1. Introduction
Nonlinear evolution equations (NLEEs) play a very important role in describing many physical features, particularly in quantum acoustic waves, quantum Langmuir waves, dense quantum plasma, ion-acoustic waves, electron thermal energy, ion plasma, solid state physics, optical fibers, chemical physics, chaos theory, mathematical physics, astrophysics, biophysics, nuclear physics, fluid mechanics, engineering problems [1, 2] etc. Nonlinear soliton solutions of these NLEEs play a fundamental role in unraveling the dynamics and describing the facts. Thus, different methods are used to search soliton solutions to these NLEEs. Some of them are the extended simple equation technique [3], the generalized Kudryashov technique [4], the first-integral technique [5], the new auxiliary equation technique [6], the Bernoulli sub-equation function technique [7, 8], the Riccati–Bernoulli sub-ODE method [9–11], the sine-Gordon expansion technique [12, 13], the sine–cosine technique [14], the generalized unified technique [15], the modified Khater method [16], the tanh–coth method [17], He's variational principle [18, 19], the B$\ddot{{\rm{a}}}$cklund transformation method [20], the unified Riccati equation expansion method [21], the extended Kudryashov method [22], the improved $(G^{\prime} /G)$-expansion method [23], the functional variable method [24], the exp-expansion technique [25], the improved $F$-expansion technique [26], the MSE technique [27–30], the enhanced MSE method [31], the Jacobi elliptic functions method [32–35], and other different techniques [36–40].
In various fields of physics, applied mathematics, and engineering, the Zakharov–Kuznetsov (ZK) equation is used. The new extended quantum ZK (QZK) and the (3 + 1)-dimensional ZK equations were derived for finite but small amplitude ion-acoustic waves in a quantum magneto-plasma by using the reductive perturbation theory and the quantum hydrodynamical model [41]. In this study, we consider the enhanced MSE technique to search bright solitons, periodic, compacton, bell-shape, parabolic, singular periodic and other general soliton solutions of the (3 + 1)-dimensional ZK equation and the new extended quantum QZK equation [42]. The ZK equation is one of the two canonical two-dimensional extensions of the Korteweg–de Vries equation, which was initially developed in two dimensions for weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma. The new extended QZK equation [43] is:
$\begin{eqnarray}{v}_{t}+\alpha v{v}_{x}+\beta \left({v}_{xxx}+{v}_{yyy}\right)+\gamma \left({v}_{xyy}+{v}_{xxy}\right)=0,\end{eqnarray}$
where $v(x,y,t)$ represents the electrostatic wave potential with the temporal variable $t$ and spatial variables $x,$ $y$ and $\alpha ,\beta ,\gamma $ are all constants. The coefficients of $\beta $ and $\gamma $ describe the multi-dimensional dispersion terms, the coefficient of $\alpha $ is the nonlinear term, and ${v}_{t}$ is the evolution with respect to time.In the subsequent, we consider the (3 + 1)-dimensional ZK [44]:
$\begin{eqnarray}{v}_{t}+\alpha v{v}_{z}+\beta {v}_{zzz}+\gamma {v}_{xxz}+\delta {v}_{yyz}=0,\end{eqnarray}$
where α is the nonlinear coefficient and $\beta ,$ $\gamma ,$ $\delta $ are the dispersion coefficients. In [44], the ZK equation was derived by putting in use the reductive perturbation technique and the quantum hydrodynamic model and some periodic, explosive, and solitary wave solutions are obtained with the help of the extended Conte's truncation technique.The exact solutions and symmetry reductions to the extended QZK equation have been derived by Lie symmetry analysis in [45]. In [46], the optimal system and conservation law were used for further investigation of the extended QZK equation. To obtain the exact travelling wave solutions to the extended QZK equation, El-Ganaini and Akbar [47] contrive the modified simplest equation technique, the extended simplest equation technique, and the simplest equation technique. Baskonus et al [48] extracted hyperbolic and complex function solutions of the extended QZK equation by using the sine-Gordon expansion technique. Raza et al [49] used the trial equation method to search soliton and periodic solutions of equation (1 ).
The multiple soliton solutions and new exact solitary wave solutions of (2 ) were attained by means of the Hirota's bilinear technique and the auxiliary equation method in [50]. Ebadi et al [51] used the adapted $F$-expansion technique, exp-function technique, and the $(G^{\prime} /G)$-expansion technique to find exact travelling wave solutions. Bhrawy et al [52] introduced the extended $F$-expansion technique for finding the complexiton solutions, singular soliton, non-topological, and topological solutions. In [53], the authors applied the extended generalized $(G^{\prime} /G)$-expansion technique to determine new exact solutions of (2 ). Lu et al [54] introduced the modified extended direct algebraic method to extract the elliptic function solutions, soliton and new exact solitary wave solutions of (2 ). New types of travelling wave solutions of (2 ) were attained by applying the generalized $(G^{\prime} /G)$-expansion technique and the improved $\tan (\phi /2)$-expansion technique in [55]. Zayed et al [56] applied the $(G^{\prime} /G,\,1/G)$-expansion method to extract further exact solutions of (2 ). Recently, Vinita and Ray [57] used the Lie symmetry analysis to study the equation (2 ) and new exact solitary wave solutions are attained and also derived the conservation laws to the QZK equation.
The enhanced modified simple equation (EMSE) method is recently developed a significant approach to determining competent solutions that is effective, compatible, and reliable to extract soliton solutions to NLEEs. This method have been used to examine the (1 + 1)-dimensional Burgers–Fisher equation with variable coefficients, the (2 + 1)-dimensional ZK equation with variable coefficient, the FitzHugh–Nagumo equation, the Burgers–Huxley equation [58], the modified Volterra equations, the Burger–Fisher equation [59], the Phi-4 equation [60], Chafee–Infante equation [61], the Gardner equation and the modified Benjamin–Bona–Mahony equation [62]. The (3 + 1)-dimensional ZK and the new extended QZK equations are important mathematical models to decipher ion-acoustic waves, quantum acoustic waves, quantum Langmuir waves, etc. To the optimal of our cognition and based on the analysis of the documents reachable in the literature obtained, the formerly introduced equations have not been investigated by the EMSE technique earlier. Thus, supported by the earlier studies, the aim of this article is to ascertain standard, realistic and far-reaching compatible solutions to these equations through the EMSE method. The feature of the solutions is that they extract bright solitons, periodic, compacton, bell-shape, parabolic shape, singular periodic and other solitons for certain values of the associated parameters.
The remaining of the article is sorted out as follows: the explanation of the enhanced MSE technique is given in section 2 . In section 3 , we use technique to search the advanced and wide-spectral soliton solutions. Some of the important graphical depictions of the solutions obtained are given in section 4 . Finally in the last section, the conclusion is given.
2. The EMSE technique
In this section, we concisely explain the EMSE technique [58–62]. Let us consider a general NLEE in the subsequent form:
$\begin{eqnarray} {\mathcal L} \left(v,{v}_{t},{v}_{x},{v}_{y},{v}_{z},{v}_{xt},{v}_{tt},{v}_{xx},{v}_{xx},{v}_{yy},{v}_{zz},\ldots \right)=0,\end{eqnarray}$
where $ {\mathcal L} $ is a polynomial with respect to the wave function $v(x,y,z,t)$ in which the highest order derivatives and nonlinear terms are involved. The following are the basic steps of this technique: The wave variable $\xi =p\left(t\right)x$ $+q\left(t\right)y$ $+r\left(t\right)z+s(t),$ where $v\left(x,y,z,t\right)=V(\xi )$ remodels equation (
$\begin{eqnarray} {\mathcal M} \left(V,(\dot{p}x+\dot{q}y+\dot{r}z+\dot{s})V^{\prime} ,\,pV^{\prime} ,qV^{\prime} ,\,\ldots \right)=0,\end{eqnarray}$
where $V=V(\xi ),$ dot specifies the differentiation related to time $t,$ and prime ($^{\prime} $) specifies the differentiation with respect to $\xi .$ The solution of the reduced equation (
$\begin{eqnarray}V\left(\xi \right)={a}_{0}\left(t\right)+\displaystyle \sum _{i=1}^{N}{a}_{i}(t){\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)}^{i},\end{eqnarray}$
where ${a}_{0}\left(t\right),$ ${a}_{1}\left(t\right),$ ${a}_{2}\left(t\right),$…,${a}_{N}(t)$ are the unknown function of ‘$t$' to be evaluated, such that ${a}_{N}(t)\ne 0,$ and $\psi ^{\prime} \left(\xi \right)\ne 0.$ The outstanding feature of this approach is that, instead of constants, the coefficients of $\psi {\left(\xi \right)}^{-i}$ are the function of $t,$ and $\psi \left(\xi \right)$ is an unknown function or not a solution of any well-known equation. The balancing theory between the nonlinear and linear terms appearing in equation (
Using the solution (
3. Solutions analysis
In this section, bright solitons, periodic, compacton, parabolic, bell-shape, singular periodic and other types of soliton solutions of the (3 + 1)-dimensional ZK and the new extended QZK equations are established via the enhanced MSE technique.
3.1. The new extended QZK equation
In this section, we derive the general solitary wave solution in terms of hyperbolic function, trigonometric function, exponential function and rational function of the new extended QZK equation using the enhance MSE technique. The wave transformation1 ) into the subsequent equation:8 ) after integrating (7 ),
$\begin{eqnarray}v\left(x,y,t\right)=V(\xi ),\,\xi =p\left(t\right)x+q\left(t\right)y+r\left(t\right),\end{eqnarray}$
remodel the equation ( $\begin{eqnarray}\begin{array}{ccc}\left(\dot{p}x+\dot{q}y+\dot{r}\right)V^{\prime} + \alpha \,p\,VV^{\prime} +\beta \left({p}^{3}+{q}^{3}\right)V^{\prime \prime\prime} \\ + \gamma pq\left(p+q\right)V^{\prime \prime\prime} =0.\end{array}\end{eqnarray}$
We attain equation ( $\begin{eqnarray}\begin{array}{ccc}2\left(\dot{p}x+\dot{q}y+\dot{r}\right)V + \alpha p{V}^{2}+2\beta \left({p}^{3}+{q}^{3}\right)V^{\prime\prime} \\ + 2\gamma pq\left(p+q\right)V^{\prime\prime} =0,\end{array}\end{eqnarray}$
setting the zero-integrating constant.Using the balancing principle between $V^{\prime\prime} $ and V2, we obtain $N=2.$
Therefore, the equation (8 ) has a solution of the following form9 ) and its necessary derivatives into equation (8 ); equating all the coefficients of ${\psi }^{-i},\,x{\psi }^{-i},\,y{\psi }^{-i},$ $i=0,$ $1,$ $2,$ $3,$ $4,$ and setting them to zero, we attain the algebraic and differential equations as follows:11 ), (12 ), (14 ), (15 ), (17 ) and (18 ) give the subsequent results:
$\begin{eqnarray}V\left(\xi \right)={a}_{0}\left(t\right)+{a}_{1}\left(t\right)\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)+{a}_{2}(t){\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)}^{2}.\end{eqnarray}$
Inserting solution ( $\begin{eqnarray}{\rm{Constant}}:p\alpha {a}_{0}^{2}+2{a}_{0}\dot{r}=0,\end{eqnarray}$
$\begin{eqnarray}x\,{\psi }^{0}:2{a}_{0}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y\,{\psi }^{0}:2{a}_{0}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }^{-1}:2p\,\alpha \,{a}_{0}\,{a}_{1}\psi ^{\prime} \left(\xi \right)+2{a}_{1}\dot{r\,}\psi ^{\prime} \left(\xi \right)+2\left({p}^{3}+{q}^{3}\right)\\ \,\times \beta \,{a}_{1}\psi ^{\prime \prime\prime} \left(\xi \right)+2\,p\,q\left(p+q\right)\,\gamma \,{a}_{1}\,\psi ^{\prime \prime\prime} \left(\xi \right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}x\,{\psi }^{-1}:2{a}_{1}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y\,{\psi }^{-1}:2{a}_{1}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }^{-2}:p\alpha \,{a}_{1}^{2}\,{\psi }^{{\prime} 2}+2\,p\,\alpha \,{a}_{0}\,{a}_{2}{\psi }^{{\prime} 2}+2\,{a}_{2}\,\dot{r}{\psi }^{{\prime} 2}\\ \,-6\,\left({p}^{3}+{q}^{3}\right)\beta \,{a}_{1}\,\psi ^{\prime} \psi ^{\prime\prime} \\ \,-6\,p\,q\,\left(p+q\right)\gamma \,{a}_{1}\,\psi ^{\prime} \psi ^{\prime\prime} +4\,\left({p}^{3}+{q}^{3}\right)\beta \,{a}_{2}\,{\psi }^{{\prime\prime} 2}\\ \,+4\,p\,q\,\left(p+q\right)\gamma \,{a}_{2}\,{\psi }^{{\prime\prime} 2}+4\,\left({p}^{3}+{q}^{3}\right)\beta \,{a}_{2}\,\psi ^{\prime} \psi ^{\prime \prime\prime} \\ \,+4\,p\,q\,\left(p+q\right)\gamma \,{a}_{2}\,\psi ^{\prime} \psi ^{\prime \prime\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}x\,{\psi }^{-2}:2{a}_{2}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y\,{\psi }^{-2}:2{a}_{2}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }^{-3}:4\left({p}^{3}+{q}^{3}\right)\beta {a}_{1}{\psi }^{{\prime} 3}+4pq\left(p+q\right)\gamma {a}_{1}{\psi }^{{\prime} 3}\\ \,+2p\alpha {a}_{1}{a}_{2}{\psi }^{{\prime} 3}-20\left({p}^{3}+{q}^{3}\right)\beta {a}_{2}{\psi }^{{\prime} 2}\psi ^{\prime\prime} \\ \,-20pq\left(p+q\right)\gamma {a}_{2}{\psi }^{{\prime} 2}\psi ^{\prime\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }^{-4}:12\left({p}^{3}+{q}^{3}\right)\beta {a}_{2}{\psi }^{{\prime} 4}+12pq\left(p+q\right)\gamma {a}_{2}{\psi }^{{\prime} 4}\\ \,+p\alpha {a}_{2}^{2}{\psi }^{{\prime} 4}=0.\end{array}\end{eqnarray}$
Equations ($\dot{p}=0$ and $\dot{q}=0.$ Thus, $p=h$ and $q=k,$ where $h$ and $k$ are integrating constants.
On the other hand, from equations (10 ) and (20 ), we found
${a}_{0}(t)=0,$ $-2\dot{r}/h\alpha $ and ${a}_{2}(t)=-\tfrac{12({h}^{3}\beta +{k}^{3}\beta +{h}^{2}k\gamma +h{k}^{2}\gamma )}{h\,\alpha },$ since ${a}_{2}\left(t\right)\ne 0.$
Equation (19 ) provides the subsequent result:
$\begin{eqnarray}\psi (\xi )=\displaystyle \frac{{c}_{1}}{{\rm{K}}}{{\rm{e}}}^{K\xi }+{c}_{2},\end{eqnarray}$
where ${c}_{1},\,{c}_{2}$ are constants of integration and $K\,=\tfrac{{a}_{1}\left(2{h}^{3}\beta +2{k}^{3}\beta +2{h}^{2}k\gamma +2h{k}^{2}\gamma +h\alpha {a}_{2}\right)}{10\left(h+k\right)\left({h}^{2}\beta -hk\beta +{k}^{2}\beta +hk\gamma \right){a}_{2}}.$The following cases should now be discussed:
When ${a}_{0}=0.$
Inserting the values of $\psi (\xi ),$ ${a}_{2}(t)$ and ${a}_{0}(t)$ into (13 ), we obtain16 ), the values of $p,$ $q,$ ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi $ are admissible to determine the soliton solutions.
$\begin{eqnarray*}{a}_{1}\left(t\right)=\pm \displaystyle \frac{12{\rm{i}}\sqrt{\dot{r}\,\left(h+k\right)}\,\sqrt{{h}^{2}\beta -hk\beta +{k}^{2}\beta +hk\gamma }}{h\alpha }.\end{eqnarray*}$
Since the values of $p,$ $q,$ ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi $ satisfy equation (Using the values of ${a}_{0},\,{a}_{1},\,{a}_{2}$ and $\psi (\xi )$ into the solution (9 ), we ascertain
$\begin{eqnarray}V\left(\xi \right)=\pm \displaystyle \frac{12\,{c}_{1}{c}_{2}\,\dot{r\,}G\,{{\rm{e}}}^{\pm G\,\xi }}{p\,\alpha \,{\left({c}_{1}{{\rm{e}}}^{\pm G\xi }\pm {c}_{2}G\right)}^{2}},\end{eqnarray}$
where $G=\tfrac{\sqrt{\dot{r}}}{\sqrt{h+k}\sqrt{hk\left(\beta -\gamma \right)-\beta \left({h}^{2}+{k}^{2}\right)}}.$By using the transformation between the exponential and hyperbolic functions, solution (22 ) is reduced to the subsequent soliton solution:
$\begin{eqnarray}\begin{array}{lll}V\left(\xi \right) & = & \pm \,\displaystyle \frac{12{c}_{1}{c}_{2}\dot{\,r}\,G}{h\alpha \,}\,\left[\left({c}_{1}\pm {c}_{2}G\right)\cosh \left(\displaystyle \frac{G\xi }{2}\right)\right.\\ & & \pm \,{\left.\left({c}_{1}\mp {c}_{2}G\right)\sinh \left(\displaystyle \frac{G\xi }{2}\right)\right]}^{-2}.\end{array}\end{eqnarray}$
Choosing the value of arbitrary constants, c1 = ±4 and ${c}_{2}=\mp \tfrac{1}{G},$ therefore, we attain $\begin{eqnarray}V\left(\xi \right)=\pm \displaystyle \frac{48\,\dot{r}}{h\alpha \,}\,{\left[\pm \,\,3\,\cosh \left(\displaystyle \frac{G\xi }{2}\right)\pm 5\,\sinh \left(\displaystyle \frac{G\xi }{2}\right)\right]}^{-2},\end{eqnarray}$
where $\xi =hx+ky+r\left(t\right).$Again, choosing ${c}_{1}=\pm 12$ and ${c}_{2}=\mp \tfrac{1}{G},$ the solution (23 ) turns to be8 )
$\begin{eqnarray}V\left(\xi \right)=\pm \displaystyle \frac{144\,\dot{r}}{h\alpha \,}\,{\left[\pm 11\,\cosh \left(\displaystyle \frac{G\xi }{2}\right)\pm 13\,\sinh \left(\displaystyle \frac{G\xi }{2}\right)\right]}^{-2}.\end{eqnarray}$
Moreover, if we assign, ${c}_{1}=\pm G,$ ${c}_{2}=\pm 1$ or ${c}_{1}=\pm 1,$ ${c}_{2}=\pm \tfrac{1}{G},$ we obtain the succeeding hyperbolic function solutions of equation ( $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{3\dot{r}}{h\alpha }{{\rm{sech}} }^{2}\left(\displaystyle \frac{G\xi }{2}\right),\end{eqnarray}$
$\begin{eqnarray}V\left(\xi \right)=\displaystyle \frac{3\dot{r}}{h\alpha }{\mathrm{csch}}^{2}\left(\displaystyle \frac{G\xi }{2}\right).\end{eqnarray}$
Thus, subject to the spatio-temporal coordinates, we attain the subsequent bright and singular soliton to the extended QZK equation as $\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{3\dot{r}}{h\alpha }{{\rm{sech}} }^{2}\left(\displaystyle \frac{G}{2}\left(hx+ky+r\left(t\right)\right)\right),\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,t\right)=\displaystyle \frac{3\dot{r}}{h\alpha }{\mathrm{csch}}^{2}\left(\displaystyle \frac{G}{2}\left(hx+ky+r\left(t\right)\right)\right),\end{eqnarray}$
where $G=\displaystyle \frac{\sqrt{\,\dot{r}}}{\sqrt{h+k}\sqrt{hk\left(\beta -\gamma \right)-\beta ({h}^{2}+{k}^{2})}}.$We attain the subsequent trigonometric function solutions, from solutions (28 ) and (29 ):
$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{3\dot{r}}{h\alpha }{\text{sec}}^{2}\left(M\left(hx+ky+r\left(t\right)\right)/2\right),\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{3\dot{r}}{h\alpha }{\csc }^{2}\left(M\left(hx+ky+r\left(t\right)\right)/2\right),\end{eqnarray}$
where $M=\tfrac{-\,\sqrt{\dot{r}}}{\sqrt{h+k}\sqrt{\beta \,({h}^{2}+{k}^{2})-hk(\beta -\gamma )}}.$When ${a}_{0}=-\tfrac{2\dot{r}}{h\,\alpha }.$
Introducing the values of ${a}_{0}(t),$ ${a}_{2}(t)$ and $\psi (\xi )$ into (13 ), yields9 ), we attain the ensuing solution
$\begin{eqnarray*}{a}_{1}(t)=\pm 12\sqrt{\dot{r}\,(h+k)}\sqrt{\beta ({h}^{2}+{k}^{2})-hk(\beta -\gamma )}/h\alpha .\end{eqnarray*}$
Substituting the values of ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi (\xi )$ into ( $\begin{eqnarray}V\left(\xi \right)=-2\dot{r}\displaystyle \frac{\left({c}_{1}^{2}\,{{\rm{e}}}^{\pm 2H\xi }\mp 4\,{c}_{1}{c}_{2}\,H\,{{\rm{e}}}^{\pm H\xi }+{c}_{2}^{2}{H}^{2}\right)}{h\,\alpha \,{\left({c}_{1}\,{{\rm{e}}}^{\pm H\xi }+{c}_{2}H\right)}^{2}},\end{eqnarray}$
where $H=\tfrac{\sqrt{\dot{r}}}{\sqrt{h+k}\sqrt{\beta ({h}^{2}+{k}^{2})-hk(\beta -\gamma )}}.$The solution (32 ) can be written as33 ), we extract
$\begin{eqnarray}\begin{array}{l}V\left(\xi \right)\\ \,=\,-\displaystyle \frac{2\dot{r}}{h\alpha }\left\{1\mp \displaystyle \frac{6H{{c}}_{1}{{c}}_{2}}{{\left\{\left({{c}}_{1}\pm {{c}}_{2}{\rm{H}}\right)\cosh \left(\displaystyle \frac{H\xi }{2}\right)\pm \left({{c}}_{1}\mp {{c}}_{2}H\right)\sinh \left(\displaystyle \frac{H\xi }{2}\right)\right\}}^{2}}\right\}.\end{array}\,\end{eqnarray}$
Setting the values ${c}_{1}=\pm 6$ and ${c}_{2}=\pm \tfrac{1}{H}$ of the arbitrary constants in ( $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\,\left\{1\mp \displaystyle \frac{36}{{\left\{\pm 7\,\cosh \left(\displaystyle \frac{H\xi }{2}\right)\pm 5\,\sinh \left(\displaystyle \frac{H\xi }{2}\right)\right\}}^{2}}\right\},\end{eqnarray}$
where $\xi =h\,x+k\,y+r\left(t\right).$Yet again, setting ${c}_{1}=\pm 18$ and ${c}_{2}=\pm \tfrac{1}{H},$ we obtain38 ) and (39 ) is considered to be:28 )–(31 ) and (38 )–(41 ) represent bright, parabolic soliton, bell-shape soliton, singular periodic soliton, and compacton.
$\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\,\left\{1\mp \displaystyle \frac{108}{{\left\{\pm 19\,\cosh \left(\displaystyle \frac{H\xi }{2}\right)\pm 17\,\sinh \left(\displaystyle \frac{H\xi }{2}\right)\right\}}^{2}}\right\}.\end{eqnarray}$
Alternatively, if we set ${c}_{1}=\pm H,\,{c}_{2}=\pm 1$ or ${c}_{1}=\pm 1,{c}_{2}=\pm \tfrac{1}{H},$ we ascertain the under mentioned soliton solutions $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1-\displaystyle \frac{3}{2}{{\rm{sech}} }^{2}\left(\displaystyle \frac{H\xi }{2}\right)\right],\end{eqnarray}$
$\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1+\displaystyle \frac{3}{2}{\mathrm{csch}}^{2}\left(\displaystyle \frac{H\xi }{2}\right)\right].\end{eqnarray}$
Subject to the temporal-spatial coordinates, we obtain the interpreted bright and singular soliton to the extended QZK equation as follows: $\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1-\displaystyle \frac{3}{2}{{\rm{sech}} }^{2}\left(\displaystyle \frac{H}{2}\left(hx+ky+r\left(t\right)\right)\right)\right],\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1+\displaystyle \frac{3}{2}{\mathrm{csch}}^{2}\left(\displaystyle \frac{H}{2}\left(hx+ky+r\left(t\right)\right)\right)\right].\end{eqnarray}$
The periodic soliton for solutions ( $\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1-\displaystyle \frac{3}{2}{\text{sec}}^{2}\left(\displaystyle \frac{G}{2}\left(hx+ky+r\left(t\right)\right)\right)\right],\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1-\displaystyle \frac{3}{2}{\csc }^{2}\left(\displaystyle \frac{G}{2}\left(hx+ky+r\left(t\right)\right)\right)\right],\end{eqnarray}$
where $G$ is provided in the above. The solutions (3.2. The (3 + 1)-dimensional ZK equation
In this section, relating to exponential function, rational function, hyperbolic function and trigonometric function, we obtain the general solitary wave solution to the (3 + 1)-dimensional ZK equation via the EMSE method. The wave variable2 ) into the ensuing equation:43 ) with zero integrating constant, we obtain
$\begin{eqnarray}\begin{array}{l}v\left(x,y,z,\,t\right)=V(\xi ),\,\xi =p\left(t\right)x+q\left(t\right)y\\ \,+r\left(t\right)z+s(t),\end{array}\end{eqnarray}$
reduces the (3 + 1)-dimensional ZK equation ( $\begin{eqnarray}\begin{array}{l}\left(\dot{p}x+\dot{q}y+\dot{r}z+\dot{s}\right)V^{\prime} +\alpha \,rVV^{\prime} \\ \,+(\beta {r}^{3}+\gamma {p}^{2}r+\delta {q}^{2}r)V^{\prime \prime\prime} =0.\end{array}\end{eqnarray}$
Integrating ( $\begin{eqnarray}\begin{array}{l}2\left(\dot{p}x+\dot{q}y+\dot{r}z+\dot{s}\right)V+\alpha r{V}^{2}\\ \,+2(\beta {r}^{3}+\gamma {p}^{2}r+\delta {q}^{2}r)V^{\prime\prime} =0.\end{array}\end{eqnarray}$
Applying the balance principle between the nonlinear term ${V}^{2}$ and highest order derivative $V^{\prime\prime} ,$ we found $N=2.$Thus, the solution of equation (44 ) accept the following form
$\begin{eqnarray}V\left(\xi \right)={a}_{0}\left(t\right)+{a}_{1}\left(t\right)\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)+{a}_{2}{\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)}^{2},\end{eqnarray}$
where ${a}_{0},$ ${a}_{1}$ and ${a}_{2}$ are an unknown function of $t,$ and $\psi ^{\prime} \left(\xi \right)\ne 0.$Putting the values of $V\left(\xi \right)$ and it's twice derivative into (44 ) and equalizing the coefficients of same power to zero, we attain some algebraic and differential equations as follows:47 )–(49 ), (51 )–(53 ) and (55 )–(57 ), it is found
$\begin{eqnarray}{\rm{Constant}}:r\alpha {a}_{0}^{2}+2{a}_{0}\dot{s}=0,\end{eqnarray}$
$\begin{eqnarray}x:2{a}_{0}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y:2{a}_{0}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}z:2{a}_{0}\dot{r}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}{\psi }^{-1}:2r\alpha {a}_{0}{a}_{1}\psi ^{\prime} & + & 2{a}_{1}\dot{s}\psi ^{\prime} +2r\left({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta \right)\\ & & \times \,{a}_{1}\psi ^{\prime \prime\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}x\,{\psi }^{-1}:2{a}_{1}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y\,{\psi }^{-1}:2{a}_{1}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}z\,{\psi }^{-1}:2{a}_{1}\dot{r}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}{\psi }^{-2}:r\alpha {a}_{1}^{2}{\psi }^{{\prime} 2} & + & 2r\alpha {a}_{0}{a}_{2}{\psi }^{{\prime} 2}+2{a}_{2}\dot{s}{\psi }^{{\prime} 2}\\ & & -\,6r\left({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta \right){a}_{1}\psi ^{\prime} \psi ^{\prime\prime} \\ & & +\,4r({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta ){a}_{2}{\psi }^{{\prime\prime} 2}\\ & & +\,4r({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta ){a}_{2}\psi ^{\prime} \psi ^{\prime \prime\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}x\,{\psi }^{-2}:2{a}_{2}\dot{p}=0,\end{eqnarray}$
$\begin{eqnarray}y\,{\psi }^{-2}:2{a}_{2}\dot{q}=0,\end{eqnarray}$
$\begin{eqnarray}z\,{\psi }^{-2}:2{a}_{2}\dot{r}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }^{-3}:4r\left({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta \right){a}_{1}{\psi }^{{\prime} 3}+2r\alpha {a}_{1}{a}_{2}{\psi }^{{\prime} 3}\\ \,-20r({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta ){a}_{2}{\psi }^{{\prime} 2}\psi ^{\prime\prime} =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}{\psi }^{-4}:12r({r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta ){a}_{2}{\psi }^{{\prime} 4}+r\alpha {a}_{2}^{2}{\psi }^{{\prime} 4}=0.\end{eqnarray}$
From equations ( $\begin{eqnarray*}\dot{p}=0,\,\dot{q}=0,\,\dot{r}=0.\end{eqnarray*}$
Integrating the above expressions with respect to time, yields $\begin{eqnarray*}p=l,\,q=m\,{\rm{and}}\,r=n,\end{eqnarray*}$
where l, m and n are the constants of integration.Equation (58 ) gives the following result:
$\begin{eqnarray}\psi ={c}_{3}\displaystyle \frac{1}{K^{\prime} }{{\rm{e}}}^{K^{\prime} \xi }+{c}_{4},\end{eqnarray}$
where ${c}_{3}$ and ${c}_{4}$ are constants of integration and $K^{\prime} \,=\tfrac{{a}_{1}\left(2{n}^{2}\beta +2{l}^{2}\gamma +2{m}^{2}\delta +\alpha {a}_{2}\right)}{10\left({n}^{2}\beta +{l}^{2}\gamma +q{m}^{2}\delta \right){a}_{2}}.$Solving equations (46 ) and (59 ) and putting the values of $p,q,r,$ we attain
$\begin{eqnarray*}{a}_{0}=0,\,-\displaystyle \frac{2\dot{s}}{n\,\alpha }\,{\rm{and}}\,{a}_{2}=-\displaystyle \frac{12({n}^{2}\beta +{l}^{2}\gamma +{m}^{2}\delta )}{\alpha }.\end{eqnarray*}$
When ${a}_{0}=0,$
Using the values of ${a}_{0},$ ${a}_{2},$ $\psi $ into the equation (50 ) and solving for ${a}_{1},$ we acquire54 ) is satisfied for the values of $p,$ $q,$ $r,$ ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi ,$ these values are acceptable for determining the soliton solutions.
$\begin{eqnarray*}{a}_{1}=\pm \displaystyle \frac{12\sqrt{{n}^{2}\beta +{l}^{2}\gamma +{m}^{2}\delta }\sqrt{-\dot{{\rm{s}}}}}{\sqrt{n}\alpha }.\end{eqnarray*}$
Inasmuch as equation (Substituting the values of ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi $ into the solution (45 ), we attain the subsequent exponential function solution
$\begin{eqnarray}V\left(\xi \right)=\mp \displaystyle \frac{12\dot{s}{c}_{3}{c}_{4}\,{{\rm{e}}}^{\pm M\,\xi }}{n\,\alpha {\left({c}_{3}\,{{\rm{e}}}^{\pm M\xi }\pm {c}_{4}M\right)}^{2}},\end{eqnarray}$
where $M=\tfrac{\sqrt{-\dot{{\rm{s}}}}}{\sqrt{n}\sqrt{{n}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta }}.$Transforming the exponential function solution (61 ) into the hyperbolic function solution, we attain62 ) turns out to be44 ) in the ensuing66 ) and (67 ), the trigonometric function solutions can be simulated as:
$\begin{eqnarray}\begin{array}{lll}V\left(\xi \right) & = & \mp \displaystyle \frac{12\dot{s}M{c}_{3}{c}_{4}}{n\alpha \,}\,\left[\left({c}_{3}\pm {c}_{4}M\right)\cosh \left(\displaystyle \frac{M\xi }{2}\right)\right.\\ & & \pm {\left.\,\left({c}_{3}\mp {c}_{4}M\right)\sinh \left(\displaystyle \frac{M\xi }{2}\right)\right]}^{-2}.\end{array}\end{eqnarray}$
Since ${c}_{3}$ and ${c}_{4}$ are integral constants, their values could be set at random. As a result, we can choose the values of ${c}_{3}$ and ${c}_{4}$ arbitrarily. Therefore, if we choose ${c}_{3}=\pm 5/2$ and ${c}_{4}=\pm 1/2M,$ the solution ( $\begin{eqnarray}V\left(\xi \right)=\mp \displaystyle \frac{30\dot{s}}{n\alpha }\,\,{\left[\pm 2\,\cosh \left(\displaystyle \frac{M\xi }{2}\right)\pm 3\,\sinh \left(\displaystyle \frac{M\xi }{2}\right)\right]}^{-2}.\end{eqnarray}$
Again, if we choose ${c}_{3}=\pm M,$ ${c}_{4}=\pm 1$ or ${c}_{3}=\pm 1,$ ${c}_{4}=\pm \tfrac{1}{M},$ we attain the hyperbolic function solutions of equation ( $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{3\dot{s}}{n\alpha }{{\rm{sech}} }^{2}\left(\displaystyle \frac{M\xi }{2}\right),\end{eqnarray}$
$\begin{eqnarray}V\left(\xi \right)=\displaystyle \frac{3\dot{s}}{n\alpha }{\mathrm{csch}}^{2}\left(\displaystyle \frac{M\xi }{2}\right).\end{eqnarray}$
Thus, the bell-shape and singular solitons to the (3 + 1)-dimensional ZK equation in relation to the space-time coordinates are $\begin{eqnarray}V\left(x,y,z,t\right)=-\displaystyle \frac{3\dot{s}}{n\alpha }{{\rm{sech}} }^{2}\left(M\left(lx+my+nz+s\left(t\right)\right)/2\right),\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,z,t\right)=\displaystyle \frac{3\dot{s}}{n\alpha }{\mathrm{csch}}^{2}\left(M\left(lx+my+nz+s\left(t\right)\right)/2\right).\end{eqnarray}$
In relation to ( $\begin{eqnarray}V\left(x,y,z,t\right)=-\displaystyle \frac{3\dot{s}}{n\alpha }{\text{sec}}^{2}\left(N\left(lx+my+nz+s\left(t\right)\right)/2\right),\end{eqnarray}$
$\begin{eqnarray}V\left(x,y,z,t\right)=-\displaystyle \frac{3\dot{s}}{n\alpha }{\csc }^{2}\left(N\left(lx+my+nz+s\left(t\right)\right)/2\right),\end{eqnarray}$
where $N=\tfrac{-\,\sqrt{\dot{{\rm{s}}}}}{\sqrt{n}\sqrt{{n}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta }}.$When ${a}_{0}=-\tfrac{2\dot{{\rm{s}}}}{n\alpha }.$
Embedding the values of ${a}_{0},$ ${a}_{2},$ $\psi $ into (50 ) and deciphering for ${a}_{1},$ yield45 ), we attain
$\begin{eqnarray*}{a}_{1}=\pm \displaystyle \frac{12\,\sqrt{{s}_{1}}\,\sqrt{{r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta }}{\sqrt{r}\alpha }.\end{eqnarray*}$
For these values of ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi ,$ from solution ( $\begin{eqnarray}V\left(\xi \right)=-\dot{\displaystyle \frac{2s}{n\alpha }}\displaystyle \frac{\left({c}_{3}^{2}\,{{\rm{e}}}^{\pm 2T\xi }\,\mp 4\,T\,{c}_{3}{c}_{4}\,{{\rm{e}}}^{\pm T\xi }+{c}_{4\,}^{2}{T}^{2}\right)}{{\left({c}_{3}\,{{\rm{e}}}^{\pm T\xi }+{c}_{4}T\right)}^{2}},\end{eqnarray}$
where $T=\tfrac{\sqrt{\dot{s}}}{\sqrt{n}\sqrt{{n}^{2}\beta +{l}^{2}\gamma +{m}^{2}\delta }}.$The solution (70 ) of the exponential function can be formulated as71 ) turns to the soliton solutions:72 ) turns into:75 ) and (76 ), the trigonometric function solutions can be constructed as:66 )–(69 ) and (75 )–(78 ) produce the bright, periodic, singular bell-shaped and plane shaped solitons which helps to understand the physical features of the (3 + 1)-dimensional ZK equation.
$\begin{eqnarray}\begin{array}{l}V\left(\xi \right)\\ \,=\,-\displaystyle \frac{2\dot{s}}{n\alpha }\left(1\mp \displaystyle \frac{6T{{c}}_{3}{{c}}_{4}}{{\left\{\left({{c}}_{3}\pm {{c}}_{4}{\rm{T}}\right)\cosh \left(\displaystyle \frac{T\xi }{2}\right)\pm \left({{c}}_{3}\mp {{c}}_{4}T\right)\sinh \left(\displaystyle \frac{T\xi }{2}\right)\,\right\}}^{2}}\right).\end{array}\,\end{eqnarray}$
For the values ${c}_{3}=\pm 6$ and ${c}_{4}=\pm \tfrac{1}{T},$ solution ( $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{s}}{n\alpha }\left(1\mp \displaystyle \frac{36}{{\left\{\pm 7\,\cosh \left(\displaystyle \frac{T\xi }{2}\right)\pm 5\,\sinh \left(\displaystyle \frac{T\xi }{2}\right)\,\right\}}^{2}}\right).\end{eqnarray}$
Furthermore, if we set ${c}_{3}=\pm T,$ ${c}_{4}=\pm 1$ or ${c}_{3}=\pm 1,$ ${c}_{4}=\pm \tfrac{1}{T},$ solution ( $\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{s}}{n\alpha }\left[1-\displaystyle \frac{3}{2}{{\rm{sech}} }^{2}\left(\displaystyle \frac{T\xi }{2}\right)\right],\end{eqnarray}$
$\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{s}}{n\alpha }\left[1+\displaystyle \frac{3}{2}{\mathrm{csch}}^{2}\left(\displaystyle \frac{T\xi }{2}\right)\right].\end{eqnarray}$
Therefore, we attain the under mentioned bell-shape and singular soliton solutions subject to space-time coordinates. $\begin{eqnarray}\begin{array}{l}V\left(x,y,z,t\right)\\ \,=\,-\displaystyle \frac{2\dot{s}}{n\alpha }\left[1-\displaystyle \frac{3}{2}{{\rm{sech}} }^{2}\left(\displaystyle \frac{T}{2}\left(lx+my+nz+s\left(t\right)\right)\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}V\left(x,y,z,t\right)\\ \,=\,-\displaystyle \frac{2\dot{s}}{n\alpha }\left[1+\displaystyle \frac{3}{2}{\mathrm{csch}}^{2}\left(\displaystyle \frac{T}{2}\left(lx+my+nz+s\left(t\right)\right)\right)\right].\end{array}\end{eqnarray}$
Relating to ( $\begin{eqnarray}\begin{array}{l}V\left(x,y,z,t\right)\\ \,=\,-\displaystyle \frac{2\dot{s}}{n\alpha }\left[1-\displaystyle \frac{3}{2}{\text{sec}}^{2}\left({\rm{i}}\displaystyle \frac{T}{2}\left(lx+my+nz+s\left(t\right)\right)\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}V\left(x,y,z,t\right)\\ \,-\,\displaystyle \frac{2\dot{s}}{n\alpha }\left[1-\displaystyle \frac{3}{2}{\csc }^{2}\left({\rm{i}}\displaystyle \frac{T}{2}\left(lx+my+nz+s\left(t\right)\right)\right)\right].\end{array}\end{eqnarray}$
The solutions (4. Results and discussion
The symbolic demonstration is very significant to understand the physical context of the solutions obtained for diverse parameter values. Therefore, we have portrayed two and three-dimensional graphics of the completed solutions to the (3 + 1)-dimensional ZK equation and the new extended QZK equation, in this section. To ascertain the physical properties of the earlier described equations, graphical descriptions of some of the solutions obtained by Matlab software are portrayed. Since the arbitrarily function $r(t)$ of the new extended QZK equation and $s(t)$ of the (3 + 1)-dimensional ZK equation exist in the solutions, therefore we may choose their values randomly for each solution. Figures 1–8 show the two and three-dimensional diagrams of the obtained solutions for the new extended QZK equation, and figures 9–16 indicate the two and three-dimensional graphs for the (3 + 1)-dimensional ZK equation. In figures 1–8, the 3D graphs are portrayed at $y=0$ and 2D graphs are depicted at $x=y=0.$ Also, in figures 9–16, the 3D graphs are depicted at $y=z=0$ and the 2D graphs are illustrated at $x=y=z=0.$
Figure 1. The 3D and 2D graphics for $r\left(t\right)=\lambda t$ of the solution ( |
Figure 2. The 3D and 2D graphics for r (t) = λt of the solution ( |
Figure 3. The 3D and 2D graphics for $r\left(t\right)=\,\sinh \left(\lambda t\right)$ of the solution ( |
Figure 4. The 3D and 2D graphics for $r\left(t\right)=\lambda t$ of the solution ( |
Figure 5. The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution ( |
Figure 6. The 3D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution ( |
Figure 7. The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution ( |
Figure 8. The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the particular solution ( |
Figure 9. The 3D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the solution ( |
Figure 10. The 3D and 2D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the general solution ( |
Figure 11. The 3D and 2D graphics for $s\left(t\right)={{\rm{e}}}^{\lambda {\rm{t}}}$ of the solution ( |
Figure 12. The 3D and 2D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the particular solution ( |
Figure 13. The 3D and 2D graphics for $s\left(t\right)=\,\tanh \left(\lambda {\rm{t}}\right)$ of the solution ( |
Figure 14. The 3D and 2D graphics for $s\left(t\right)=\,\cos \left(\lambda {\rm{t}}\right)$ of the particular solution ( |
Figure 15. The 3D and 2D graphics for $s\left(t\right)=\lambda {\rm{t}}$ of the particular solution ( |
Figure 16. The 3D and 2 graphics for $s\left(t\right)=\,\cos (\lambda {\rm{t}})$ of the solution ( |
Since $r\left(t\right)$ is an arbitrary function, without loss of generality we have considered $r\left(t\right)=\lambda {\rm{t}}.$ For the parametric values $h=2,\,k=1,\,\alpha =6,\,\beta =2,\,\gamma =2$ and $\lambda =2,$ the solution (24 ) demonstrate the periodic soliton and designated in figure 1. Periodic solitons perform a dynamic onward in characterizing the electrostatic wave potential in plasma physics. The space-time range for 3D graph is $-10\leqslant x,t\leqslant 10.$ Also, the two-dimensional plot with the time range $-15\leqslant t\leqslant 15$ is depicted for $\lambda =2,$ $2.1,$ $2.2$ which are identified by three different colors.
Figure 2 describes the periodic soliton nature represented by the solution (28 ). The suitable values of parameters with $r\left(t\right)=\lambda t$ are taken as $h=0.5,$ $k=0.2,$ $\alpha =0.3,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =0.5$ for 3D and 2D graphs. The two-dimensional graph is portrayed at $\lambda =0.5,$ $0.506,$ $0.509$ with different colors. The space-time range of 3D and 2D graphs is $-5\leqslant x,t\leqslant 5.$
Also, the 3D graph in figure 3 reflects the bright solitonic nature for $r\left(t\right)=\,\sinh \left(\lambda t\right)$ of the solution (28 ) with the appropriate parametric values $h=0.7,$ $k=-1,$ $\alpha =-1,$ $\beta =2,$ $\gamma =3$ and $\lambda =0.2.$ The bright solitons are worthwhile to understand the behavior of charged carriers in quantum plasmas. The space-time range of 3D and 2D graphics is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is drawn at $\lambda =0.2,$ $0.21,$ $0.22$ which are identified by diverse colors.
On the contrary, the solution (29 ) for the parametric values $h=4,$ $k=4,$ $\alpha =3,$ $\beta =-0.1,$ $\gamma =0.3$ and $\lambda =4$ represents pulse like singular periodic soliton, presented in figure 4 for $r\left(t\right)=\lambda t.$ The singular solitons are traveling wave solutions having discontinuities. The time-space range for 3D graph is $-5\leqslant x,t\leqslant 5,$ for 2D graph the time range is $-10\leqslant t\leqslant 10$ at $\lambda =4,$ $5,$ $6.$
Furthermore, the solution shapes in figure 5 show the compacton drawn from the solution (30 ) for $r\left(t\right)=\lambda {\rm{t}}$ with the particular values $h=2.8,$ $k=3,$ $\alpha =3,$ $\beta =2,$ $\gamma =1.9$ and $\lambda =2.$ Compactons are a sort of soliton with a dense support that can be used to describe ion-acoustic waves, electrostatics wave, quantum acoustic waves in plasma physics. The space-time range for 3D graph is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is shown at $\lambda =2,$ $2.1,$ $2.2$ in different colors with the time range $-10\leqslant t\leqslant 10.$
Figure 6 indicates the periodic soliton for $r\left(t\right)=\lambda \,{\rm{t}}$ sketched from the solution (34 ) with the particular values $h=0.7,$ $k=1,$ $\alpha =0.4,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =-1.$ The time-space range of 3D graph for real and imaginary parts of the solution (34 ) is $-10\leqslant x,t\leqslant 10.$
Figure 7, describes the bell-shape soliton portrayed from the solution (38 ) for $r\left(t\right)=\lambda {\rm{t}}$ with the parametric values $h=0.7,$ $k=1,$ $\alpha =0.4,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =1.5.$ Bell-shaped solitons, which have infinite wings on both sides, are another sort of solitary wave solution useful signal management. The time-space range for 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is portrayed at $\lambda =1.5,$ $2,$ $2.5.$
The solution (40 ) exhibits the parabolic soliton for the values of $h=9,$ $k=-0.8,$ $\alpha =2,$ $\beta =4,$ $\gamma =6$ and $\lambda =-0.1$ with $r\left(t\right)=\lambda {\rm{t}}$ sketched in figure 8. The space-time range of 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is depicted at $\lambda =-0.1.$
The 3D graph in figure 9 shows the soliton nature for $s\left(t\right)=\,\sin \left(\lambda t\right)$ of the particular solution (63 ) with the suitable parametric values $l=-0.1,$ $m=0.3,$ $n=1,$ $\delta =0.8,$ $\gamma =0.6,$ $\beta =0.9,$ $\alpha =0.4$ and $\lambda =0.2.$ The space-time range of 3D graph for real and imaginary parts of the solution (63 ) is $-10\leqslant x,t\leqslant 10.$
On the other hand, figure 10 reflects the periodic soliton nature for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the particular solution (66 ) with the particular values $l=0.6,$ $m=0.8,$ $n=1,$ $\alpha =2,$ $\beta =4,$ $\gamma =2,$ $\delta =1$ and $\lambda =1.$ The space-time range of 3D and 2D graph is $-8\leqslant x,t\leqslant 8.$ Also, the 2D graph is depicted at $\lambda =1,$ $1.2,$ $1.4$ which are specified by different colors.
Furthermore, the solution shapes in figure 11 is the bright-like soliton profile is interpreted from the solution (66 ) for $s\left(t\right)={{\rm{e}}}^{\lambda {\rm{t}}}$ with the suitable parametric values $l=5,$ $m=0.5,$ $n=2,$ $\alpha =3,$ $\beta =1,$ $\gamma =2,$ $\delta =2$ and, $\lambda =-0.4.$ This solution is descend from left to right asymptotic state. The space-time range for 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ Also, the two-dimensional plot is illustrated at $\lambda =-0.4,$ $-0.5,$ $-0.6.$
Figure 12 is the singular soliton characterized by the solution (67 ). The appropriate values of parameters with $s\left(t\right)=\,\sin (\lambda t)$ are taken as $l=1,$ $m=0.2,$ $n=0.9,$ $\delta =1,$ $\gamma =1.5,$ $\beta =2.5,$ $\alpha =1.5$ and $\lambda =0.1$ for 3D and 2D graphs. The two-dimensional graph is portrayed at $\lambda =0.1,$ $0.2,$ $0.3$ with different colors. The space-time range of 3D and 2D graphs is $-5\leqslant x,t\leqslant 5.$
Figure 13 represents the soliton like solution for $s\left(t\right)=\,\tanh (\lambda t)$ of the solution (68 ) with suitable parametric values $l=5,$ $m=5,$ $n=2,$ $\delta =1,$ $\gamma =2,$ $\beta =1,$ $\alpha =3$ and $\lambda =0.4.$ The space-time range for 3D and 2D graphs is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is drawn at $\lambda =0.4,$ $0.5,$ $0.6$ which are specified by different colors.
Figure 14, is periodic soliton depicted from the solution (75 ) for $s\left(t\right)=\,\cos (\lambda {\rm{t}})$ with the suitable parametric values $l=4,$ $m=7,$ $n=2.5,$ $\alpha =3.5,$ $\beta =2.5,$ $\gamma =3.5,$ $\delta =8$ and $\lambda =0.6.$ The space-time range for 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ The two-dimensional plot is shown at $\lambda =0.6,$ $0.7,$ $0.8$ with different colors.
Figure 15 signifies the singular soliton for $s\left(t\right)=\lambda {\rm{t}}$ constructed from the solution (76 ) with the particular values $l=1,$ $m=7,$ $n=5,$ $\alpha =3,$ $\beta =0.5,$ $\gamma =1.5,$ $\delta =1$ and $\lambda =1.$ The time- space range of 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ Also, the two-dimensional plot is depicted at $\lambda =1,$ $2,$ $3$ which is shown in different colors.
The solution shape in figure 16 shows the plane soliton which is drawn from the solution (77 ) for $s\left(t\right)=\,\cos (\lambda {\rm{t}})$ with the parametric values $l=-0.9,$ $m=7,$ $n=2.5,$ $\alpha =3.5,$ $\beta =2.5,$ $\gamma =3.5,$ $\delta =8$ and $\lambda =0.1.$ The space time range for 3D and 2D graph is $-5\leqslant x,t\leqslant 5.$ Also, the two-dimensional plot is drawn at λ = 0.1, $0.2,$ $0.3$ which are identified by three different colors.
For simplicity, some graphical representations of the achieved solutions are not reported here. Figures 1–16 indicate the periodic, bright, singular periodic, bell-shaped, compacton, parabolic, plane-shaped solitons etc describe the physical behavior of instances modulated by extended QZK equation and (3 + 1)-dimensional ZK equation.
5. Conclusion
In this study, we have effectively contrived the enhanced MSE technique to search for standard, definitive, and large-scale soliton solutions of the new extended QZK and (3 + 1)-dimensional ZK equations. The solutions are revealed in rational function, trigonometric function, hyperbolic function, exponential function and their integration. Bright solitons, periodic solitons, compacton, bell-shaped, parabolic, singular periodic solitons, plane shaped solitons, and some distinct and general solitons have been established. The soliton solutions that are similar to the preceding solutions validate this study, and the generic soliton solutions will enrich the literature and be valuable in future research. Two and three-dimensional graphs of these solutions are portrayed to help understand physical phenomena, namely ion-acoustic waves, dense quantum plasma, electron thermal energies, ion plasmas, quantum acoustic waves, quantum Langmuir waves, etc. This study confirms that the enhanced MSE technique is an effective and useful mathematical tool and is applicable to other NLEEs in physics, engineering and mathematical physics.