional [17]. Shivamoggi in Shivamoggi [18] dispensed a detailed discourse which brought to play the analytical features of (2). Besides, Nawaz et al. [19] instituted pertinent solutions to the ZK equations which possess full nonlinear dispersion property via the homotopy analysis approach. The author in Gu and Qi [9] performed the symmetry reduction of (1) and also secured exact solutions to the equation by utilizing exp(−ϕ(z))-expansion as well as complex techniques. We reveal here that considering some special cases of gZKe (1), some other popular and well-recognized NLEEs are secured. For example, suppose we take ${{r}_{2}}={{r}_{4}}={{r}_{5}}=0$, then gZK (1) reverts back to the (2+1)-D ZK-MEW equation [20] and if we let ${{r}_{1}}={{r}_{3}}={{r}_{4}}={{r}_{6}}=0$, then (1) modifies to the famous Korteweg-de Vries equation [21]. Moreover, if ${{r}_{4}}={{r}_{5}}={{r}_{6}}=0$, we see that gZK (1) alters to the modified Zakharov–Kuznetsov equation [22] and if we let ${{r}_{3}}={{r}_{4}}={{r}_{6}}=0$, then (1) converts to the Gardner equation [23].
Seeking exact solutions to NPDEs, has been the major focal point of scientists over the years. Thus, some of the developed productive and functional techniques in gaining viable analytical as well as numerical solutions to NLPEs are outlined in this literature as extended homoclinic test approach [24], tanh-coth approach [25], generalized unified technique [26], bifurcation technique [27], Adomian decomposition approach [28], homotopy perturbation technique [29], Painlévé expansion [30], Cole-Hopf transformation approach [31], Bäcklund transformation [32], rational expansion method [33], F-expansion technique [34], extended simplest equation method [35], Hirota technique [36], Lie symmetry analysis [37], [38], the (G′/G)−expansion method [39], Darboux transformation [40] and sine-Gordon equation expansion technique [41].
Our research work investigates the (3 + 1)-dimensional generalized Zakharov-Kuznetsov equation with dual power-law nonlinearity ((3 + 1)D-gZKe) given as
${{u}_{t}}+\alpha {{u}^{n}}{{u}_{x}}+\beta {{u}^{2n}}{{u}_{x}}+\kappa {{u}_{xxx}}+\sigma {{u}_{txx}}+\epsilon {{u}_{xyy}}+\gamma {{u}_{xzz}}=0,$
where parameters $\alpha,\beta,\kappa,\sigma,\epsilon,\gamma $ and n are nonzero constants. We note that gZK (1) is a particular case of (3 + 1)D-gZKe (3) and so (1) is contained in (3). This study carries out explicit investigations on the solutions of the (3 + 1)D-gZKe (3) using Lie group approach. Thus, the paper is cataloged as follows. Section 2 presents the systematic way of securing Lie symmetries of the underlying equation while Section 3 reveals the exact solutions of (3 + 1)D-gZKe (3) via direct integration as well as extended Jacobi elliptic function expansion approaches. Furthermore, we present the graphical representations of the solutions and discuss their applications in various fields of science and engineering with particular interest in ocean engineering in Section 4. Conclusively, in Section 5, we construct the conservation laws via Noether’s theorem as well as the standard multiplier technique and discuss the results. Concluding remarks follow.
2. Symmetry analysis
This Section exhibits the systematic structure of Lie algorithm in computing Lie point symmetries of (3 + 1)D-gZKe (3) before it is engaged in the construction of closed-form solutions.
2.1. Lie point symmetries of (3 + 1)D-gZKe (3)
The symmetry group of the (3 + 1)-dimensional generalized Zakharov-Kuznetsov equation with dual power-law nonlinearity (3) is calculated by the vector field which is of a formal structure
$\begin{matrix} \text{ }\!\!\Sigma\!\!\text{ } & = & {{\xi }^{1}}\left( t,x,y,z,u \right)\frac{\partial }{\partial t}+{{\xi }^{2}}\left( t,x,y,z,u \right)\frac{\partial }{\partial x}+{{\xi }^{3}}\left( t,x,y,z,u \right)\frac{\partial }{\partial y} \\ \ & {} & +{{\xi }^{4}}\left( t,x,y,z,u \right)\frac{\partial }{\partial z}+\eta \left( t,x,y,z,u \right)\frac{\partial }{\partial u}, \\ \end{matrix}$
with ${{\xi }^{i}},i=1,\ldots,4$ alongside η are infinitesimals of Σ depending on t, x, y, z and u. The vector (Σ) is regarded as a Lie point symmetry of (3 + 1)D-gZKe (3) if
$\text{P}{{\text{r}}^{\left( 3 \right)}}\text{ }\!\!\Sigma\!\!\text{ }\left( {{u}_{t}}+\alpha {{u}^{n}}{{u}_{x}}+\beta {{u}^{2n}}{{u}_{x}}+\kappa {{u}_{xxx}}+\sigma {{u}_{txx}}+\epsilon {{u}_{xyy}}+\gamma {{u}_{xzz}} \right)=0$
Whenever ${{u}_{t}}+\alpha {{u}^{n}}{{u}_{x}}+\beta {{u}^{2n}}{{u}_{x}}+\kappa {{u}_{xxx}}+\sigma {{u}_{txx}}+\epsilon {{u}_{xyy}}+\gamma {{u}_{xzz}}=0$. Here Pr$^{\left( 3 \right)}\text{ }\!\!\Sigma\!\!\text{ }$ denotes he third prolongation of Σ and is defined as
$\begin{matrix} \text{P}{{\text{r}}^{\left( 3 \right)}}\text{ }\!\!\Sigma\!\!\text{ } & = & \text{ }\!\!\Sigma\!\!\text{ }+{{\zeta }^{t}}{{\partial }_{{{u}_{t}}}}+{{\zeta }^{x}}{{\partial }_{{{u}_{x}}}}+{{\zeta }^{y}}{{\partial }_{{{u}_{y}}}}+{{\zeta }^{z}}{{\partial }_{{{u}_{z}}}}+{{\zeta }^{xxx}}{{\partial }_{{{u}_{xxx}}}}+{{\zeta }^{txx}}{{\partial }_{{{u}_{txx}}}} \\ {} & {} & +{{\zeta }^{xyy}}{{\partial }_{{{u}_{xyy}}}}+{{\zeta }^{xzz}}{{\partial }_{{{u}_{xzz}}}} \\ \end{matrix}$
with the coefficient functions of $\text{P}{{\text{r}}^{\left( 3 \right)}}\text{ }\!\!\Sigma\!\!\text{ }$ expressed as
$\begin{array}{l} \zeta^{t}=D_{t}(\eta)-u_{t} D_{t}\left(\xi^{1}\right)-u_{x} D_{t}\left(\xi^{2}\right)-u_{y} D_{t}\left(\xi^{3}\right)-u_{z} D_{t}\left(\xi^{4}\right) \\ \zeta^{x}=D_{x}(\eta)-u_{t} D_{x}\left(\xi^{1}\right)-u_{x} D_{x}\left(\xi^{2}\right)-u_{y} D_{x}\left(\xi^{3}\right)-u_{z} D_{x}\left(\xi^{4}\right) \\ \zeta^{y}=D_{y}(\eta)-u_{t} D_{y}\left(\xi^{1}\right)-u_{x} D_{y}\left(\xi^{2}\right)-u_{y} D_{y}\left(\xi^{3}\right)-u_{z} D_{y}\left(\xi^{4}\right) \\ \zeta^{z}=D_{z}(\eta)-u_{t} D_{z}\left(\xi^{1}\right)-u_{x} D_{z}\left(\xi^{2}\right)-u_{y} D_{z}\left(\xi^{3}\right)-u_{z} D_{z}\left(\xi^{4}\right) \\ \zeta^{t x x}=D_{x}\left(\zeta^{t x}\right)-u_{t t x} D_{x}\left(\xi^{1}\right)-u_{x x t} D_{x}\left(\xi^{2}\right)-u_{x y t} D_{x}\left(\xi^{3}\right)-u_{x z t} D_{x}\left(\xi^{4}\right) \\ \zeta^{x x x}=D_{x}\left(\zeta^{x x}\right)-u_{x x t} D_{x}\left(\xi^{1}\right)-u_{x x x} D_{x}\left(\xi^{2}\right)-u_{x x y} D_{x}\left(\xi^{3}\right)-u_{x x z} D_{x}\left(\xi^{4}\right) \\ \zeta^{x y y}=D_{x}\left(\zeta^{y y}\right)-u_{y y t} D_{x}\left(\xi^{1}\right)-u_{x y y} D_{x}\left(\xi^{2}\right)-u_{y y y} D_{x}\left(\xi^{3}\right)-u_{y y z} D_{x}\left(\xi^{4}\right) \\ \zeta^{x z z}=D_{z}\left(\zeta^{x z}\right)-u_{x z t} D_{z}\left(\xi^{1}\right)-u_{x x z} D_{z}\left(\xi^{2}\right)-u_{x y z} D_{z}\left(\xi^{3}\right)-u_{x z z} D_{z}\left(\xi^{4}\right) \end{array}$
as well as the total derivatives presented as
$\begin{matrix} {} & {{D}_{t}}={{\partial }_{t}}+{{u}_{t}}{{\partial }_{u}}+{{u}_{tt}}{{\partial }_{{{u}_{t}}}}+{{u}_{xt}}{{\partial }_{{{u}_{x}}}}+\ldots, \\ {} & {{D}_{x}}={{\partial }_{x}}+{{u}_{x}}{{\partial }_{u}}+{{u}_{xx}}{{\partial }_{{{u}_{x}}}}+{{u}_{xt}}{{\partial }_{{{u}_{t}}}}+\ldots \\ {} & {{D}_{y}}={{\partial }_{y}}+{{u}_{y}}{{\partial }_{u}}+{{u}_{yy}}{{\partial }_{{{u}_{y}}}}+{{u}_{yt}}{{\partial }_{{{u}_{t}}}}+\ldots, \\ {} & {{D}_{z}}={{\partial }_{z}}+{{u}_{z}}{{\partial }_{u}}+{{u}_{zz}}{{\partial }_{{{u}_{z}}}}+{{u}_{zt}}{{\partial }_{{{u}_{t}}}}+\ldots. \\ \end{matrix}$
If we expand Eq. (4) and disintegrate the result over diverse derivatives of u, one secures the following twenty-one overdetermined system of linear PDEs
$\begin{matrix} {} & \xi _{u}^{1}=0,\quad \xi _{x}^{1}=0,\quad \xi _{y}^{1}=0,\quad \xi _{z}^{1}=0,\quad \xi _{t}^{1}=0,\quad \xi _{t}^{2}=0,\quad \xi _{u}^{2}=0,\quad \xi _{y}^{2}=0, \\ {} & \xi _{x}^{2}=0,\quad \xi _{z}^{2}=0,\quad \xi _{t}^{3}=0,\quad \xi _{u}^{3}=0,\quad \xi _{x}^{3}=0,\quad \xi _{y}^{3}=0,\quad \xi _{u}^{4}=0,\quad \xi _{x}^{4}=0, \\ {} & \xi _{t}^{4}=0,\quad \xi _{z}^{4}=0,\quad \eta =0,\quad \xi _{yy}^{4}=0,\quad \gamma \xi _{z}^{3}+\epsilon \xi _{y}^{4}=0, \\ \end{matrix}$
whose solution can be secured without much tedious calculation. Consequently, the values of ξ1, ξ2, ξ3, ξ4 and η are
${{\xi }^{1}}={{\mathbf{c}}_{1}},\quad {{\xi }^{2}}={{\mathbf{c}}_{5}},\quad {{\xi }^{3}}={{\mathbf{c}}_{2}}z+{{\mathbf{c}}_{3}},\quad {{\xi }^{4}}=-\frac{\gamma {{\mathbf{c}}_{2}}}{\epsilon }y+{{\mathbf{c}}_{4}},\quad \eta =0,$
with ${{\mathbf{c}}_{i}},i=1,\ldots,5$ regarded as arbitrary constants. Therefore, we have five Lie point symmetries presented as
${{\text{ }\!\!\Sigma\!\!\text{ }}_{1}}=\frac{\partial }{\partial t},\quad {{\text{ }\!\!\Sigma\!\!\text{ }}_{2}}=\frac{\partial }{\partial x},\quad {{\text{ }\!\!\Sigma\!\!\text{ }}_{3}}=\frac{\partial }{\partial y},\quad {{\text{ }\!\!\Sigma\!\!\text{ }}_{4}}=\frac{\partial }{\partial z},\quad {{\text{ }\!\!\Sigma\!\!\text{ }}_{5}}=\epsilon z\frac{\partial }{\partial y}-\gamma y\frac{\partial }{\partial z}.$
Thus, we can state the following theorem.
Theorem 2.1 The (3 + 1)D-gZKe (3) admits a five-dimensional Lie algebra spanned by the vectors ${{\text{ }\!\!\Sigma\!\!\text{ }}_{1}},\ldots,{{\text{ }\!\!\Sigma\!\!\text{ }}_{5}}$ given in (7).
2.2. Symmetry reduction and group-invariant solutions
Case 1 Vectors Σ1, Σ2, Σ3, Σ4
Contemplating the linear combinations of the translation generators Σ1, Σ2, Σ3 and Σ4 as $\text{ }\!\!\Sigma\!\!\text{ }=b{{\text{ }\!\!\Sigma\!\!\text{ }}_{1}}+{{\text{ }\!\!\Sigma\!\!\text{ }}_{2}}+a{{\text{ }\!\!\Sigma\!\!\text{ }}_{3}}+{{\text{ }\!\!\Sigma\!\!\text{ }}_{4}}$ with nonzero constant parameters a and b, we reduce (3 + 1)D-gZKe (3) to a PDE expressed with regards to three independent variables. The invariants associated with the Lagrangian system
$\frac{dt}{b}=\frac{dx}{1}=\frac{dy}{a}=\frac{dz}{1}=\frac{du}{0}$
which is relative to Σ are
$w=t-bx,\quad g=y-at,\quad f=t-bz,\quad \text{whereas}\quad \phi =u.$
In this case, we regard w, g and h as the current independent variables as well as ϕ as the recent dependent variable. Thus, the invariants (8), transform Eq. (3) into
$\begin{matrix} {} & {{\phi }_{f}}+{{\phi }_{w}}-a{{\phi }_{g}}-\alpha b{{\phi }_{w}}{{\phi }^{n}}-b\beta {{\phi }_{w}}{{\phi }^{2n}}-a{{b}^{2}}\sigma {{\phi }_{wwg}}-{{b}^{3}}\gamma {{\phi }_{wff}} \\ {} & -{{b}^{3}}\kappa {{\phi }_{www}}+{{b}^{2}}\sigma {{\phi }_{wwf}}+{{b}^{2}}\sigma {{\phi }_{www}}-b\epsilon {{\phi }_{wgg}}=0. \\ \end{matrix}$
NPDE (9) presented in three independent variables shall be further reduced via the Lie symmetry operators of the equation. Here, Eq. (9) possesses three generators that are transnational, namely,
${{\text{ }\!\!\Upsilon\!\!\text{ }}_{1}}=\frac{\partial }{\partial w},\quad {{\text{ }\!\!\Upsilon\!\!\text{ }}_{2}}=\frac{\partial }{\partial g},\quad {{\text{ }\!\!\Upsilon\!\!\text{ }}_{3}}=\frac{\partial }{\partial f}.$
We invoke the combination of Υ1, Υ2, Υ3 as Υ=Υ1+Υ2+Υ3 to reduce Eq. (9). Consequently, we present invariants related to characteristic equations of Υ as
$r=w-g,\quad s=g-f,\quad \text{with}\quad \phi =\theta.$
Handling r and s as the most recent independent variables as well as θ as the current dependent variable, (9) becomes
$\begin{matrix} {} & a{{\theta }_{r}}-{{\theta }_{s}}+{{\theta }_{r}}-a{{\theta }_{s}}-\alpha b{{\theta }_{r}}{{\theta }^{n}}-b\beta {{\theta }_{r}}{{\theta }^{2n}}-a{{b}^{2}}\sigma {{\theta }_{rrs}}+a{{b}^{2}}\sigma {{\theta }_{rrr}}-{{b}^{3}}\gamma {{\theta }_{rss}} \\ {} & -{{b}^{3}}\kappa thet{{a}_{rrr}}-{{b}^{2}}\sigma {{\theta }_{rrs}}+{{b}^{2}}\sigma {{\theta }_{rrr}}-b\epsilon {{\theta }_{rss}}+2b\epsilon {{\theta }_{rrs}}-b\epsilon {{\theta }_{rrr}}=0 \\ \end{matrix}$
and this is a NPDE expressed in two independent variables r and s. We now invoke the symmetries of (11) in order to transform it into a nonlinear ordinary differential equation (NODE). Lie symmetries of (11) include
${{\text{ }\!\!\Gamma\!\!\text{ }}_{1}}=\frac{\partial }{\partial r},\quad {{\text{ }\!\!\Gamma\!\!\text{ }}_{2}}=\frac{\partial }{\partial s}$
and their combined $\text{ }\!\!\Gamma\!\!\text{ }={{\text{ }\!\!\Gamma\!\!\text{ }}_{1}}+c{{\text{ }\!\!\Gamma\!\!\text{ }}_{2}}$, (c a constant), guarantees the invariants
$\xi =s-cr,\quad \text{ }\!\!\Phi\!\!\text{ }=\theta,$
thus yielding a group-invariant solution expressed as $\text{ }\!\!\Phi\!\!\text{ }=\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)$. Consequently, Eq. (11) reduces to the third-order NODE
$\begin{matrix} {} & bc\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}{{\text{ }\!\!\Phi\!\!\text{ }}^{\prime\prime \prime }}\left( \xi \right) \\ {} & +\alpha bc{{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{n}}+bc\beta {{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2n}} \\ {} & -\left( 1+a \right)\left( 1+c \right){{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)=0, \\ \end{matrix}$
with $\xi =bcx+\left( 1+c \right)y+bz-\left( 1+a \right)\left( 1+c \right)t$.
Case 2 Vector field Σ5
The invariants for the vector field Σ5 are obtained by securing the solution of the associated Lagrangian system
$\frac{dt}{0}=\frac{dx}{0}=\frac{dy}{\epsilon z}=\frac{dz}{-\gamma y}=\frac{du}{0}.$
Consequently, we have
$g=t,\quad f=x,\quad w=\frac{1}{2}\left( \epsilon {{z}^{2}}-\gamma {{y}^{2}} \right),\quad u=\phi,$
which transforms (3 + 1)D-gZKe (3) into the NPDE
$\begin{matrix} {} & {} & {{\phi }_{g}}+\alpha {{\phi }_{f}}{{\phi }^{n}}+\beta {{\phi }_{f}}{{\phi }^{2n}}+\sigma {{\phi }_{gff}}+\kappa {{\phi }_{fff}}+{{\gamma }^{2}}{{y}^{2}}\epsilon {{\phi }_{wwf}} \\ {} & {} & +\gamma {{z}^{2}}{{\epsilon }^{2}}{{\phi }_{wwf}}=0. \\ \end{matrix}$
Application of Lie algorithm to secure the generators of (17) yields
${{\text{ }\!\!\Upsilon\!\!\text{ }}_{1}}=\frac{\partial }{\partial w},\quad {{\text{ }\!\!\Upsilon\!\!\text{ }}_{2}}=\frac{\partial }{\partial g},\quad {{\text{ }\!\!\Upsilon\!\!\text{ }}_{3}}=\frac{\partial }{\partial f},$
which we combine linearly as $\text{ }\!\!\Upsilon\!\!\text{ }={{\text{ }\!\!\Upsilon\!\!\text{ }}_{1}}+\nu {{\text{ }\!\!\Upsilon\!\!\text{ }}_{2}}+{{\text{ }\!\!\Upsilon\!\!\text{ }}_{3}}$ to get the invariants
$r=f-\nu w,\quad s=f-\nu g,\quad \phi =\theta.$
Applying invariants (18) transforms (17) into
$\begin{matrix} {} & \alpha {{\theta }_{s}}{{\theta }^{n}}+\alpha {{\theta }_{r}}{{\theta }^{n}}+\beta {{\theta }_{s}}{{\theta }^{2n}}+\beta {{\theta }_{r}}{{\theta }^{2n}}+\kappa {{\theta }_{sss}}+3\kappa {{\theta }_{rss}}+3\kappa {{\theta }_{rrs}} \\ {} & +\kappa {{\theta }_{rrr}}-\nu \sigma {{\theta }_{sss}}-2\nu \sigma {{\theta }_{rss}}-\nu \sigma {{\theta }_{rrs}}-\nu {{\theta }_{s}}+{{\gamma }^{2}}{{\nu }^{2}}{{y}^{2}}\epsilon {{\theta }_{rrs}} \\ {} & +{{\gamma }^{2}}{{\nu }^{2}}{{y}^{2}}\epsilon {{\theta }_{rrr}}+\gamma {{\nu }^{2}}{{z}^{2}}{{\epsilon }^{2}}{{\theta }_{rrs}}+\gamma {{\nu }^{2}}{{z}^{2}}{{\epsilon }^{2}}{{\theta }_{rrr}}=0. \\ \end{matrix}$
Eq. (19) produces the translation generators ${{\text{ }\!\!\Gamma\!\!\text{ }}_{1}}=\partial /\partial r$, ${{\text{ }\!\!\Gamma\!\!\text{ }}_{2}}=\partial /\partial s$. Utilizing their linear combination ${{\text{ }\!\!\Gamma\!\!\text{ }}_{1}}+\mu {{\text{ }\!\!\Gamma\!\!\text{ }}_{2}}$ gives
$\xi =r-\mu s,\quad \text{ }\!\!\Phi\!\!\text{ }=\theta $
as invariants, which eventually converts (19) to the NODE
$\begin{matrix} {} & \left[ \nu \left\{ {{\left( \mu -1 \right)}^{2}}\mu \sigma -{{\gamma }^{2}}\left( \mu -1 \right)\nu {{y}^{2}}\epsilon +\gamma \nu {{z}^{2}}\epsilon \left( \epsilon -\gamma \mu \right)-\kappa {{\left( \mu -1 \right)}^{3}} \right\} \right]{{\text{ }\!\!\Phi\!\!\text{ }}^{\prime\prime \prime }}\left( \xi \right) \\ {} & +\alpha \left( 1-\mu \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{n}}{{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)+\beta \left( 1-\mu \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2n}}{{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)+\mu \nu {{\text{ }\!\!\Phi\!\!\text{ }}^{\prime }}\left( \xi \right)=0, \\ \end{matrix}$
with $\xi =\mu \nu t+\left( 1-\mu \right)x+\left( 1/2 \right)\nu \left( \gamma {{y}^{2}}-\epsilon {{z}^{2}} \right)$
3. Exact solutions of (3 + 1)D-gZKe (3)
This section presents the exact solutions of (3 + 1)D-gZKe (3) through the adoption of some techniques. We shall first do the direct integration of the achieved NODEs presented in (14) and (21).
3.1. Exact solution of (3) via the NODE (14)
In this subsection, we present the direct integration of NODE (14) in order to seek some soliton solutions of (3 + 1)D-gZKe (3). Integration of (14) twice with respect to ξ yields
$\begin{matrix} {} & \frac{1}{2}bc\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}{{\text{ }\!\!\Phi\!\!\text{ }}^{{{\prime }^{2}}}}\left( \xi \right) \\ {} & +\frac{\alpha bc}{\left( n+1 \right)\left( n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{n+2}}+\frac{bc\beta }{\left( 2n+1 \right)\left( 2n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2n+2}} \\ {} & -\frac{1}{2}\left( 1+a \right)\left( 1+c \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2}}+{{K}_{0}}\text{ }\!\!\Phi\!\!\text{ }+{{K}_{1}}=0, \\ \end{matrix}$
where $n\ne -1,-2,-1/2$ and K0, K1 being constants of integration. We consider the following cases to secure some soliton solutions of (3 + 1)D-gZKe (3).
3.1.1. Soliton solution type I
Suppose we contemplate a special case of (22) by taking constants K0 and K1 to be zero. Eq. (22) then becomes
$\begin{matrix} {} & bc\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}{{\text{ }\!\!\Phi\!\!\text{ }}^{{{\prime }^{2}}}}\left( \xi \right) \\ {} & -\left( 1+a \right)\left( 1+c \right)\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2}}+\frac{2\alpha bc}{\left( n+1 \right)\left( n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{n+2}} \\ {} & +\frac{2bc\beta }{\left( 2n+1 \right)\left( 2n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2n+2}}=0. \\ \end{matrix}$
Letting $\text{ }\!\!\Phi\!\!\text{ }={{V}^{1/n}}$ transforms Eq. (23) to
$\begin{matrix} {} & bc\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}{{V}^{{{\prime }^{2}}}}\left( \xi \right) \\ {} & -\left( 1+a \right)\left( 1+c \right)V{{\left( \xi \right)}^{2}}+\frac{2\alpha bc}{\left( n+1 \right)\left( n+2 \right)}V{{\left( \xi \right)}^{3}} \\ {} & +\frac{2bc\beta }{\left( 2n+1 \right)\left( 2n+2 \right)}V{{\left( \xi \right)}^{4}}=0, \\ \end{matrix}$
which we rewrite as
${{V}^{\prime 2}}=A{{V}^{2}}-B{{V}^{3}}-C{{V}^{4}},$
where
$\begin{matrix} {} & A=\frac{\left( 1+a \right)\left( 1+c \right){{n}^{2}}}{bc\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}}, \\ {} & B=\frac{2\alpha bc{{n}^{2}}}{bc\left( n+1 \right)\left( n+2 \right)\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}}, \\ {} & C=\frac{2\beta bc{{n}^{2}}}{2bc\left( n+1 \right)\left( 2n+1 \right)\left\{ {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right\}}. \\ \end{matrix}$
By assuming
$V\left( \xi \right)=\frac{1}{\lambda +Q\left( \xi \right)}$
Eq. (25) becomes
$\begin{matrix} {} & {{Q}^{\prime 2}}-\left( A{{\lambda }^{2}}-B{{\lambda }^{3}}-C{{\lambda }^{4}} \right){{Q}^{4}}-\left( 2A\lambda -3B{{\lambda }^{2}}-4C{{\lambda }^{3}} \right){{Q}^{3}} \\ {} & -\left( A-3B\lambda -6C{{\lambda }^{2}} \right){{Q}^{2}}+\left( B+4C\lambda \right)Q+C=0. \\ \end{matrix}$
Equating the coefficients of Q4 and Q2 to zero yields the values of λ and C as
$\lambda =\frac{5A}{3B},\quad C=-\frac{6{{B}^{2}}}{25A}.$
Utilizing the outcomes makes (28) to become
${{Q}^{\prime 2}}=-\frac{5{{A}^{2}}}{9B}{{Q}^{3}}+\frac{3B}{5}Q+\frac{6{{B}^{2}}}{25A},$
which can be written as
${{R}^{\prime 2}}=4{{R}^{3}}-{{H}_{2}}R-{{H}_{3}}$
with the representations that
$R\left( r \right)=Q\left( \xi \right),\quad r=\frac{A}{6}\sqrt{\frac{-5}{B}}\xi,\quad {{H}_{2}}=\frac{108{{B}^{2}}}{25{{A}^{2}}}\quad {{H}_{3}}=\frac{216{{B}^{3}}}{125{{A}^{3}}},$
$R\left( r \right)=\wp \left( r,{{H}_{2}},{{H}_{3}} \right)$
and so the general solution to first-order ODE (25) can be expressed as
$V\left( \xi \right)={{\left\{ \wp \left( \frac{A}{6}\sqrt{\frac{-5}{B}}\xi,{{H}_{2}},{{H}_{3}} \right) \right\}}^{-1}}+\frac{5A}{3B}.$
The above then has the soliton solutions [44]
$\begin{matrix} {} & {} & V\left( \xi \right)=5A{{\left[ B\left\{ 3-\text{cos}{{\text{h}}^{2}}\left( \frac{\sqrt{A}}{2}\xi \right) \right\} \right]}^{-1}}, \\ {} & {} & V\left( \xi \right)=5A{{\left[ B\left\{ 3+\text{sin}{{\text{h}}^{2}}\left( \frac{\sqrt{A}}{2}\xi \right) \right\} \right]}^{-1}}, \\ \end{matrix}$
where A>0. Thus, reverting to the basic variables, we can describe solutions to the dual-power law nonlinearity Eq. (3) as the bright soliton solutions of the structure
$u\left( t,x,y,z \right)={{\left[ \frac{5\left( n+1 \right)\left( n+2 \right)\left( 1+a \right)\left( 1+c \right)}{2\alpha bc\left\{ 3-\text{cos}{{\text{h}}^{2}}\left( \sqrt{\frac{\left( 1+a \right)\left( 1+c \right){{n}^{2}}}{4bc\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right)}}\xi \right) \right\}} \right]}^{\frac{1}{n}}},$
$u\left( t,x,y,z \right)={{\left[ \frac{5\left( n+1 \right)\left( n+2 \right)\left( 1+a \right)\left( 1+c \right)}{2\alpha bc\left\{ 3+\text{sin}{{\text{h}}^{2}}\left( \sqrt{\frac{\left( 1+a \right)\left( 1+c \right){{n}^{2}}}{4bc\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right)}}\xi \right) \right\}} \right]}^{\frac{1}{n}}},$
where we have constraint conditions $6{{\alpha }^{2}}{{b}^{2}}{{c}^{2}}\left( 2n+1 \right)\left( 2n+2 \right)+25\beta bc{{\left( \left( n+1 \right)\left( n+2 \right) \right)}^{2}}=0$ and $\left( 1+a \right)\left( 1+c \right)\left( bc\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right) \right)>0$ with $\xi =bcx-\left( 1+a \right)\left( 1+c \right)t+\left( 1+c \right)y+bz$. We depict the dynamics of solutions (36) and (37) for some particular cases of n in Fig. 1, Fig. 2, Fig. 3, Fig. 4.
Fig. 1. Solitary wave depiction of solution (36) at t=0 and x=0.5 for n=1. |
Fig. 2. Solitary wave depiction of solution (36) at t=1 and x=2 for n=2. |
Fig. 3. Solitary wave depiction of solution (37) at t=1 and x=1 for n=1. |
Fig. 4. Solitary wave depiction of solution (37) at t=−1 and x=2 for n=2. |
3.1.2. Soliton solution type II
We seek to secure other forms of soliton solutions for (3 + 1)D-gZKe (3) by assuming that solution of (25) can be presented in the structure
$V\left( \xi \right)=\frac{1}{S\left( \xi \right)}.$
Thus Eq. (25) transforms into
${{S}^{\prime 2}}=A{{S}^{2}}-BS-C.$
Next, we contemplate a first-order ODE which is in the form of a polynomial of degree four, given as
${{S}^{\prime 2}}=Q\left( S \right)={{c}_{0}}{{S}^{4}}+4{{c}_{1}}{{S}^{3}}+6{{c}_{2}}{{S}^{2}}+4{{c}_{3}}S+{{c}_{4}},$
whose solution can be expressed with regards to the Weierstrass elliptic function
$S\left( \xi \right)={{Q}_{0}}+\frac{1}{4}{Q}'\left( {{S}_{0}} \right){{\left[ \wp \left( \xi,{{g}_{2}},{{g}_{3}} \right)-\frac{1}{24}{Q}''\left( {{S}_{0}} \right) \right]}^{-1}},$
with invariants g2 and g3 stated as ${{g}_{2}}={{c}_{0}}{{c}_{4}}-4{{c}_{1}}{{c}_{3}}+3c_{2}^{2}$, and ${{g}_{3}}={{c}_{0}}{{c}_{2}}{{c}_{4}}+2{{c}_{1}}{{c}_{2}}{{c}_{3}}-c_{2}^{3}-{{c}_{0}}c_{3}^{2}-c_{1}^{2}{{c}_{4}}$. We note that the prime (’) connotes the derivative with regards to S and S0 is a root of polynomal Q(S) in (40). Here, ODE (39) possesses second-order polynomial structured as
$Q\left( S \right)=A{{S}^{2}}-BS-C.$
The roots of (42) are expressed as
$S_{0}^{+}=\frac{B+\sqrt{{{B}^{2}}+4AC}}{2A}\quad \text{and}\quad S_{0}^{-}=\frac{B-\sqrt{{{B}^{2}}+4AC}}{2A}.$
On the basis of the invariants given earlier, ${{g}_{2}}={{A}^{2}}/12$ and ${{g}_{3}}=-{{A}^{3}}/216$. Therefore the solutions of (42) can be expressed as
$S\left( \xi \right)=\frac{B}{2A}\pm \frac{\sqrt{{{B}^{2}}+4AC}}{2A}\left[ \frac{12\wp \left( \xi,{{g}_{2}},{{g}_{3}} \right)+5A}{12\wp \left( \xi,{{g}_{2}},{{g}_{3}} \right)-A} \right].$
$\begin{matrix} {} & {} & \wp \left( \xi,12{{r}^{2}},-8{{r}^{3}} \right)=r-3r\text{cos}{{\text{h}}^{-2}}\left[ {{\left( 3r \right)}^{1/2}}\xi \right], \\ {} & {} & \wp \left( \xi,12{{r}^{2}},-8{{r}^{3}} \right)=r+3r\text{sin}{{\text{h}}^{-2}}\left[ {{\left( 3r \right)}^{1/2}}\xi \right]. \\ \end{matrix}$
Thus, from the relation (45) one can secure soliton solutions for (39) via cosh as well as sinh functions expressed in the form
$S\left( \xi \right)=\frac{B}{2A}\pm \frac{\sqrt{{{B}^{2}}+4AC}}{2A}\text{cosh}\left( \sqrt{A}\xi \right),$
$S\left( \xi \right)=\frac{B}{2A}\pm \frac{\sqrt{-\left( {{B}^{2}}+4AC \right)}}{2A}\text{sinh}\left( \sqrt{A}\xi \right),$
with the criteria that A>0 and ${{B}^{2}}+4AC>0$ in the case of (46) as well as A>0 and B2+4AC<0 in the case of (47). Consequently, reverting to the original variables, Eq. (3) possesses bright and singular soliton solutions structured as
$u\left( t,x,y,z \right)={{\left[ \frac{\left( n+1 \right)\left( n+2 \right)\left( 2n+1 \right)\left( 1+a \right)\left( 1+c \right)}{\alpha bc\left( 2n+1 \right)\mp \sqrt{q}\text{cosh}\left( \sqrt{\frac{\left( 1+a \right)\left( 1+c \right){{n}^{2}}}{bc\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right)}}\xi \right)} \right]}^{\frac{1}{n}}},$
$u\left( t,x,y,z \right)={{\left[ \frac{\left( n+1 \right)\left( n+2 \right)\left( 2n+1 \right)\left( 1+a \right)\left( 1+c \right)}{\alpha bc\left( 2n+1 \right)\mp \sqrt{-q}\text{sinh}\left( \sqrt{\frac{\left( 1+a \right)\left( 1+c \right){{n}^{2}}}{bc\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right)}}\xi \right)} \right]}^{\frac{1}{n}}},$
with $bc\left( 1+a \right)\left( 1+c \right)\left( {{b}^{2}}\left( {{c}^{2}}\kappa +\gamma \right)-\sigma bc\left( a+1 \right)\left( c+1 \right)+{{\left( c+1 \right)}^{2}}\epsilon \right)>0$ and $\xi =bcx+\left( 1+c \right)y+bz-\left( 1+a \right)\left( 1+c \right)t$. We declare here that soliton solution (48) is valid when $q={{\alpha }^{2}}{{b}^{2}}{{c}^{2}}{{\left( 2n+1 \right)}^{2}}+b\beta c\left( a+1 \right)\left( c+1 \right)\left( 2n+1 \right)\left( n+1 \right){{\left( n+2 \right)}^{2}}>0$ whereas the validity of singular soliton solution (49) requires that $q={{\alpha }^{2}}{{b}^{2}}{{c}^{2}}{{\left( 2n+1 \right)}^{2}}+b\beta c\left( a+1 \right)\left( c+1 \right)\left( 2n+1 \right)\left( n+1 \right){{\left( n+2 \right)}^{2}}0$. We reveal the streaming behaviour of (48) and (49) for some cases of n in Fig. 5, Fig. 6, Fig. 7, Fig. 8.
Fig. 5. Solitary wave depiction of solution (48) at t=5 and x=1 for n=1. |
Fig. 6. Solitary wave depiction of solution (48) at t=10 and x=−10 for n=2. |
Fig. 7. Solitary wave depiction of solution (49) at t=1 and x=2 for n=1. |
Fig. 8. Solitary wave depiction of solution (49) at t=3 and x=−4 for n=2. |
3.2. Exact solution of (3) via the NODE (21)
Integrating (21) repeatedly and taking the constants to be zero as demonstrated earlier in the previous subsection, we obtain the first-order NODE
$\begin{matrix} {} & \frac{1}{2}\left( \nu \left( {{\left( \mu -1 \right)}^{2}}\mu \sigma -{{\gamma }^{2}}\left( \mu -1 \right)\nu {{y}^{2}}\epsilon +\gamma \nu {{z}^{2}}\epsilon \left( \epsilon -\gamma \mu \right)-\kappa {{\left( \mu -1 \right)}^{3}} \right) \right){{\text{ }\!\!\Phi\!\!\text{ }}^{{{\prime }^{2}}}}\left( \xi \right) \\ {} & +\frac{\alpha \left( 1-\mu \right)}{\left( n+1 \right)\left( n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{n+2}}+\frac{\beta \left( 1-\mu \right)}{\left( 2n+1 \right)\left( 2n+2 \right)}\text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2n+2}}+\frac{1}{2}\mu \nu \text{ }\!\!\Phi\!\!\text{ }{{\left( \xi \right)}^{2}}=0. \\ \end{matrix}$
Integrating Eq. (50) directly and reverting to the basic variables, one achieves
$\frac{1}{nP}\left[ \ln \left( \frac{{{u}^{n}}}{P\left( 1+P \right)\sqrt{{{P}^{2}}-\beta \left( \mu -1 \right)\left( n+2 \right){{u}^{2n}}+\left( Q-2\alpha R \right){{u}^{n}}}} \right) \right]=\mp \sqrt{\frac{\mu \nu }{PS}}\xi +{{C}_{0}},$
in which we have $P=\sqrt{\mu \nu \left( n+1 \right)\left( n+2 \right)\left( 2n+1 \right)}$, $Q=\alpha \left( \mu -1 \right)\left( -2n-1 \right)$, $R=\left( \mu -1 \right)\left( 2n+1 \right)$, $S=\kappa {{\left( \mu -1 \right)}^{3}}+\nu \left( -{{\left( \mu -1 \right)}^{2}}\mu \sigma +{{\gamma }^{2}}\left( \mu -1 \right)\nu {{y}^{2}}\epsilon +\gamma \nu {{z}^{2}}\epsilon \left( \gamma \mu -\epsilon \right) \right)$, $\xi =\mu \nu t+\left( 1-\mu \right)x+1/2\gamma \nu {{y}^{2}}-1/2\nu \epsilon {{z}^{2}}$ and C0 as constant of integration.
3.3. Exact solution of (3 + 1)D-gZKe (3) with the aid of extended Jacobi elliptic function expansion technique
We seek to achieve solutions of (3) with regards to the main copolar trio of Jacobian elliptic functions, meaning, elliptic cosine cn(ξ|ω), elliptic sine sn(ξ|ω) and delta amplitude dn(ξ|ω), with parameter ω, 0≤ω≤1. The rest of the Jacobian functions can be presented with reference to at least one of the elliptic functions in this copolar trio. For instance, see [42], [43]. Jacobi elliptic functions are of value in that they relapse to trigonometric, hyperbolic and by extension exponential functions, see Table 3.3.
Table 3.3 Copolar trio for ω=0 and ω=1. |
| ω=0 | ω=1 | ||||
|---|---|---|---|---|---|
| sn(ξ|ω) | sinω | tanhω | |||
| cn(ξ|ω) | cosω | sechω | |||
| dn(ξ|ω) | 1 | sechω |
We declare here that function W(ξ) satisfies the first-order NODEs
${W}'\left( \xi \right)+{{\left\{ \left( 1-{{W}^{2}}\left( \xi \right) \right)\left( 1-\omega +\omega {{W}^{2}}\left( \xi \right) \right) \right\}}^{1/2}}=0,$
${W}'\left( \xi \right)-{{\left\{ \left( 1-{{W}^{2}}\left( \xi \right) \right)\left( 1-\omega {{W}^{2}}\left( \xi \right) \right) \right\}}^{1/2}}=0,$
${W}'\left( \xi \right)+{{\left\{ \left( 1-{{W}^{2}}\left( \xi \right) \right)\left( \omega -1+{{W}^{2}}\left( \xi \right) \right) \right\}}^{1/2}}=0,$
whose solutions are expressed accordingly with regards to the Jacobi elliptic cosine, sine as well as delta amplitude functions, respectively, as
$W\left( \xi \right)=\text{cn}(\xi |\omega ),W\left( \xi \right)=\text{sn}(\xi |\omega ),\text{and}W\left( \xi \right)=\text{dn}(\xi |\omega ).$
Suppose the third-order NODE (14) owns a solution of the structure
$\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)=\underset{i=-m}{\overset{m}{\mathop \sum }}\,{{B}_{i}}W{{\left( \xi \right)}^{i}},$
where we aim to obtain the value of positive integer m by adopting balancing procedure, see [47], then we contemplate the subsequent solitary wave solution pathways.
3.3.1. Cnoidal wave solutions
Here, contemplating the NODE (14), we consider the case when n=1. Thus, balancing procedure produces m=1 and then (56) assumes structure
$\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)={{B}_{-1}}W{{\left( \xi \right)}^{-1}}+{{B}_{0}}+{{B}_{1}}W\left( \xi \right).$
Substituting the value of $\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)$ from (57) into (14) in conjunction with (52), we secure an algebraic equation, which splits over various powers of $W\left( \xi \right)$ and yields a system of eleven algebraic equations:
$\begin{array}{l} \begin{array}{l} 2 \beta \omega p B_{0} B_{1}^{2}+\alpha \omega p B_{1}^{2}=0 \\ -4 \beta \omega p B_{0} B_{1}^{2}-2 \alpha \omega p B_{1}^{2}+2 \beta p B_{0} B_{1}^{2}+\alpha p B_{1}^{2}=0, \\ -2 \beta \omega p B_{-1}^{2} B_{0}-\alpha \omega p B_{-1}^{2}+2 \beta p B_{-1}^{2} B_{0}+\alpha p B_{-1}^{2}=0, \\ 4 \beta \omega p B_{-1}^{2} B_{0}+2 \alpha \omega p B_{-1}^{2}-2 \beta p B_{-1}^{2} B_{0}-\alpha p B_{-1}^{2}=0 \text {, } \\ -6 \epsilon \omega^{2} p q^{2} B_{1}-6 \gamma k^{2} \omega^{2} p B_{1}-6 \kappa \omega^{2} p^{3} B_{1}-6 \omega^{2} p^{2} r \sigma B_{1}+\beta \omega p B_{1}^{3}=0, \\ -2 \beta \omega p B_{-1}^{2} B_{0}+2 \beta \omega p B_{0} A_{1}^{2}-\alpha \omega p B_{-1}^{2}+\alpha \omega p B_{1}^{2}-2 \beta p B_{0} B_{1}^{2}-\alpha p B_{1}^{2}=0, \\ 6 \epsilon \omega^{2} p q^{2} B_{-1}+6 \gamma k^{2} \omega^{2} p B_{-1}+6 \kappa \omega^{2} p^{3} B_{-1}+6 \omega^{2} p^{2} r \sigma B_{-1}-\beta \omega p B_{-1}^{3} \\ -12 \epsilon \omega p q^{2} B_{-1}-12 \gamma k^{2} \omega p B_{-1}-12 \kappa \omega p^{3} B_{-1}-12 \omega p^{2} r \sigma B_{-1}+\beta p B_{-1}^{3} \\ +6 \epsilon p q^{2} B_{-1}+6 \gamma k^{2} p B_{-1}+6 \kappa p^{3} B_{-1}+6 p^{2} r \sigma B_{-1}=0 \text {, } \\ 14 \epsilon \omega^{2} p q^{2} B_{1}+14 \gamma k^{2} \omega^{2} p B_{1}+14 \kappa \omega^{2} p^{3} B_{1}+14 \omega^{2} p^{2} r \sigma B_{1}+\beta \omega p B_{-1} B_{1}^{2} \\ +\beta \omega p B_{0}^{2} B_{1}-2 \beta \omega p B_{1}^{3}-7 \epsilon \omega p q^{2} B_{1}-7 \gamma k^{2} \omega p B_{1}-7 \kappa \omega p^{3} B_{1}-7 \omega p^{2} r \sigma B_{1} \\ +\alpha \omega p B_{0} B_{1}+\beta p B_{1}^{3}+\omega r B_{1}=0, \\ -14 \epsilon \omega^{2} p q^{2} B_{-1}-14 \gamma k^{2} \omega^{2} p B_{-1}-14 \kappa \omega^{2} p^{3} B_{-1}-14 \omega^{2} p^{2} r \sigma B_{-1}+2 \beta \omega p B_{-1}^{3} \\ -\beta \omega p B_{-1}^{2} B_{1}-\beta \omega p B_{-1} B_{0}^{2}+21 \epsilon \omega p q^{2} B_{-1}+21 \gamma k^{2} \omega p B_{-1}+21 \kappa \omega p^{3} B_{-1} \\ +21 \omega p^{2} r \sigma B_{-1}-\alpha \omega p B_{-1} B_{0}-\beta p B_{-1}^{3}+\beta p B_{-1}^{2} B_{1}+\beta p B_{-1} B_{0}^{2}-7 \epsilon p q^{2} B_{-1} \\ -7 \gamma k^{2} p B_{-1}-7 \kappa p^{3} B_{-1}-7 p^{2} r \sigma B_{-1}+\alpha p B_{-1} B_{0}-\omega r B_{-1}+r B_{-1}=0, \\ -2 \epsilon \omega^{2} p q^{2} B_{-1}-10 \epsilon \omega^{2} p q^{2} B_{1}-2 \gamma k^{2} \omega^{2} p B_{-1}-10 \gamma k^{2} \omega^{2} p B_{1}-2 \kappa \omega^{2} p^{3} B_{-1} \\ -10 \kappa \omega^{2} p^{3} B_{1}-2 \omega^{2} p^{2} r \sigma B_{-1}-10 \omega^{2} p^{2} r \sigma B_{1}-\beta \omega p B_{-1}^{2} B_{1}-\beta \omega p B_{-1} B_{0}^{2} \\ -2 \beta \omega p B_{-1} B_{1}^{2}-2 \beta \omega p B_{0}^{2} B_{1}+\beta \omega p B_{1}^{3}+\epsilon \omega p q^{2} B_{-1}+10 \epsilon \omega p q^{2} B_{1}+\gamma k^{2} \omega p B_{-1} \\ +10 \gamma k^{2} \omega p B_{1}+10 \kappa \omega p^{3} B_{1}+\omega p^{2} r \sigma B_{-1}+10 \omega p^{2} r \sigma B_{1}-\alpha \omega p B_{-1} B_{0} \\ -2 \alpha \omega p B_{0} B_{1}+\beta p B_{-1} B_{1}^{2}+\beta p B_{0}^{2} B_{1}-\beta p B_{1}^{3}-\epsilon p q^{2} B_{1}-\gamma k^{2} p B_{1}-\kappa p^{3} B_{1} \\ +\kappa \omega p^{3} B_{-1}-p^{2} r \sigma B_{1}+\alpha p B_{0} B_{1}-\omega r B_{-1}-2 \omega r B_{1}+r B_{1}=0, \\ 10 \epsilon \omega^{2} p q^{2} B_{-1}+2 \epsilon \omega^{2} p q^{2} B_{1}+10 \gamma k^{2} \omega^{2} p B_{-1}+2 \gamma k^{2} \omega^{2} p B_{1}+10 \kappa \omega^{2} p^{3} B_{-1} \\ +2 \kappa \omega^{2} p^{3} B_{1}+10 \omega^{2} p^{2} r \sigma B_{-1}+2 \omega^{2} p^{2} r \sigma B_{1}-\beta \omega p B_{-1}^{3}+2 \beta \omega p B_{-1}^{2} B_{1} \\ +2 \beta \omega p B_{-1} B_{0}^{2}+\beta \omega p B_{-1} B_{1}^{2}-10 \epsilon \omega p q^{2} B_{-1}-3 \epsilon \omega p q^{2} B_{1}-10 \gamma k^{2} \omega p B_{-1} \\ -3 \gamma k^{2} \omega p B_{1}-10 \kappa \omega p^{3} B_{-1}-3 \kappa \omega p^{3} B_{1}-10 \omega p^{2} r \sigma B_{-1}-3 \omega p^{2} r \sigma B_{1} \\ +2 \alpha \omega p B_{-1} B_{0}+\alpha \omega p B_{0} B_{1}-\beta p B_{-1}^{2} B_{1}-\beta p B_{-1} B_{0}^{2}-\beta p B_{-1} B_{1}^{2}-\beta p B_{0}^{2} B_{1} \\ +\beta \omega p B_{0}^{2} B_{1}+\epsilon p q^{2} B_{-1}+\epsilon p q^{2} B_{1}+\gamma k^{2} p B_{-1}+\gamma k^{2} p B_{1}+\kappa p^{3} B_{-1}+\kappa p^{3} B_{1} \\ +p^{2} r \sigma B_{-1}+p^{2} r \sigma B_{1}-\alpha p B_{-1} B_{0}-\alpha p B_{0} B_{1}+2 \omega r B_{-1}+\omega r B_{1}-r B_{-1} \\ -r B_{1}=0. \end{array}\\ 2 \beta \omega p B_{0} B_{1}^{2}+\alpha \omega p B_{1}^{2}=0\\ -4 \beta \omega p B_{0} B_{1}^{2}-2 \alpha \omega p B_{1}^{2}+2 \beta p B_{0} B_{1}^{2}+\alpha p B_{1}^{2}=0,\\ -2 \beta \omega p B_{-1}^{2} B_{0}-\alpha \omega p B_{-1}^{2}+2 \beta p B_{-1}^{2} B_{0}+\alpha p B_{-1}^{2}=0,\\ 4 \beta \omega p B_{-1}^{2} B_{0}+2 \alpha \omega p B_{-1}^{2}-2 \beta p B_{-1}^{2} B_{0}-\alpha p B_{-1}^{2}=0 \text {, }\\ -6 \epsilon \omega^{2} p q^{2} B_{1}-6 \gamma k^{2} \omega^{2} p B_{1}-6 \kappa \omega^{2} p^{3} B_{1}-6 \omega^{2} p^{2} r \sigma B_{1}+\beta \omega p B_{1}^{3}=0,\\ -2 \beta \omega p B_{-1}^{2} B_{0}+2 \beta \omega p B_{0} A_{1}^{2}-\alpha \omega p B_{-1}^{2}+\alpha \omega p B_{1}^{2}-2 \beta p B_{0} B_{1}^{2}-\alpha p B_{1}^{2}=0,\\ 6 \epsilon \omega^{2} p q^{2} B_{-1}+6 \gamma k^{2} \omega^{2} p B_{-1}+6 \kappa \omega^{2} p^{3} B_{-1}+6 \omega^{2} p^{2} r \sigma B_{-1}-\beta \omega p B_{-1}^{3}\\ -12 \epsilon \omega p q^{2} B_{-1}-12 \gamma k^{2} \omega p B_{-1}-12 \kappa \omega p^{3} B_{-1}-12 \omega p^{2} r \sigma B_{-1}+\beta p B_{-1}^{3}\\ +6 \epsilon p q^{2} B_{-1}+6 \gamma k^{2} p B_{-1}+6 \kappa p^{3} B_{-1}+6 p^{2} r \sigma B_{-1}=0 \text {, }\\ 14 \epsilon \omega^{2} p q^{2} B_{1}+14 \gamma k^{2} \omega^{2} p B_{1}+14 \kappa \omega^{2} p^{3} B_{1}+14 \omega^{2} p^{2} r \sigma B_{1}+\beta \omega p B_{-1} B_{1}^{2}\\ +\beta \omega p B_{0}^{2} B_{1}-2 \beta \omega p B_{1}^{3}-7 \epsilon \omega p q^{2} B_{1}-7 \gamma k^{2} \omega p B_{1}-7 \kappa \omega p^{3} B_{1}-7 \omega p^{2} r \sigma B_{1}\\ +\alpha \omega p B_{0} B_{1}+\beta p B_{1}^{3}+\omega r B_{1}=0,\\ -14 \epsilon \omega^{2} p q^{2} B_{-1}-14 \gamma k^{2} \omega^{2} p B_{-1}-14 \kappa \omega^{2} p^{3} B_{-1}-14 \omega^{2} p^{2} r \sigma B_{-1}+2 \beta \omega p B_{-1}^{3}\\ -\beta \omega p B_{-1}^{2} B_{1}-\beta \omega p B_{-1} B_{0}^{2}+21 \epsilon \omega p q^{2} B_{-1}+21 \gamma k^{2} \omega p B_{-1}+21 \kappa \omega p^{3} B_{-1}\\ +21 \omega p^{2} r \sigma B_{-1}-\alpha \omega p B_{-1} B_{0}-\beta p B_{-1}^{3}+\beta p B_{-1}^{2} B_{1}+\beta p B_{-1} B_{0}^{2}-7 \epsilon p q^{2} B_{-1}\\ -7 \gamma k^{2} p B_{-1}-7 \kappa p^{3} B_{-1}-7 p^{2} r \sigma B_{-1}+\alpha p B_{-1} B_{0}-\omega r B_{-1}+r B_{-1}=0,\\ -2 \epsilon \omega^{2} p q^{2} B_{-1}-10 \epsilon \omega^{2} p q^{2} B_{1}-2 \gamma k^{2} \omega^{2} p B_{-1}-10 \gamma k^{2} \omega^{2} p B_{1}-2 \kappa \omega^{2} p^{3} B_{-1}\\ -10 \kappa \omega^{2} p^{3} B_{1}-2 \omega^{2} p^{2} r \sigma B_{-1}-10 \omega^{2} p^{2} r \sigma B_{1}-\beta \omega p B_{-1}^{2} B_{1}-\beta \omega p B_{-1} B_{0}^{2}\\ -2 \beta \omega p B_{-1} B_{1}^{2}-2 \beta \omega p B_{0}^{2} B_{1}+\beta \omega p B_{1}^{3}+\epsilon \omega p q^{2} B_{-1}+10 \epsilon \omega p q^{2} B_{1}+\gamma k^{2} \omega p B_{-1}\\ +10 \gamma k^{2} \omega p B_{1}+10 \kappa \omega p^{3} B_{1}+\omega p^{2} r \sigma B_{-1}+10 \omega p^{2} r \sigma B_{1}-\alpha \omega p B_{-1} B_{0}\\ -2 \alpha \omega p B_{0} B_{1}+\beta p B_{-1} B_{1}^{2}+\beta p B_{0}^{2} B_{1}-\beta p B_{1}^{3}-\epsilon p q^{2} B_{1}-\gamma k^{2} p B_{1}-\kappa p^{3} B_{1}\\ +\kappa \omega p^{3} B_{-1}-p^{2} r \sigma B_{1}+\alpha p B_{0} B_{1}-\omega r B_{-1}-2 \omega r B_{1}+r B_{1}=0,\\ 10 \epsilon \omega^{2} p q^{2} B_{-1}+2 \epsilon \omega^{2} p q^{2} B_{1}+10 \gamma k^{2} \omega^{2} p B_{-1}+2 \gamma k^{2} \omega^{2} p B_{1}+10 \kappa \omega^{2} p^{3} B_{-1}\\ +2 \kappa \omega^{2} p^{3} B_{1}+10 \omega^{2} p^{2} r \sigma B_{-1}+2 \omega^{2} p^{2} r \sigma B_{1}-\beta \omega p B_{-1}^{3}+2 \beta \omega p B_{-1}^{2} B_{1}\\ +2 \beta \omega p B_{-1} B_{0}^{2}+\beta \omega p B_{-1} B_{1}^{2}-10 \epsilon \omega p q^{2} B_{-1}-3 \epsilon \omega p q^{2} B_{1}-10 \gamma k^{2} \omega p B_{-1}\\ -3 \gamma k^{2} \omega p B_{1}-10 \kappa \omega p^{3} B_{-1}-3 \kappa \omega p^{3} B_{1}-10 \omega p^{2} r \sigma B_{-1}-3 \omega p^{2} r \sigma B_{1}\\ +2 \alpha \omega p B_{-1} B_{0}+\alpha \omega p B_{0} B_{1}-\beta p B_{-1}^{2} B_{1}-\beta p B_{-1} B_{0}^{2}-\beta p B_{-1} B_{1}^{2}-\beta p B_{0}^{2} B_{1}\\ +\beta \omega p B_{0}^{2} B_{1}+\epsilon p q^{2} B_{-1}+\epsilon p q^{2} B_{1}+\gamma k^{2} p B_{-1}+\gamma k^{2} p B_{1}+\kappa p^{3} B_{-1}+\kappa p^{3} B_{1}\\ +p^{2} r \sigma B_{-1}+p^{2} r \sigma B_{1}-\alpha p B_{-1} B_{0}-\alpha p B_{0} B_{1}+2 \omega r B_{-1}+\omega r B_{1}-r B_{-1}\\ -r B_{1}=0. \end{array}$
Employing the Maple package to solve the above system of equations furnishes
$\begin{matrix} {} & {{B}_{-1}}=0,{{B}_{0}}=-\frac{\alpha }{2\beta },{{B}_{1}}=\pm \frac{1}{\beta }\sqrt{\frac{3{{\alpha }^{2}}\omega p-12\omega r\beta }{4\omega p-2p}}, \\ {} & \omega =\frac{4\beta \gamma {{k}^{2}}p+4\beta \kappa {{p}^{3}}+4\beta {{p}^{2}}r\sigma +{{\alpha }^{2}}p+4\beta p{{q}^{2}}\epsilon -4\beta r}{8\beta p\left( \gamma {{k}^{2}}+\kappa {{p}^{2}}+pr\sigma +{{q}^{2}}\epsilon \right)}. \\ \end{matrix}$
Thus, we gain the solution of (3) related to (58) as
$u\left( t,x,y,z \right)={{B}_{0}}+{{B}_{1}}\text{cn}(\xi |\omega )$
Fig. 9. Solitary wave depiction of periodic solution (59) at t=1 and x=0 for n=1. |
Fig. 10. Solitary wave depiction of periodic solution (59) at t=1 and x=1 for n=1. |
3.3.2. Snoidal wave solutions
We consider the NODE (14), for n=1 and as shown earlier, the balancing procedure yields m=1. Then the assumed solution (56) becomes
$\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)={{B}_{-1}}W{{\left( \xi \right)}^{-1}}+{{B}_{0}}+{{B}_{1}}W\left( \xi \right).$
Following the same procedure as in the previous section but invoking (53), we obtain the values of B−1, B0, B1 and ω as
$\begin{matrix} {} & {{B}_{-1}}=0,{{B}_{0}}=-\frac{\alpha }{2\beta },{{B}_{1}}=\mp \frac{1}{\beta }\sqrt{\frac{3{{\alpha }^{2}}\omega p-12\beta \omega r}{2\omega p+2p}}, \\ {} & \omega =\frac{-4\beta \gamma {{k}^{2}}p-4\beta \kappa {{p}^{3}}-4\beta {{p}^{2}}r\sigma -{{\alpha }^{2}}p-4\beta p{{q}^{2}}\epsilon +4\beta r}{4\beta p\left( \gamma {{k}^{2}}+\kappa {{p}^{2}}+pr\sigma +2{{q}^{2}}\epsilon \right)}. \\ \end{matrix}$
Hence the solution of (3) related to (61) is
$u\left( t,x,y,z \right)={{B}_{0}}+{{B}_{1}}\text{sn}(\xi |\omega ),$
Fig. 11. Solitary wave depiction of periodic solution (62) at t=1 and x=2 for n=1. |
Fig. 12. Solitary wave depiction of periodic solution (62) at t=−1 and x=6 for n=1. |
3.3.3. Dnoidal wave solutions
We consider the NODE (14), for n=1 and as shown earlier, using the balancing procedure we get m=1 and so the assumed solution (56) is
$\text{ }\!\!\Phi\!\!\text{ }\left( \xi \right)={{B}_{-1}}W{{\left( \xi \right)}^{-1}}+{{B}_{0}}+{{B}_{1}}W\left( \xi \right).$
Following the same procedure as in the previous section but using (54), the values of B−1, B0, B1 and ω are
$\begin{matrix} {} & {{B}_{-1}}=0,{{B}_{0}}=-\frac{\alpha }{2\beta },{{B}_{1}}=\frac{1}{2\beta }\sqrt{-\frac{6{{\alpha }^{2}}p-24\beta r}{p\left( \omega -2 \right)}}, \\ {} & \omega =\frac{8\beta \gamma {{k}^{2}}p+8\beta \kappa {{p}^{3}}+8\beta {{p}^{2}}r\sigma +8\epsilon \beta p{{q}^{2}}-{{\alpha }^{2}}p+4\beta r}{4\beta p\left( \gamma {{k}^{2}}+{{p}^{2}}\kappa +r\sigma p+{{q}^{2}}\epsilon \right)}. \\ \end{matrix}$
Hence, the solution of (3) corresponding to (64) is
$u\left( t,x,y,z \right)={{B}_{0}}+{{B}_{1}}\text{dn}(\xi |\omega ),$
4. Graphical depiction and application of results
Solitary waves are referred to as localized gravity waves whose coherence are maintained and thus ensure their visibility through properties of nonlinear hydrodynamics. A solitary wave usually possess a finite amplitude and also propagates with a constant speed as well as constant shape. Moreover, solitons are a special class of solitary waves possessing an elastic scattering property: they are found to constantly retain their shapes along with speed even after experiencing collision with one another. The results secured for (3 + 1)D-gZKe (3) in this paper are solitonic as can be seen in their various graphical representations by considering some particular cases of n. Therefore, we depicted the dynamics of bright soliton solution (36) with Fig. 1 in the structure of 3D plot, density plot and 2D plots with dissimilar values of the employed constant parameters as ϵ=1, κ=1, γ=10, α=1, a=105, b=0.5, c=20, σ=1 for n=1 with −5≤y, z≤5 where t=0 and x=0.5. Besides, Fig. 2 further gives the streaming pattern of (36) in 3D, density alongside 2D plots for n=2 with unalike values of the included parameters as ϵ=0, κ=1, γ=10, α=1, a=105 b=1.5, c=20, σ=1 where −5≤y, z≤5 as well as t=1 and x=2. Furthermore, bright soliton solution (37) is represented with 3D, density and 2D plots form in Fig. 3 with diverse values of the included parameters as ϵ=1, κ=1, γ=10, α=1, a=105 b=0.4, c=20, σ=1 for n=1 with −5≤y, z≤5 where t=1 and x=1. Solitary wave solution (37) is also exhibited graphically in Fig. 4 in 3D plot, density plot alongside 2D plot with varying values of the parameters as ϵ=0, κ=1, γ=10, α=1, a=105 b=1.8, c=20, σ=1 for n=2 where −5≤y, z≤5 with t=−1 and x=2. We further depicted soliton solution (48) via 3D plot, density plot together with 2D plot in Fig. 5 with diverse parameter values as ϵ=0, β=0, κ=4, γ=10, α=8, a=−15, b=5, c=300, σ=1 for n=1 where −5≤y, z≤5 with t=5 and x=1. Fig. 6 shows the streaming behaviour of (48) with 3D plot, density plot as well as 2D plot format with varying values of the involved parameters as ϵ=0, κ=−40, β=0, γ=100, α=80, a=10 b=5, c=300, σ=10 for n=2 where −5≤y, z≤5 with t=10 and x=−10. Singular soliton solution (49) is depicted with 3D plot, density plot and 2D plot in Fig. 7 with unalike values of parameters as ϵ=1, κ=1, β=0.6, γ=10, α=1, a=105 b=20, c=20, σ=10 for n=1 where −5≤y, z≤5 with t=1 and x=2. Moreover, we displayed the dynamics of the singular solution in Fig. 8 in 3D, density and 2D plots for n=2 with varying values of parameters as ϵ=0, κ=1, β=0.9, γ=10, α=1, a=105 b=20, c=20, σ=1 where −5≤y, z≤5 with t=3 and x=−4. Cnoidal wave solution (59) of (3) which is a periodic soliton solution is represented with 3D, density together with 2D plots in Fig. 9 with dissimilar values of parameters as ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 with −10≤y, z≤10 whereas t=1 and x=0. Besides, Fig. 10 depicted (59) further with 3D plot, density plot and 2D plot with diverse values of parameters as ϵ=1, κ=1, β=1.01, γ=0, α=1, p=−1 q=1, r=1, σ=1 with −15≤y, z≤15 where t=1 and x=1. Snoidal wave solution (62) (periodic soliton) is considered and depicted in Fig. 11 in 3D, density and 2D format with varying values of parameters as ϵ=1, κ=1, β=1, γ=0, α=0, p=−1 q=1, r=1, σ=1 where −6≤y, z≤6 with t=1 and x=2. In addition, periodic soliton solution (62) is further represented in Fig. 12 via 3D plot, density plot as well as 2D plot with dissimilar values of the parameters as ϵ=1, κ=1, β=1.01, γ=0, α=1, p=−1 q=1, r=1, σ=1 where −10≤y, z≤10 alongside t=−1 and x=6. The depiction of dnoidal wave solution (65) is given via Fig. 13 in 3D, density as well as 2D plots with unalike parametric values ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 whereas −5≤y, z≤5 together with variables t=1.3 and x=0. Finally, we exhibited the wave influence of solitary wave solution (65) in Fig. 14 via 3D plot, density along with 2D plots using diverse involved parameters with assigned values ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 whereas −3≤y, z≤3 where we have variables t=−0.4 with x=1.
Fig. 13. Solitary wave depiction of periodic solution (65) at t=1.3 and x=0 for n=1. |
Fig. 14. Solitary wave depiction of periodic solution (65) at t=−0.4 and x=1 for n=1. |
4.1. Application of results in science and engineering
We observe that various hyperbolic-trigonometric solutions are secured in this study as we known from Table 3.3 that when ω→0, cn(ξ|ω)→cos(ξ), sn(ξ|ω)→sin(ξ) and dn(ξ|ω)→1 while as ω→1, cn(ξ|ω)→sech(ξ), sn(ξ|ω)→tanh(ξ) and dn(ξ|ω)→sech(ξ). Hyperbolic functions possess diverse, well-recognized uses in applied science, particularly in the theory of charts (mercator’s projection), as well as in mechanics (strains). In addition, their applications to electrical engineering within recent years have become evident. Wherever a line, or series of lines having uniform linear constants is/are met with, an immediate field of usefulness for hyperbolic functions presents itself, especially in a case of high-frequency alternating-current lines [48]. Besides, they are used to describe the shape of a rope hanging from two points and the theory of electric conductors. They are also useful in unravelling the formation of satellite rings around planets and application regarding the theory of special relativity. Moreover, most of the solitary wave solutions obtained exhibit constant multi-peak soliton wave profiles as is observed in their respective three-dimensional wave depictions. Moreover, they maintain periodic transverse waves with uniform amplitude and frequency as is evidently revealed via their two-dimensional profiles. Transverse waves (see the sketch of the wave in Fig. 15) possess motion in which all points on a wave oscillate along paths at right angles to the direction of the wave’s advance. Moreover, these waves may also appear complex, in which case we have the loops representing them, comprising two or more sine or cosine curves. Application of these waves is found in seismic S (secondary) waves, ripples on the water surface and electromagnetic (such as radio and light) waves. Lately, added to the fact that transverse oscillations are useful in the electromagnetic spectrum (ranging from the shorter wavelengths such as in gamma as well as X-rays to the longer wavelengths that includes microwaves together with broadcast radio waves) (see Fig. 16) which by extension has its application in the field of medicine and remote sensing, solar physicists have discovered transverse oscillations in the Hα on-disk and quiet-solar chromospheric mottles. In this case, transverse wave properties such as, velocity, amplitude and phase speed are engaged in estimating plasma parameters along waveguide by imploring magnetohydrodynamics technique [50]. These oscillations are later interpreted in terms of propagating and standing magnetohydrodynamic kink waves. Wave characteristics including the maximum transverse velocity amplitude and the phase speed are measured as a function of distance along the structures length. Furthermore, solar magneto-seismology is applied to these measured parameters with a view to achieving diagnostic information on key plasma parameters (such as density, magnetic field, flow speed, temperature) of these localized wave-guides. Thus, solitary wave solutions as well as transverse waves like the type secured in the results from the study of (3 + 1)D-gZKe (3) applies in physical sciences and engineering.
Fig. 15. A schematic representation of a labeled transverse wave. |
Fig. 16. Graphical portrayal of the electromagnetic spectrum [49]. |
4.2. Solitary waves and their impacts in ocean engineering
Ocean engineering in detail is defined as a branch of technological investigation dealing with the design as well as operations of man-made systems existent in the ocean along with other marine bodies. It makes provision for an essential link that is existent between other disciplines of oceanography inclusive of chemical alongside physical oceanography, marine biology as well as geophysics and marine geology. The innovations in the equipment design and instrumentation accomplished by ocean engineers have restructured the oceanography field. In the same vein, the oceanographers’ point of interest has been to drive the demand for design skills as well as the technical expertise of ocean engineers. Further, in an actual sense, ocean engineering combines several other types of engineering which include a merge of electrical, mechanical, acoustical, civil along with chemical engineering skills and techniques in conjunction with a fundamental understanding of how the oceans operate. In addition to designing and building instruments that must outlast the wear and tear of frequent use, ocean engineers must also design instruments that can pull through the harsh realities surrounding the ocean environment.
The ocean engineers are equipped with a detailed understanding regards to ocean policies alongside various areas of concern for technological requirements. These along with other modern navigators have further established engineering wonders that are found across the seas. There have been in existence, various technologically advanced systems and new products which have been developed in various countries across the globe under the adjuration of the ocean engineering departments. In addition, communications, underwater navigations together with positioning have been active areas of research in this branch of technology. Besides this fact, on the priority list of the companies as well as diverse firms all over who deal in this field of engineering are the development and creation of an unmanned underwater vehicle [53].
Internal solitary waves also called internal soliton [54] are all-pervasive in marginal seas and coastal regions of the world’s oceans. In most cases, they evolve from the nonlinear steepening of internal tides which are generated when tidal currents flow over abrupt topography. This includes shelf breaks or submarine ridges. Internal solitary waves constitute a major hazard to submarine navigation and marine engineering. It also has significant impacts on marine ecosystems alongside fishery activity [56]. See Fig. 17, Fig. 18, Fig. 19, Fig. 20 for dynamic views of ocean internal solitary waves.
Fig. 17. Schematic representation of Internal Solitary Wave (ISW) sea surface manifestations in synthetic aperture radar images with regards to an unperturbed background. A comparison between the mode-1 and mode-2 vertical structures is illustrated for waves propagating rightwards along the ocean pycnocline. From top to bottom, a measure for ${{\sigma }_{0}}$ is seen to depart from the background owing to the convergence and divergence patterns induced by the waves’ surface currents-all of which reverse between mode-1 and mode-2 waves [55]. |
Fig. 18. Echogram representation of a large amplitude internal solitary wave [58]. The thick white bell-shaped structure represent the profile of a KdV solitary wave which can be calculated using the background stratifications. |
Fig. 19. Schematic plot of remote sensing mechanism of internal solitary waves [60]. |
Fig. 20. Echogram view of Internal Solitary Waves (ISWs) [58]. |
Various researchers have been able to study the significance of solitary waves concerning the work of ocean engineers such as underwater navigations. For instance, in Zhang and Chwang [57], the authors investigated the phenomenon of a succession of upstream-advancing solitary waves generated by underwater disturbances that steadily move with a trans-critical velocity in two-dimensional shallow water channels. It was established that on utilizing a certain numerical technique, connectivities between celerity and amplitude, as well as that, between the period of generation of solitary waves and amplitude can be accurately simulated and these are in good agreement with predictions of theoretical formulae. Moreover, they examined the dependence of solitary wave radiation on the blockage as well as on the body shape. This consequently furnishes collateral evidence of the experimental findings which reveals that the blockage performs a key role in the generation of solitary waves. Thus, the period of generation decreases while the amplitude increases as the blockage coefficient also increase. Besides, it was discovered that the shape of an underwater object in a viscous flow, has a significant impact on the generation of solitary waves due to the viscous effect in the boundary layer. It was also revealed that, if a change in body shape results in increasing the region of the viscous boundary layer, the viscous effect is further enhanced and so does the disturbance force. In consequence, the amplitudes of solitary waves increase [57] (see Fig. 18).
Furthermore, in Apel et al. [59], nonlinear internal waves in the ocean are discussed. The discussion was addressed from the viewpoint of soliton theory and also the standpoint of experimental measurements. They first began with a brief description of theoretical models for internal solitary waves in the ocean and also contemplated copious nonlinear analytical solutions of some nonlinear wave equations such as the well-known Boussinesq as well as Korteweg-de Vries equations. In addition, certain generalizations such as effects of cubic nonlinearity, cylindrical divergence, Earths rotation, shear flows, dissipation, and others were also examined with the outline of recent theoretical models for strongly nonlinear internal waves. The data utilized comprised radar and optical images, stationary measurements of waveforms, dispersion characteristics and propagation speeds with the various highlights of action of internal solitons on sound wave propagation. In consequence, examples of experimental evidence for the existence of solitons in the upper ocean were established and presented. Some recent theoretical and observational results that are useful for mainstream oceanographers and theoreticians are explicated also in the study.
Moreover, diverse aspects of internal solitary waves inclusive of field data, internal solitary wave action on acoustic wave propagation together with theoretical models were outlined. Various natural observations ascertained that extensive internal solitary wave or their trains usually referred to as solibores are in existence both in shallow as well as deep ocean areas. Also, on many occasions, their parameters are close to those that were theoretically predicted. In another development, they were able to produce transport particles, strong vertical mixing which has an effect on turbulence along with biological life and even interferes possibly with underwater navigation. In addition, they affect surface waves and consequently create surface “slicks” which are visible by radars, optical devices and even sometimes by the naked eye. They also have some level of influence on the propagation of acoustic signals in water which may result in the formation of specific conditions for ducting sound waves. In the last decades, substantial theoretical and observational investigations have established that internal solitary waves are a widespread phenomenon throughout the oceans and most especially in coastal regions where they are generated through barotropic tides [61]. They perhaps, mop up an appreciable part of the total tidal energy. In consequence, internal solitons are, as a matter of fact, the “extremes” of the internal wave spectrum, with their magnitudes reaching possibly several dozens of meters.
5. Conserved quantities of (3 + 1)D-gZKe (3)
This section furnishes the derivation of conserved currents for the (3 + 1)D-gZKe (3). We intend to construct the Lagrangian of the equation using the Helmholtz criteria and consequently engage Noether’s theorem [62] to secure its conserved vectors. In addition, we shall also employ the general multipier technique [38], [63] to achieve the conserved quantities of the equation under study and then compare our results in both cases. We present a brief outline of these techniques and some useful definitions.
5.1. Preliminaries
$\text{ }\!\!\Xi\!\!\text{ }\equiv \left( \text{ }\!\!\Xi\!\!\text{ }\left( t,x,u,\partial u,\ldots,{{\partial }^{\mathcal{N}}}u \right),\ldots,{{\text{ }\!\!\Xi\!\!\text{ }}^{\mathcal{M}}}\left( t,x,u,\partial u,\ldots,{{\partial }^{\mathcal{N}}}u \right) \right)=0,$
with t, $x=\left( {{x}^{1}},\ldots,{{x}^{n}} \right)$ standing for the independent variables, n≥1 and $u=\left( {{u}^{1}},\ldots,{{u}^{m}} \right)$ representing the dependent variables with m≥1. Besides, $\partial u=\left( {{u}_{t}},{{u}_{{{x}^{1}}}}\ldots,{{u}_{{{x}^{n}}}} \right)$ denotes the partial derivatives of u with regards to the given t, x, whereas ${{\partial }^{k}}u,k\ge 2$ denotes the kth-order partial derivatives. We also note that the space of all locally smooth solutions which are associated with $u\left( t,x \right)$ of the system is represented with Δ.
A local conservation law of any presented system of PDE (66) is given as a local continuity equation
$\left( {{D}_{t}}{{C}^{t}}+{{D}_{x}}\cdot {{C}^{x}} \right){{|}_{\text{ }\!\!\Delta\!\!\text{ }}}=0,$
which holds for the system on the entire solution space Δ with $\left( {{D}_{t}},{{D}_{x}} \right)$ standing for the total derivatives of (t, x) accordingly and Div=Dx· which is the spatial divergence denoting the vector dot product. Furthermore, ${{C}^{t}}\left( t,x,u,\partial u,\ldots,{{\partial }^{r}}u \right)$ denotes the conserved density while ${{C}^{x}}=\left( {{C}^{1}}\left( t,x,u,\partial u,\ldots,{{\partial }^{r}}u \right),\ldots,{{C}^{n}}\left( t,x,u,\partial u,\ldots,{{\partial }^{r}}u \right) \right)$ represents the spatial flux. Thus, the pair $\left( {{C}^{t}},{{C}^{x}} \right)={{\text{ }\!\!\Phi\!\!\text{ }}^{*}}$ is referred to as the conserved current.
A system of PDE given in (66) is said to be locally variational if it can be expressed by the Euler-Lagrange equations
$0=\text{ }\!\!\Xi\!\!\text{ }\equiv {{E}_{u}}{{\left( L \right)}^{t}},$
(where t stands for the transpose) for some differential function $\mathcal{L}\left( t,x,u,\partial u,\ldots,{{\partial }^{k}}u \right)$ which we call a Lagrangian. The operator Eu is defined as
${{E}_{u}}\equiv \frac{\delta }{{{\delta }_{u}}}=\frac{\partial }{\partial u}-{{D}_{t}}\frac{\partial }{\partial {{u}_{t}}}-{{D}_{x}}\frac{\partial }{\partial {{u}_{x}}}+{{D}_{t}}{{D}_{x}}\frac{\partial }{\partial {{u}_{tx}}}+\ldots.$
Next, we consider a Lemma
Lemma 5.1 The relation $\text{ }\!\!\Xi\!\!\text{ }={{E}_{u}}{{\left( L \right)}^{t}}$ holds for some Lagrangian $\mathcal{L}\left( t,x,u,\partial u,\ldots,{{\partial }^{k}}u \right)$ iff
${{\delta }_{v}}{{\text{ }\!\!\Xi\!\!\text{ }}^{t}}=\delta _{v}^{*}{{\text{ }\!\!\Xi\!\!\text{ }}^{t}}$
also holds for all differential functions $v\left( t,x \right)$. See the proof in Anco [64].
The criteria presented in (70) for a system of PDE Ξ=0 to be locally variational asserts that the linearization of Ξt must be self-adjoint. Thus, we have it that
$\frac{\partial \text{ }\!\!\Xi\!\!\text{ }}{\partial \left( {{\partial }^{k}}u \right)}={{\left( -1 \right)}^{k}}{{\left( E_{u}^{\left( k \right)}\left( \text{ }\!\!\Xi\!\!\text{ } \right) \right)}^{t}},k=0,1,\ldots,\mathcal{N}$
with $\mathcal{N}$ regarded as the differential order of the system of PDE Ξ=0. These equations are regarded as the Helmholtz conditions/criteria. We note here that the presence of the included transpose means that the Helmholtz criteria can not hold if Ξ as well as u possess different number of components. Therefore, it is necessary that a system of PDE has differential order $\mathcal{N}$ to be even as well as number $\mathcal{M}$ of the PDEs to be same with m number of dependent variables for it to be locally variational. Moreover, $E_{v}^{k}$ stands for higher Euler operators that are expressed as
$\begin{matrix} E_{v}^{\left( s \right)}\left( \text{ }\!\!\Xi\!\!\text{ } \right) & = & \frac{\partial \text{ }\!\!\Xi\!\!\text{ }}{\partial \left( {{\partial }^{s}}v \right)}-\left( \begin{matrix} s+1 \\ s \\ \end{matrix} \right)D\cdot \frac{\partial \text{ }\!\!\Xi\!\!\text{ }}{\partial \left( {{\partial }^{s+1}}v \right)}+\ldots \\ {} & {} & +\left( \begin{matrix} w \\ s \\ \end{matrix} \right){{\left( -D \right)}^{w-s}}\cdot \frac{\partial \text{ }\!\!\Xi\!\!\text{ }}{\partial \left( {{\partial }^{s}}v \right)}, \\ \end{matrix}$
where s=1,…,w. The moment the Helmholtz criteria (71) are satisfied, we can recover a Lagrangian from system $\text{ }\!\!\Xi\!\!\text{ }=\left( {{\text{ }\!\!\Xi\!\!\text{ }}^{1}},\ldots,{{\text{ }\!\!\Xi\!\!\text{ }}^{\mathcal{M}}} \right)$ via the general homotopy integral formula presented as
$\mathcal{L}=\underset{0}{\overset{1}{\mathop \int }}\,{{\partial }_{\lambda }}{{u}_{\left( \lambda \right)}}{{\text{ }\!\!\Xi\!\!\text{ }}^{t}}{{|}_{u={{u}_{\left( \lambda \right)}}}}d\lambda.$
Remark 5.1 We can add a complete divergence to $\mathcal{L}$ in (73) in order to secure an equivalent Lagrangian that possesses the lowest possible differential order, that is, $\mathcal{N}/2$.
For a local variational principle given in (68), a variational symmetry also called divergence symmetry [64] is a generator whose prolongation satisfies the invariance condition
$\text{Pr }\!\!\Omega\!\!\text{ }\left( \mathcal{L} \right)={{\xi }^{1}}{{D}_{t}}\mathcal{L}+{{\xi }^{i}}\cdot {{D}_{x}}\mathcal{L}+{{D}_{t}}{{\text{ }\!\!\Psi\!\!\text{ }}^{t}}+{{D}_{x}}\cdot {{\text{ }\!\!\Psi\!\!\text{ }}^{x}},i=2,3,4$
(where $\text{ }\!\!\Omega\!\!\text{ }={{\xi }^{1}}\partial /\partial t+{{\xi }^{i}}\partial /\partial x$) for some differential vector function ${{\text{ }\!\!\Psi\!\!\text{ }}^{x}}$ as well as differential scalar function ${{\text{ }\!\!\Psi\!\!\text{ }}^{t}}$.
A determining condition to gain all multipliers to the characteristic equation $Q=({{Q}_{1}}\left( t,x,u,\partial u,\ldots,{{\partial }^{r}}u \right),\ldots,$ ${{Q}_{\mathcal{M}}}\left( t,x,u,\partial u,\ldots,{{\partial }^{r}}u \right){{)}^{t}}$ is given as [64]
${{E}_{u}}\left( \text{ }\!\!\Xi\!\!\text{ }Q \right)\equiv \delta _{Q}^{*}\text{ }\!\!\Xi\!\!\text{ }+\delta _{\text{ }\!\!\Xi\!\!\text{ }}^{*}Q=0,$
where Eu is the Euler–Lagrange operator as earlier defined in (69). This condition which is needed to hold identically in jet space $J=\left( t,x,u,\partial u,{{\partial }^{2}}u,\ldots \right)$ is seen to be sufficient and necessary for Q to be a multiplier. There exists a corresponding conserved current for each multiplier Q using the explicit relation presented in the Lemma given as
Lemma 5.2 For a regular system of PDE stated in (66), each multiplier Q furnishes a conserved current which is explicitly presented by a homotopy integral given as
$\begin{matrix} {} & {{\overset{}{\mathop{C}}\,}^{t}}=\underset{0}{\overset{1}{\mathop \int }}\,\left( \underset{l=0}{\overset{k-1}{\mathop \sum }}\,{{\partial }_{\lambda }}{{\partial }^{l}}{{u}_{\left( \lambda \right)}}\cdot \left( {{E}_{{{\partial }^{l}}{{\partial }_{t}}u}}\left( \text{ }\!\!\Xi\!\!\text{ }Q \right) \right){{|}_{u={{u}_{\left( \lambda \right)}}}} \right)d\lambda +{{D}_{x}}\cdot \text{ }\!\!\Theta\!\!\text{ }, \\ {} & {{\overset{}{\mathop{C}}\,}^{x}}=\underset{0}{\overset{1}{\mathop \int }}\,\left( \underset{l=0}{\overset{k-1}{\mathop \sum }}\,{{\partial }_{\lambda }}{{\partial }^{l}}{{u}_{\left( \lambda \right)}}\cdot \left( {{E}_{{{\partial }^{l}}{{\partial }_{t}}u}}\left( \text{ }\!\!\Xi\!\!\text{ }Q \right) \right){{|}_{u={{u}_{\left( \lambda \right)}}}} \right)d\lambda -{{D}_{t}}\cdot \text{ }\!\!\Theta\!\!\text{ }+{{D}_{x}}\cdot \text{ }\!\!\Gamma\!\!\text{ }, \\ \end{matrix}$
along a homotopy curve ${{u}_{\left( \lambda \right)}}\left( t,x \right)$,(where $\text{ }\!\!\Theta\!\!\text{ }\left( t,x,u,\partial u,\ldots,{{\partial }^{r-1}}u \right)$ as well as function $\text{ }\!\!\Gamma\!\!\text{ }\left( t,x,u,\partial u,\ldots,{{\partial }^{r-1}}u \right)$ are some differential vector functions) together with ${{u}_{\left( 1 \right)}}=u$ such that $\left( \text{ }\!\!\Xi\!\!\text{ }Q \right){{|}_{u={{u}_{0}}}}$ is nonsingular. In this case $k=\max \left( r,\mathcal{N} \right)$.
5.2. The standard multiplier technique
The multiplier method is advantageous in the sense that it works for any PDE either with or without variational principle(s). In other words, it does not require the availability of variational principle before the conserved vectors of a given PDE are obtained via the multiplier method. We consider the first-order multiplier
$\text{ }\!\!\Lambda\!\!\text{ }\left( t,x,y,z,u,{{u}_{t}},{{u}_{x}},{{u}_{y}},{{u}_{z}} \right)$
in order to obtain the conservation laws of (3). Using the relation in (75) alongside Lie algorithm, one secures the value of Λ as
$\text{ }\!\!\Lambda\!\!\text{ }\left( t,x,y,z,u,{{u}_{t}},{{u}_{x}},{{u}_{y}},{{u}_{z}} \right)={{C}_{1}}u+H\left( y,z \right),$
where C1 is an arbitrary constant and $H\left( y,z \right)$ is an arbitrary function depending on y and z. Thus, using Lemma 5.2 one can achieve conserved vectors corresponding to C1 and $H\left( y,z \right)$ accordingly as
$\begin{array}{l} \begin{array}{l} T^{1}=\frac{1}{6}\left[2 u u_{x x}-u_{x}^{2}\right] \sigma+\frac{1}{2} u^{2}, \\ X^{1}=\frac{1}{12 n^{3}+42 n^{2}+42 n+12}\left\{\left(6 \beta n^{2}+15 \beta n+6 \beta\right) u^{2 n+2}+\left(12 \alpha n^{2}\right.\right. \\ +18 \alpha n+6 \alpha) u^{n+2}+\left[\left(4 \epsilon u_{y y}+4 \gamma u_{z z}+12 \kappa u_{x x}+8 \sigma u_{t x}\right) u-2 \gamma u_{z}^{2}\right. \\ \left.-6 \kappa u_{x}^{2}-4 \sigma u_{x} u_{t}-2 \epsilon u_{y}^{2}\right] n^{3}+\left[\left(14 \epsilon u_{y y}+14 \gamma u_{z z}+42 \kappa u_{x x}\right.\right. \\ \left.\left.+28 \sigma u_{t x}\right) u-7 \gamma u_{z}^{2}-21 \kappa u_{x}^{2}-14 \sigma u_{x} u_{t}-7 \epsilon u_{y}^{2}\right] n^{2}+\left[\left(14 \epsilon u_{y y}\right.\right. \\ \left.+14 \gamma u_{z z}+42 \kappa u_{x x}+28 \sigma u_{t x}\right) u-7 \gamma u_{z}^{2}-21 \kappa u_{x}^{2}-14 \sigma u_{x} u_{t} \\ \left.-7 \epsilon u_{y}^{2}\right] n+\left(4 \epsilon u_{y y}+4 \gamma u_{z z}+12 \kappa u_{x x}+8 \sigma u_{t x}\right) u-2 \gamma u_{z}^{2} \\ \left.-6 \kappa u_{x}^{2}-4 \sigma u_{x} u_{t}-2 \epsilon u_{y}^{2}\right\} \text {, } \\ Y^{1}=\frac{2}{3} u \epsilon u_{x y}-\frac{1}{3} \epsilon u_{x} u_{y}, \\ Z^{1}=\frac{2}{3} u \gamma u_{x z}-\frac{1}{3} \gamma u_{x} u_{z} \\ T^{2}=\frac{1}{3} H(y, z)\left[\sigma u_{x x}+3 u\right] \\ X^{2}=\frac{1}{6 n^{3}+21 n^{2}+21 n+6}\left\{\left(3 \beta n^{2}+9 \beta n+6 \beta\right) H(y, z) u^{2 n+1}+\left[2 \epsilon n^{3} u\right.\right. \\ \left.+7 \epsilon n^{2} u+7 \epsilon n u+2 u \epsilon\right] H_{y y}(y, z)+\left(2 \gamma n^{3} u+7 \gamma n^{2} u+7 \gamma n u\right. \\ +2 u \gamma) H_{z z}(y, z)+\left(6 \alpha n^{2}+15 n \alpha+6 \alpha\right) H(y, z) u^{n+1}+\left[-2 \epsilon n^{3} u y\right. \\ \left.-7 \epsilon n^{2} u_{y}-7 \epsilon n u_{y}-2 \epsilon u_{y}\right] H_{y}(y, z)+\left[-2 \gamma n^{3} u_{z}-7 \gamma n^{2} u_{z}-7 \gamma n u_{z}\right. \\ \left.-2 \gamma u_{z}\right] H_{z}(y, z)+\left\{\left(2 \epsilon u_{y y}+2 \gamma u_{z z}+6 \kappa u_{x x}+4 \sigma u_{t x}\right) n^{3}+\left[7 \epsilon u_{y y}\right.\right. \\ \left.+7 \gamma u_{z z}+21 \kappa u_{x x}+14 \sigma u_{t x}\right] n^{2}+\left(7 \epsilon u_{y y}+7 \gamma u_{z z}+21 \kappa u_{x x}\right. \\ \left.\left.\left.+14 \sigma u_{t x}\right) n+2 \gamma u_{z z}+6 \kappa u_{x x}+4 \sigma u_{t x}+2 \epsilon u_{y y}\right\} H(y, z)\right\} \\ Y^{2}=-\frac{1}{3} \epsilon\left[H_{y}(y, z) u_{x}-2 H(y, z) u_{x y}\right] \\ Z^{2}=-\frac{1}{3} \gamma\left[H_{z}(y, z) u_{x}-2 H(y, z) u_{x z}\right] \end{array}\\ T^{1}=\frac{1}{6}\left[2 u u_{x x}-u_{x}^{2}\right] \sigma+\frac{1}{2} u^{2},\\ X^{1}=\frac{1}{12 n^{3}+42 n^{2}+42 n+12}\left\{\left(6 \beta n^{2}+15 \beta n+6 \beta\right) u^{2 n+2}+\left(12 \alpha n^{2}\right.\right.\\ +18 \alpha n+6 \alpha) u^{n+2}+\left[\left(4 \epsilon u_{y y}+4 \gamma u_{z z}+12 \kappa u_{x x}+8 \sigma u_{t x}\right) u-2 \gamma u_{z}^{2}\right.\\ \left.-6 \kappa u_{x}^{2}-4 \sigma u_{x} u_{t}-2 \epsilon u_{y}^{2}\right] n^{3}+\left[\left(14 \epsilon u_{y y}+14 \gamma u_{z z}+42 \kappa u_{x x}\right.\right.\\ \left.\left.+28 \sigma u_{t x}\right) u-7 \gamma u_{z}^{2}-21 \kappa u_{x}^{2}-14 \sigma u_{x} u_{t}-7 \epsilon u_{y}^{2}\right] n^{2}+\left[\left(14 \epsilon u_{y y}\right.\right.\\ \left.+14 \gamma u_{z z}+42 \kappa u_{x x}+28 \sigma u_{t x}\right) u-7 \gamma u_{z}^{2}-21 \kappa u_{x}^{2}-14 \sigma u_{x} u_{t}\\ \left.-7 \epsilon u_{y}^{2}\right] n+\left(4 \epsilon u_{y y}+4 \gamma u_{z z}+12 \kappa u_{x x}+8 \sigma u_{t x}\right) u-2 \gamma u_{z}^{2}\\ \left.-6 \kappa u_{x}^{2}-4 \sigma u_{x} u_{t}-2 \epsilon u_{y}^{2}\right\} \text {, }\\ Y^{1}=\frac{2}{3} u \epsilon u_{x y}-\frac{1}{3} \epsilon u_{x} u_{y},\\ Z^{1}=\frac{2}{3} u \gamma u_{x z}-\frac{1}{3} \gamma u_{x} u_{z}\\ T^{2}=\frac{1}{3} H(y, z)\left[\sigma u_{x x}+3 u\right]\\ X^{2}=\frac{1}{6 n^{3}+21 n^{2}+21 n+6}\left\{\left(3 \beta n^{2}+9 \beta n+6 \beta\right) H(y, z) u^{2 n+1}+\left[2 \epsilon n^{3} u\right.\right.\\ \left.+7 \epsilon n^{2} u+7 \epsilon n u+2 u \epsilon\right] H_{y y}(y, z)+\left(2 \gamma n^{3} u+7 \gamma n^{2} u+7 \gamma n u\right.\\ +2 u \gamma) H_{z z}(y, z)+\left(6 \alpha n^{2}+15 n \alpha+6 \alpha\right) H(y, z) u^{n+1}+\left[-2 \epsilon n^{3} u y\right.\\ \left.-7 \epsilon n^{2} u_{y}-7 \epsilon n u_{y}-2 \epsilon u_{y}\right] H_{y}(y, z)+\left[-2 \gamma n^{3} u_{z}-7 \gamma n^{2} u_{z}-7 \gamma n u_{z}\right.\\ \left.-2 \gamma u_{z}\right] H_{z}(y, z)+\left\{\left(2 \epsilon u_{y y}+2 \gamma u_{z z}+6 \kappa u_{x x}+4 \sigma u_{t x}\right) n^{3}+\left[7 \epsilon u_{y y}\right.\right.\\ \left.+7 \gamma u_{z z}+21 \kappa u_{x x}+14 \sigma u_{t x}\right] n^{2}+\left(7 \epsilon u_{y y}+7 \gamma u_{z z}+21 \kappa u_{x x}\right.\\ \left.\left.\left.+14 \sigma u_{t x}\right) n+2 \gamma u_{z z}+6 \kappa u_{x x}+4 \sigma u_{t x}+2 \epsilon u_{y y}\right\} H(y, z)\right\}\\ Y^{2}=-\frac{1}{3} \epsilon\left[H_{y}(y, z) u_{x}-2 H(y, z) u_{x y}\right]\\ Z^{2}=-\frac{1}{3} \gamma\left[H_{z}(y, z) u_{x}-2 H(y, z) u_{x z}\right] \end{array}$
5.3. Noether’s theorem using the Helmholtz condition
The classical Noether’s theorem [38], [62] is a technique that achieves the conserved currents of a PDE associated with variational principles. Helmholtz criteria stipulate categorically in clear terms that a PDE must be even-ordered amongst other factors for it to possess a variational principle. We observe that (3 + 1)D-gZKe (3) is odd-ordered and so contravenes the criteria presented by Helmholtz, thus necessitating an action to be taken with a view to recovering an even-ordered form of the equation. Introduction of $u={{v}_{x}}$ into (3) achieves that purpose and hence, we get
${{G}^{*}}\equiv {{v}_{tx}}+\alpha v_{x}^{n}{{v}_{xx}}+\beta v_{x}^{2n}{{v}_{xx}}+\kappa {{v}_{xxxx}}+\sigma {{v}_{txxx}}+\epsilon {{v}_{xxyy}}+\gamma {{v}_{xxzz}}=0,$
which readily has a variational principle. Thus employing the Helmholtz criteria in (71), we get some selected terms from ${{G}^{*}}$ via the estimation of (77), that is
$\begin{aligned} (k=0) \frac{\partial G^{*}}{\partial v} & \equiv E_{v}\left(G^{*}\right)=0 \\ (k=1) \frac{\partial G^{*}}{\partial v_{t}} & \equiv-E_{v}^{(t)}\left(G^{*}\right)=-\left(D_{x}(1)+D_{y}(1)+D_{z}(1)\right) \\ \frac{\partial G^{*}}{\partial v_{x}} & \equiv-E_{v}^{(x)}\left(G^{*}\right)=-\alpha n v_{x}^{n-1} v_{x x}-2 \beta n v_{x}^{2 n-1} v_{x x} \\ \frac{\partial G^{*}}{\partial v_{y}} & \equiv-E_{v}^{(y)}\left(G^{*}\right)=-\left(D_{t}(1)+D_{x}(1)+D_{z}(1)\right) \\ \frac{\partial G^{*}}{\partial v_{z}} & \equiv-E_{v}^{(z)}\left(G^{*}\right)=-\left(D_{t}(1)+D_{x}(1)+D_{y}(1)\right) \\ (k=2) \frac{\partial G^{*}}{\partial v_{x t}} & \equiv E_{v}^{(x t)}\left(G^{*}\right)=1 \\ \frac{\partial G^{*}}{\partial v_{x x}} & \equiv E_{v}^{(x x)}\left(G^{*}\right)=\alpha v_{x}^{n}+\beta v_{x}^{2 n} \\ (k=4) \frac{\partial G^{*}}{\partial v_{t x x x}} & \equiv E_{v}^{(t x x x)}\left(G^{*}\right)=\sigma \\ \frac{\partial G^{*}}{\partial v_{x x x x}} & \equiv E_{v}^{(x x x x)}\left(G^{*}\right)=\kappa \\ \frac{\partial G^{*}}{\partial v_{x x y y}} & \equiv E_{v}^{(x x y y)}\left(G^{*}\right)=\epsilon \\ \frac{\partial G^{*}}{\partial v_{x x z z}} & \equiv E_{v}^{(x x z z)}\left(G^{*}\right)=\gamma \end{aligned}$
Thus, the (3 + 1)D-gZKe (3) is locally variational under the earlier given transformation. Having ascertained that a Lagrangian $\left( \mathcal{L} \right)$ is recovered for (77) whose equivalent minimal differential order is presented as
$\begin{matrix} \mathcal{L} & = & -\frac{1}{2}{{v}_{t}}{{v}_{x}}-\frac{\alpha v_{x}^{n+2}}{\left( n+1 \right)\left( n+2 \right)}-\frac{\beta v_{x}^{2n+2}}{\left( 2n+1 \right)\left( 2n+2 \right)} \\ {} & {} & +\frac{1}{2}\kappa v_{xx}^{2}+\frac{1}{2}\sigma {{v}_{xx}}{{v}_{tx}}+\frac{1}{2}\epsilon {{v}_{xx}}{{v}_{yy}}+\frac{1}{2}\gamma {{v}_{xx}}{{v}_{zz}}, \\ \end{matrix}$
we give a Lemma.
Lemma 5.3 The (3 + 1)D-gZKe (3) forms the Euler-Lagrange equation with the functional
$J\left( v \right)=\underset{0}{\overset{\infty }{\mathop \int }}\,\underset{0}{\overset{\infty }{\mathop \int }}\,\underset{0}{\overset{\infty }{\mathop \int }}\,\underset{0}{\overset{\infty }{\mathop \int }}\,\mathcal{L}\left( t,x,y,z,{{v}_{t}},{{v}_{x}},{{v}_{tx}},{{v}_{xx}},{{v}_{yy}},{{v}_{zz}} \right)dtdxdydz,$
where the conforming function of Lagrangian $\mathcal{L}$ is stated as
$\begin{matrix} \mathcal{L} & = & -\frac{1}{2}{{v}_{t}}{{v}_{x}}-\frac{\alpha v_{x}^{n+2}}{\left( n+1 \right)\left( n+2 \right)}-\frac{\beta v_{x}^{2n+2}}{\left( 2n+1 \right)\left( 2n+2 \right)} \\ {} & {} & +\frac{1}{2}\kappa v_{xx}^{2}+\frac{1}{2}\sigma {{v}_{xx}}{{v}_{tx}}+\frac{1}{2}\epsilon {{v}_{xx}}{{v}_{yy}}+\frac{1}{2}\gamma {{v}_{xx}}{{v}_{zz}}. \\ \end{matrix}$
It is worthy of note to declare here that Lagrangian (78) conforms with the Euler-Lagrangian Eq. (69) and that can easily be ascertained. We achieve variational symmetry Ω by employing symmetry invariance condition expressed as
$\begin{matrix} {} & {} & \text{P}{{\text{r}}^{\left( 2 \right)}}\text{ }\!\!\Omega\!\!\text{ }\mathcal{L}+\mathcal{L}\left[ {{D}_{t}}\left( {{\xi }^{1}} \right)+{{D}_{x}}\left( {{\xi }^{2}} \right)+{{D}_{y}}\left( {{\xi }^{3}} \right)+{{D}_{z}}\left( {{\xi }^{4}} \right) \right] \\ {} & {} & ={{D}_{t}}\left( {{B}^{t}} \right)+{{D}_{x}}\left( {{B}^{x}} \right)+{{D}_{y}}\left( {{B}^{y}} \right)+{{D}_{z}}\left( {{B}^{z}} \right), \\ \end{matrix}$
where second prolongation Pr(2)Ω of Ω can be recovered by (74) with the gauge functions Bt, Bx, By and Bz depending on $\left( t,x,y,z,u \right)$. Separating the monomials in the expanded form of (80) secures a system of sixty-one linear PDEs which are
$\begin{array}{l} \xi_{v}^{1}=0, \quad \xi_{x}^{1}=0, \quad \xi_{v}^{2}=0, \quad \xi_{v}^{3}=0, \quad \xi_{x}^{3}=0, \quad \xi_{v}^{4}=0, \quad \xi_{x}^{4}=0 \\ \eta_{x}=0, \quad \eta_{x x}=0, \quad \xi_{v x}^{1}=0, \quad \xi_{v x}^{3}=0, \quad \xi_{v x}^{4}=0, \quad \xi_{v v}^{1}=0, \quad \xi_{x x}^{1}=0 \\ \xi_{v v}^{2}=0, \quad \xi_{v v}^{3}=0, \quad \xi_{x x}^{3}=0, \quad \xi_{v v}^{4}=0, \quad \xi_{x x}^{4}=0, \quad \xi_{t}^{2}=0, \quad \xi_{t}^{3}=0 \\ \xi_{t}^{4}=0, \quad \xi_{x x}^{3}=0, \quad \alpha \eta_{t}+2 B_{v}^{x}=0, \quad \xi_{y}^{4}+\gamma \xi_{z}^{3}=0, \quad B_{v}^{y}=0, \quad B_{v}^{z}=0 \\ 2 \xi_{y}^{1}+\sigma \xi_{x}^{3}=0, \quad \eta_{v v}-2 \xi_{v x}^{2}=0, \quad 2 \xi_{v}^{1}+B_{v}^{t}=0, \quad \xi_{v}^{1}+B_{v}^{t}=0 \\ \eta_{v v}-2 \xi_{v y}^{3}=0, \quad 2 \eta_{v x}-\xi_{x x}^{2}=0, \quad \sigma \xi_{v}^{2}+\kappa \xi_{v}^{1}=0, \quad B_{x}^{x}+B_{y}^{y}+B_{z}^{z}=0 \\ \sigma \xi_{x}^{4}+2 \gamma \xi_{z}^{1}=0, \quad \gamma \xi_{v z}^{3}+\xi_{v y}^{4}=0, \quad \sigma \xi_{v x}^{3}+\xi_{v y}^{1}=0, \quad \eta_{v v}-2 \xi_{v z}^{4}=0 \\ (\epsilon-2) \xi_{v}^{1}-\epsilon B_{v}^{t}=0,2 \gamma \xi_{v z}^{1}+\sigma \xi_{v x}^{4}=0,2 \eta_{v y}-\xi_{y y}-\sigma \xi_{v t}^{3}-2 \kappa \xi_{x x}^{3}-\gamma \xi_{z z}^{3}=0 \\ 2 \alpha \xi_{v}^{1}-\xi_{v}^{1}+B_{v}^{t}=0, \quad 2 \xi_{y}^{2}+\sigma \xi_{t}^{3}+2 \kappa \xi_{x}^{3}=0, \quad(2 \gamma n-\beta+2 \gamma) \xi_{v}^{1}+\beta B_{v}^{t}=0 \\ \sigma \xi_{t}^{4}+4 \kappa \xi_{x}^{4}+2 \gamma \xi_{z}^{2}=0, \quad(\gamma n-\alpha+2 \gamma) \xi_{v}^{1}+\alpha B_{v}^{t}=0, \quad \sigma \xi_{v}^{2}+\kappa \xi_{v}^{1}+\kappa B_{v}^{t}=0 \\ \sigma \xi_{v t}^{3}+2 \xi_{v y}^{2}+4 \kappa \xi_{v x}^{3}=0, \quad-\sigma \xi_{v t}^{2}+2 \kappa\left(\eta_{v v}-2 \xi_{v x}^{2}\right)=0, \quad 4 \kappa \xi_{v x}^{4}+2 \gamma \xi_{v z}^{2}+\sigma \xi_{v t}^{4}=0 \\ 2 \kappa \eta_{x x}+\gamma \eta_{z z}+\sigma \eta_{t x}+\eta_{y y}=0, \quad \xi_{x}^{2}+\xi_{z}^{4}-2 \eta_{v}-\xi_{t}^{t}-\xi_{y}^{3}+B_{t}^{t}=0 \\ \sigma\left(\eta_{v v}-\xi_{v x}^{2}\right)-4 \kappa \xi_{v x}^{1}-\sigma \xi_{v t}^{1}=0, \quad 4 \kappa \xi_{x}^{1}+\sigma\left(2 \xi_{x}^{2}-\xi_{y}^{3}-\xi_{z}^{4}-2 \eta_{v}+B_{t}^{t}\right)=0 \\ (\alpha+1) \xi_{t}^{1}+(\alpha-1) \xi_{x}^{2}-2 \alpha \eta_{v}+B_{t}^{t}-\xi_{y}^{3}-\xi_{z}^{4}=0, \sigma \xi_{v v}^{x}+2 \kappa \xi_{v v}^{1}=0 \\ 2 \gamma \eta_{v z}-\xi_{y y}^{4}-\sigma \xi_{t x}^{4}-2 \kappa \xi_{x x}^{4}-\gamma \xi_{z z}^{4}=0, \sigma \eta_{v x}-\xi_{y y}^{1}-\sigma \xi_{t x}^{1}-2 \kappa \xi_{x x}^{1}-\gamma \xi_{z z}=0 \\ (\gamma(2 n+2)-\beta) \xi_{x}^{2}-2 \gamma(n+1) \eta_{v}+\beta\left(B_{t}^{t}-\xi_{t}^{1}-\xi_{y}^{3}-\xi_{z}^{4}\right)=0 \\ (\gamma(n+2)-\alpha) \xi_{x}^{2}-\gamma(n+2) \eta_{v}+\alpha\left(B_{t}^{t}-\xi_{t}^{1}-\xi_{y}^{3}-\xi_{z}^{4}\right)=0 \\ (\epsilon-2) \xi_{x}^{2}+(\epsilon-2) \xi_{y}^{3}-\epsilon B_{t}^{t}+\epsilon \xi_{t}^{t}+\epsilon \xi_{z}^{4}+\eta_{v}=0, \\ \kappa \xi_{t}^{1}-\sigma \xi_{t}^{2}-\kappa\left(B_{t}^{t}-2 \eta_{v}+3 \xi_{x}^{2}-\xi_{y}^{3}-\xi_{z}^{4}\right)=0 \\ \sigma \eta_{t v}-\xi_{y y}^{2}-\sigma \xi_{t x}^{2}+4 \kappa \eta_{v x}-2 \kappa \xi_{x x}^{2}-\gamma \xi_{z z}^{2}=0 \end{array}$
Solving the above systems procures the solution
$\begin{matrix} {} & {{\xi }^{1}}={{\mathbf{c}}_{\mathbf{4}}},\quad {{\xi }^{2}}={{\mathbf{c}}_{\mathbf{5}}},\quad {{\xi }^{3}}={{\mathbf{c}}_{\mathbf{1}}}z+{{\mathbf{c}}_{\mathbf{2}}},\quad {{\xi }^{4}}=-{{\mathbf{c}}_{\mathbf{1}}}\gamma y+{{\mathbf{c}}_{\mathbf{3}}}, \\ {} & \eta =F\left( t,z-y \right)+G\left( t,z+y \right),\quad {{B}^{t}}={{F}^{5}}\left( x,y,z \right),\quad {{B}^{z}}={{F}^{5}}\left( t,x,y,z \right), \\ {} & {{B}^{x}}=-\int \left( F_{y}^{1}+F_{z}^{2} \right)dx-\frac{1}{2}\alpha \left( {{F}_{t}}+{{G}_{t}} \right)v+{{F}^{3}}\left( t,y,z \right), \\ {} & {{B}^{y}}={{F}^{4}}\left( t,x,y,z \right), \\ \end{matrix}$
with ${{\mathbf{c}}_{\mathbf{i}}},i=1,\ldots,5$ regarded as arbitrary constants. Arbitrary functions ${{F}^{1}},\ldots,{{F}^{5}}$ only add to the trivial parts of the conserved vectors and so we set them to zero. Consequently, the achieved solutions furnish the Noether symmetries given as,
$\begin{array}{l} \begin{array}{l} \Omega_{1}=\frac{\partial}{\partial t}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0 \\ \Omega_{2}=\frac{\partial}{\partial x}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0, \\ \Omega_{3}=\frac{\partial}{\partial y}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0, \\ \Omega_{4}=\frac{\partial}{\partial z}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0, \\ \Omega_{5}=z \frac{\partial}{\partial y}-\gamma y \frac{\partial}{\partial z}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0, \\ \Omega_{F}=F(t, y-z) \frac{\partial}{\partial v} \quad B^{t}=0, \quad B^{x}=-\frac{1}{2} \alpha F_{t} v, \quad B^{y}=0, \quad B^{z}=0, \\ \Omega_{G}=G(t, y+z) \frac{\partial}{\partial v} \quad B^{t}=0, \quad B^{x}=-\frac{1}{2} \alpha G_{t} v, \quad B^{y}=0, \quad B^{z}=0, \end{array}\\ \Omega_{1}=\frac{\partial}{\partial t}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0\\ \Omega_{2}=\frac{\partial}{\partial x}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0,\\ \Omega_{3}=\frac{\partial}{\partial y}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0,\\ \Omega_{4}=\frac{\partial}{\partial z}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0,\\ \Omega_{5}=z \frac{\partial}{\partial y}-\gamma y \frac{\partial}{\partial z}, \quad B^{t}=0, \quad B^{x}=0, \quad B^{y}=0, \quad B^{z}=0,\\ \Omega_{F}=F(t, y-z) \frac{\partial}{\partial v} \quad B^{t}=0, \quad B^{x}=-\frac{1}{2} \alpha F_{t} v, \quad B^{y}=0, \quad B^{z}=0,\\ \Omega_{G}=G(t, y+z) \frac{\partial}{\partial v} \quad B^{t}=0, \quad B^{x}=-\frac{1}{2} \alpha G_{t} v, \quad B^{y}=0, \quad B^{z}=0, \end{array} $
with function $F\left( t,z-y \right)$ satisfying second order PDE $\gamma {{F}_{zz}}-\epsilon {{F}_{yy}}=0$ as well as $G\left( t,y+z \right)$ satisfying $\gamma {{G}_{zz}}+\epsilon {{G}_{yy}}=0$. We now secure conserved vectors for (3 + 1)D-gZKe (3) by employing the Noether theorem, (78) and (81) as well as the relation given in Sarlet [65] and then reverting to the basic variable, we gain
$\begin{array}{l} \begin{array}{l} C_{1}^{t}=\frac{1}{4} \sigma u_{x x} \int u_{t} d x+\frac{1}{2} \kappa u_{x}^{2}-\frac{\alpha}{(n+1)(n+2)} u^{n+2}-\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2} \\ +\frac{1}{2} \epsilon u_{x} \int u_{y y} d x+\frac{1}{2} \gamma u_{x} \int u_{z z} d x+\frac{1}{4} \sigma u_{t} u_{x}, \\ C_{1}^{x}=\frac{3}{4} \sigma u_{t x} \int u_{t} d x-\frac{1}{4} \sigma u_{x} \int u_{t t} d x-\frac{1}{2} \sigma u_{t}^{2}-\kappa u_{t} u_{x}-\frac{1}{2} \epsilon u_{t} \int u_{y y} d x \\ -\frac{1}{2} \gamma u_{t} \int u_{z z} d x+\frac{1}{2}\left(\int u_{t} d x\right)^{2}+\kappa u_{x x} \int u_{t} d x+\frac{1}{2} \epsilon u_{y y} \int u_{t} d x \\ +\frac{1}{2} \gamma u_{z z} \int u_{t} d x+\frac{\alpha}{n+1} u^{n+1} \int u_{t} d x+\frac{\beta}{2 n+1} u^{2 n+1} \int u_{t} d x, \\ C_{1}^{y}=\frac{1}{2} \epsilon u_{x y} \int u_{t} d x-\frac{1}{2} \epsilon u_{x} \int u_{t y} d x \\ C_{1}^{z}=\frac{1}{2} \gamma u_{x z} \int u_{t} d x-\frac{1}{2} \gamma u_{x} \int u_{t z} d x \\ \mathcal{C}_{2}^{t}=\frac{1}{2} u^{2}-\frac{1}{4} \sigma u_{x}^{2}+\frac{1}{4} \sigma u u_{x x}, \\ C_{2}^{x}=\frac{3}{4} \sigma u u_{t x}-\frac{1}{4} \sigma u_{t} u_{x}+\kappa u u_{x x}-\frac{1}{2} \kappa u_{x}^{2}+\frac{1}{2} \epsilon u u_{y y}-\frac{\alpha}{(n+1)(n+2)} u^{n+2} \\ +\frac{1}{2} \gamma u u_{z z}-\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2}+\frac{\beta}{2 n+1} u^{2 n+2}+\frac{\alpha}{n+1} u^{n+2} \\ C_{2}^{y}=\frac{1}{2} \epsilon u u_{x y}-\frac{1}{2} \epsilon u_{x} u_{y} \\ C_{2}^{z}=\frac{1}{2} \gamma u u_{x z}-\frac{1}{2} \gamma u_{x} u_{z} \\ C_{3}^{t}=\frac{1}{2} u \int u_{y} d x-\frac{1}{4} \sigma u_{x} u_{y}+\frac{1}{4} \sigma u_{x x} \int u_{y} d x, \\ C_{3}^{x}=\frac{3}{4} \sigma u_{t x} \int u_{y} d x-\frac{1}{4} \sigma u_{x} \int u_{t y} d x+\frac{1}{2}\left(\int u_{t} d x\right)\left(\int u_{y} d x\right) \\ -\kappa u_{x} u_{y}+\frac{1}{2} \epsilon u_{y y} \int u_{y} d x-\frac{1}{2} \epsilon u_{y} \int u_{y y} d x+\frac{1}{2} \gamma u_{z z} \int u_{y} d x \\ -\frac{1}{2} \sigma u_{t} u_{y}+\frac{\beta}{2 n+1} u^{2 n+1} \int u_{y} d x+\frac{\alpha}{n+1} u^{n+1} \int u_{y} d x \\ +\kappa u_{x x} \int u_{y} d x-\frac{1}{2} \gamma u_{y} \int u_{z z} d x \end{array}\\ C_{1}^{t}=\frac{1}{4} \sigma u_{x x} \int u_{t} d x+\frac{1}{2} \kappa u_{x}^{2}-\frac{\alpha}{(n+1)(n+2)} u^{n+2}-\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2}\\ +\frac{1}{2} \epsilon u_{x} \int u_{y y} d x+\frac{1}{2} \gamma u_{x} \int u_{z z} d x+\frac{1}{4} \sigma u_{t} u_{x},\\ C_{1}^{x}=\frac{3}{4} \sigma u_{t x} \int u_{t} d x-\frac{1}{4} \sigma u_{x} \int u_{t t} d x-\frac{1}{2} \sigma u_{t}^{2}-\kappa u_{t} u_{x}-\frac{1}{2} \epsilon u_{t} \int u_{y y} d x\\ -\frac{1}{2} \gamma u_{t} \int u_{z z} d x+\frac{1}{2}\left(\int u_{t} d x\right)^{2}+\kappa u_{x x} \int u_{t} d x+\frac{1}{2} \epsilon u_{y y} \int u_{t} d x\\ +\frac{1}{2} \gamma u_{z z} \int u_{t} d x+\frac{\alpha}{n+1} u^{n+1} \int u_{t} d x+\frac{\beta}{2 n+1} u^{2 n+1} \int u_{t} d x,\\ C_{1}^{y}=\frac{1}{2} \epsilon u_{x y} \int u_{t} d x-\frac{1}{2} \epsilon u_{x} \int u_{t y} d x\\ C_{1}^{z}=\frac{1}{2} \gamma u_{x z} \int u_{t} d x-\frac{1}{2} \gamma u_{x} \int u_{t z} d x\\ \mathcal{C}_{2}^{t}=\frac{1}{2} u^{2}-\frac{1}{4} \sigma u_{x}^{2}+\frac{1}{4} \sigma u u_{x x},\\ C_{2}^{x}=\frac{3}{4} \sigma u u_{t x}-\frac{1}{4} \sigma u_{t} u_{x}+\kappa u u_{x x}-\frac{1}{2} \kappa u_{x}^{2}+\frac{1}{2} \epsilon u u_{y y}-\frac{\alpha}{(n+1)(n+2)} u^{n+2}\\ +\frac{1}{2} \gamma u u_{z z}-\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2}+\frac{\beta}{2 n+1} u^{2 n+2}+\frac{\alpha}{n+1} u^{n+2}\\ C_{2}^{y}=\frac{1}{2} \epsilon u u_{x y}-\frac{1}{2} \epsilon u_{x} u_{y}\\ C_{2}^{z}=\frac{1}{2} \gamma u u_{x z}-\frac{1}{2} \gamma u_{x} u_{z}\\ C_{3}^{t}=\frac{1}{2} u \int u_{y} d x-\frac{1}{4} \sigma u_{x} u_{y}+\frac{1}{4} \sigma u_{x x} \int u_{y} d x,\\ C_{3}^{x}=\frac{3}{4} \sigma u_{t x} \int u_{y} d x-\frac{1}{4} \sigma u_{x} \int u_{t y} d x+\frac{1}{2}\left(\int u_{t} d x\right)\left(\int u_{y} d x\right)\\ -\kappa u_{x} u_{y}+\frac{1}{2} \epsilon u_{y y} \int u_{y} d x-\frac{1}{2} \epsilon u_{y} \int u_{y y} d x+\frac{1}{2} \gamma u_{z z} \int u_{y} d x\\ -\frac{1}{2} \sigma u_{t} u_{y}+\frac{\beta}{2 n+1} u^{2 n+1} \int u_{y} d x+\frac{\alpha}{n+1} u^{n+1} \int u_{y} d x\\ +\kappa u_{x x} \int u_{y} d x-\frac{1}{2} \gamma u_{y} \int u_{z z} d x \end{array}$
$\begin{aligned} C_{3}^{y}= & \frac{1}{2} \sigma u_{t} u_{x}-\frac{1}{2} u \int u_{t} d x+\frac{1}{2} \epsilon u_{x y} \int u_{y} d x+\frac{1}{2} \kappa u_{x}^{2}+\frac{1}{2} \gamma u_{x} \int u_{z z} d x \\ & -\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2}-\frac{\alpha}{(n+1)(n+2)} u^{n+2} \\ C_{3}^{z}= & \frac{1}{2} \gamma u_{x z} \int u_{y} d x-\frac{1}{2} \gamma u_{x} \int u_{y z} d x \\ C_{4}^{t}= & \frac{1}{2} u \int u_{z} d x-\frac{1}{4} \sigma u_{x} u_{z}+\frac{1}{4} \sigma u_{x x} \int u_{z} d x \\ C_{4}^{x}= & \frac{3}{4} \sigma u_{t x} \int u_{z} d x-\frac{1}{4} \sigma u_{x} \int u_{t z} d x+\frac{1}{2}\left(\int u_{t} d x\right)\left(\int u_{z} d x\right) \\ & +\kappa u_{x x} \int u_{z} d x-\kappa u_{x} u_{z}+\frac{1}{2} \epsilon u_{y y} \int u_{z} d x+\frac{1}{2} \gamma u_{z z} \int u_{z} d x \\ & -\frac{1}{2} \sigma u_{t} u_{z}-\frac{1}{2} \epsilon u_{z} \int u_{y y} d x-\frac{1}{2} \gamma u_{z} \int u_{z z} d x+\frac{\beta}{2 n+1} u^{2 n+1} \int u_{z} d x \\ & +\frac{\alpha}{n+1} u^{n+1} \int u_{z} d x, \\ C_{4}^{y}= & \frac{1}{2} \epsilon u_{x y} \int u_{z} d x-\frac{1}{2} \epsilon u_{x} \int u_{y z} d x, \\ C_{4}^{z}= & \frac{1}{2} \epsilon u_{x} \int u_{y y} d x-\frac{\alpha}{(n+1)(n+2)} u^{n+2}-\frac{\beta}{(2 n+1)(2 n+2)} u^{2 n+2} \\ & +\frac{1}{2} \kappa u_{x}^{2}+\frac{1}{2} \gamma u_{x z} \int u_{z} d x-\frac{1}{2} u \int u_{t} d x+\frac{1}{2} \sigma u_{t} u_{x} \\ C_{5}^{t}= & \frac{1}{4} \gamma \sigma y u_{x} u_{z}-\frac{1}{2} \gamma y u \int u_{z} d x+\frac{1}{2} z u \int u_{y} d x-\frac{1}{4} \sigma z u_{x} u_{y} \\ & -\frac{1}{4} \gamma \sigma y u_{x x} \int u_{z} d x+\frac{1}{4} \sigma z u_{x x} \int u_{y} d x \end{aligned} $
$\begin{array}{l} \begin{array}{l} C_{5}^{x}=\frac{\alpha}{n+1} z u^{n+1} \int u_{y} d x-\frac{\alpha \gamma}{n+1} y u^{n+1} \int u_{z} d x-\frac{\gamma \beta}{2 n+1} y u^{2 n+1} \int u_{z} d x \\ +\frac{\beta}{2 n+1} z u^{2 n+1} \int u_{y} d x+\frac{1}{2} \gamma^{2} y u_{z} \int u_{z z} d x+\frac{1}{2} \gamma \epsilon y u_{z} \int u_{y y} d x \\ -\frac{1}{2} \gamma^{2} y u_{z z} \int u_{z} d x+\frac{1}{2} \gamma z u_{z z} \int u_{y} d x-\frac{1}{2} \gamma z u_{y} \int u_{z z} d x-\kappa z u_{x} u_{y} \\ -\frac{1}{2} \epsilon z u_{y} \int u_{y y} d x-\frac{1}{2} \gamma \epsilon y u_{y y} \int u_{z} d x+\frac{1}{2} \epsilon z u_{y y} \int u_{y} d x+\gamma \kappa y u_{x} u_{z} \\ -\gamma \kappa y u_{x x} \int u_{z} d x+\kappa z u_{x x} \int u_{y} d x-\frac{1}{2} \gamma y\left(\int u_{t} d x\right)\left(\int u_{z} d x\right) \\ +\frac{1}{2} z\left(\int u_{t} d x\right)\left(\int u_{y} d x\right)+\frac{1}{4} \gamma \sigma y u_{x} \int u_{t x} d x+\frac{1}{2} \gamma \sigma y u_{t} u_{z} \\ -\frac{1}{4} \sigma z u_{x} \int u_{t y} d x-\frac{1}{2} \sigma z u_{t} u_{y}-\frac{3}{4} \gamma \sigma y u_{t x} \int u_{z} d x+\frac{3}{4} \sigma z u_{t x} \int u_{y} d x, \\ C_{5}^{y}=\frac{1}{2} \gamma \epsilon u_{x} \int u_{z} d x-\frac{\alpha}{(n+1)(n+2)} z u^{n+2}-\frac{\beta}{(2 n+1)(2 n+2)} z u^{2 n+2} \\ +\frac{1}{2} \gamma z u_{x} \int u_{z z} d x+\frac{1}{2} \gamma \epsilon y u_{x} \int u_{y z} d x+\frac{1}{2} \kappa z u_{x}^{2}-\frac{1}{2} \gamma \epsilon y u_{x y} \int u_{z} d x \\ +\frac{1}{2} \epsilon z u_{x y} \int u_{y} d x-\frac{1}{2} z u \int u_{t} d x+\frac{1}{2} \sigma z u_{t} u_{x}, \\ C_{5}^{z}=-\frac{1}{2} \gamma u_{x} \int u_{y} d x+\frac{\alpha \gamma}{(n+1)(n+2)} y u^{n+2}+\frac{\beta \gamma}{(2 n+1)(2 n+2)} y u^{2 n+2} \\ -\frac{1}{2} \gamma z u_{x} \int u_{y z} d x-\frac{1}{2} \gamma \epsilon y u_{x} \int u_{y y} d x-\frac{1}{2} \gamma \kappa y u_{x}^{2}-\frac{1}{2} \gamma^{2} y u_{x z} \int u_{z} d x \\ +\frac{1}{2} \gamma z u_{x z} \int u_{y} d x+\frac{1}{2} \gamma y u \int u_{t} d x-\frac{1}{2} \gamma \sigma y u_{t} u_{x} \\ C_{F}^{t}=-\frac{1}{2} u F(t, z-y), \\ C_{F}^{x}=-\frac{1}{2} F(t, z-y) \int u_{t} d x-\frac{\alpha}{n+1} u^{n+2} F(t, z-y)-\sigma u_{t x} F(t, z-y) \\ -\frac{\beta}{2 n+1} u^{2 n+1} F(t, z-y)-\frac{1}{2} \epsilon u_{y y} F(t, z-y)-\frac{1}{2} \gamma u_{z z} F(t, z-y) \\ -\kappa u_{x x} F(t, z-y)+\frac{1}{2} F_{t}(t, z-y) \int u d x, \\ C_{F}^{y}=-\frac{1}{2} \epsilon u_{x y} F(t, z-y)-\frac{1}{2} \epsilon u_{x} F_{y}(t, z-y), \\ C_{F}^{z}=\frac{1}{2} \gamma u_{x} F_{z}(t, z-y)-\frac{1}{2} \gamma u_{x z} F(t, z-y) ; \end{array}\\ C_{5}^{x}=\frac{\alpha}{n+1} z u^{n+1} \int u_{y} d x-\frac{\alpha \gamma}{n+1} y u^{n+1} \int u_{z} d x-\frac{\gamma \beta}{2 n+1} y u^{2 n+1} \int u_{z} d x\\ +\frac{\beta}{2 n+1} z u^{2 n+1} \int u_{y} d x+\frac{1}{2} \gamma^{2} y u_{z} \int u_{z z} d x+\frac{1}{2} \gamma \epsilon y u_{z} \int u_{y y} d x\\ -\frac{1}{2} \gamma^{2} y u_{z z} \int u_{z} d x+\frac{1}{2} \gamma z u_{z z} \int u_{y} d x-\frac{1}{2} \gamma z u_{y} \int u_{z z} d x-\kappa z u_{x} u_{y}\\ -\frac{1}{2} \epsilon z u_{y} \int u_{y y} d x-\frac{1}{2} \gamma \epsilon y u_{y y} \int u_{z} d x+\frac{1}{2} \epsilon z u_{y y} \int u_{y} d x+\gamma \kappa y u_{x} u_{z}\\ -\gamma \kappa y u_{x x} \int u_{z} d x+\kappa z u_{x x} \int u_{y} d x-\frac{1}{2} \gamma y\left(\int u_{t} d x\right)\left(\int u_{z} d x\right)\\ +\frac{1}{2} z\left(\int u_{t} d x\right)\left(\int u_{y} d x\right)+\frac{1}{4} \gamma \sigma y u_{x} \int u_{t x} d x+\frac{1}{2} \gamma \sigma y u_{t} u_{z}\\ -\frac{1}{4} \sigma z u_{x} \int u_{t y} d x-\frac{1}{2} \sigma z u_{t} u_{y}-\frac{3}{4} \gamma \sigma y u_{t x} \int u_{z} d x+\frac{3}{4} \sigma z u_{t x} \int u_{y} d x,\\ C_{5}^{y}=\frac{1}{2} \gamma \epsilon u_{x} \int u_{z} d x-\frac{\alpha}{(n+1)(n+2)} z u^{n+2}-\frac{\beta}{(2 n+1)(2 n+2)} z u^{2 n+2}\\ +\frac{1}{2} \gamma z u_{x} \int u_{z z} d x+\frac{1}{2} \gamma \epsilon y u_{x} \int u_{y z} d x+\frac{1}{2} \kappa z u_{x}^{2}-\frac{1}{2} \gamma \epsilon y u_{x y} \int u_{z} d x\\ +\frac{1}{2} \epsilon z u_{x y} \int u_{y} d x-\frac{1}{2} z u \int u_{t} d x+\frac{1}{2} \sigma z u_{t} u_{x},\\ C_{5}^{z}=-\frac{1}{2} \gamma u_{x} \int u_{y} d x+\frac{\alpha \gamma}{(n+1)(n+2)} y u^{n+2}+\frac{\beta \gamma}{(2 n+1)(2 n+2)} y u^{2 n+2}\\ -\frac{1}{2} \gamma z u_{x} \int u_{y z} d x-\frac{1}{2} \gamma \epsilon y u_{x} \int u_{y y} d x-\frac{1}{2} \gamma \kappa y u_{x}^{2}-\frac{1}{2} \gamma^{2} y u_{x z} \int u_{z} d x\\ +\frac{1}{2} \gamma z u_{x z} \int u_{y} d x+\frac{1}{2} \gamma y u \int u_{t} d x-\frac{1}{2} \gamma \sigma y u_{t} u_{x}\\ C_{F}^{t}=-\frac{1}{2} u F(t, z-y),\\ C_{F}^{x}=-\frac{1}{2} F(t, z-y) \int u_{t} d x-\frac{\alpha}{n+1} u^{n+2} F(t, z-y)-\sigma u_{t x} F(t, z-y)\\ -\frac{\beta}{2 n+1} u^{2 n+1} F(t, z-y)-\frac{1}{2} \epsilon u_{y y} F(t, z-y)-\frac{1}{2} \gamma u_{z z} F(t, z-y)\\ -\kappa u_{x x} F(t, z-y)+\frac{1}{2} F_{t}(t, z-y) \int u d x,\\ C_{F}^{y}=-\frac{1}{2} \epsilon u_{x y} F(t, z-y)-\frac{1}{2} \epsilon u_{x} F_{y}(t, z-y),\\ C_{F}^{z}=\frac{1}{2} \gamma u_{x} F_{z}(t, z-y)-\frac{1}{2} \gamma u_{x z} F(t, z-y) ; \end{array} $
$\begin{aligned} C_{G}^{t}= & -\frac{1}{2} u G(t, z+y) \\ C_{G}^{x}= & -\frac{1}{2} G(t, z+y) \int u_{t} d x-\frac{\alpha}{n+1} u^{n+2} G(t, z+y)-\sigma u_{t x} G(t, z+y) \\ & -\frac{\beta}{2 n+1} u^{2 n+1} G(t, z+y)-\frac{1}{2} \epsilon u_{y y} G(t, z+y)-\frac{1}{2} \gamma u_{z z} G(t, z+y) \\ & -\kappa u_{x x} G(t, z+y)+\frac{1}{2} G_{t}(t, z+y) \int u d x \\ C_{G}^{y}= & -\frac{1}{2} \epsilon u_{x y} G(t, z+y)+\frac{1}{2} \epsilon u_{x} G_{y}(t, z+y) \\ C_{G}^{z}= & -\frac{1}{2} \gamma u_{x z} G(t, z+y)+\frac{1}{2} \gamma u_{x} G_{z}(t, z+y) \end{aligned}$
Remark. We acquired two local conserved vectors by invoking the homotopy formula via the standard multiplier approach (which is effective in securing differential equations with or without variational principles) while using the Noether theorem (which works only for differential equations with variational principles) as it were, we found seven conserved vectors. In addition, the seven vectors comprise one local as well as six non-local conserved vectors of first integrals. However, we can infer some similarities between the results from both techniques such as the inclusion of arbitrary functions alongside the presence of local conserved vectors. These functions attest to the fact that (3 + 1)D-gZKe (3) possesses infinite-dimensional conserved vectors which is an important feature of any “KdV-based” equation. Moreover, we can deduce that a particular differential equation can have as many numbers of conservation laws as possible. Nonlocal conserved vectors characterize a larger part of the results from the Noether’s theorem implying that a given differential equation can have both local as well as nonlocal conserved vectors.
6. Conclusions
In this paper, we carried out a study on higher-dimensional generalized Zakharov-Kuznetsov equation with dual power-law nonlinearity which has applications in nonlinear sciences and engineering. We gained exact solutions for the underlying Eq. (3) via Lie group theory together with direct integration. More general exact solutions of (3) via extended Jacobi functions expansion technique were achieved and periodic, hyperbolic, as well as rational function solutions were also derived. The application of these functions in engineering and sciences are immense such as in electrical and electronics, medicine, marine and so on. We complete that by utilizing suitable graphical depictions to portray the dynamics of the various achieved results. Furthermore, we derived conserved currents of the underlying Eq. (3) by invoking Noether’s theorem using the Helmholtz criteria. In addition, we adopted the homotopy formula to secure the conserved vectors of the equation and then compared the outcomes of the two techniques. Wave depictions via appropriate choice of parametric values in this study reveal a monumental existence of transverse waves which is highly applicable in physics. We presented the application of the results obtained in science and engineering among which it has been observed that in investigating chromospheric fine structures near the solar disk center, transverse wave properties such as, velocity, amplitude and phase speed are engaged in estimating plasma parameters along waveguide via magnetohydrodynamics technique. Therefore, we want to add to this end that the diverse results secured in this article may be of interest to scientists working particularly in the fields of nonlinear sciences as well as engineering, especially in ocean engineering because of their value and significance.
Declaration of Competing Interest
Authors declare that they have no conflict of interest.
Acknowledgements
The authors thank the North-West University, Mafikeng campus for its continued support. We also thank the editor and reviewers for their positive suggestions, which helped to improve the paper.
