Bright, periodic, compacton and bell-shape soliton solutions of the extended QZK and (3【-逻*辑*与-】nbsp
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M Ali Akbar1, Md Abdul Kayum1,2, M S Osman,3,*1Department of Applied Mathematics, University of Rajshahi, Bangladesh 2Department of Computer Science & Engineering, North Bengal International University, Rajshahi, Bangladesh 3Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
First author contact:*Author to whom any correspondence should be addressed. Received:2021-06-08Revised:2021-07-12Accepted:2021-08-03Online:2021-08-31
Abstract The (3+1)-dimensional Zakharov–Kuznetsov (ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy, ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation (EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic, trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bell-shape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering. Keywords：(3 + 1)-dimensional ZK;the extended QZK equation;enhanced modified simple equation method;soliton solutions;NLEEs
PDF (2079KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article M Ali Akbar, Md Abdul Kayum, M S Osman. Bright, periodic, compacton and bell-shape soliton solutions of the extended QZK and (3+1)-dimensional ZK equations. Communications in Theoretical Physics, 2021, 73(10): 105003- doi:10.1088/1572-9494/ac1a6c
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 (3) into a nonlinear differential equation in the follow way:
$\begin{eqnarray} {\mathcal M} \left(V,(\dot{p}x+\dot{q}y+\dot{r}z+\dot{s})V^{\prime} ,\,pV^{\prime} ,qV^{\prime} ,\,\ldots \right)=0,\end{eqnarray}$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 (4), in agreement with the EMSE method can be put into the form:
$\begin{eqnarray}V\left(\xi \right)={a}_{0}\left(t\right)+\displaystyle \sum _{i=1}^{N}{a}_{i}(t){\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)}^{i},\end{eqnarray}$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 (4), results the balance number $N$ arise in solution (5).
Using the solution (5) and its necessary derivatives into equation (4), together with the balance number $N,$ yield a polynomial of $\psi {\left(\xi \right)}^{-i},$ where $i=1,$ $2,$ $3,$…,$N.$ Collecting all the terms of same power and equating them to zero, it is attained a system of differential and algebraic equations, can be estimated to find ${a}_{0},$ ${a}_{i},$ $p,$ $q,$ $r,$ $s$ and $\psi (\xi ).$ Thus, we can establish inclusive, fresh and standard soliton solutions of (3) by substituting the above values into the solution (5).
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 transformation$\begin{eqnarray}v\left(x,y,t\right)=V(\xi ),\,\xi =p\left(t\right)x+q\left(t\right)y+r\left(t\right),\end{eqnarray}$remodel the equation (1) into the subsequent 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 (8) after integrating (7),$\begin{eqnarray}\begin{array}{ccc}2\left(\dot{p}x+\dot{q}y+\dot{r}\right)V + \alpha p{V}^{2}+2\beta \left({p}^{3}+{q}^{3}\right)V^{\prime\prime} \\ + 2\gamma pq\left(p+q\right)V^{\prime\prime} =0,\end{array}\end{eqnarray}$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 form$\begin{eqnarray}V\left(\xi \right)={a}_{0}\left(t\right)+{a}_{1}\left(t\right)\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)+{a}_{2}(t){\left(\displaystyle \frac{\psi ^{\prime} \left(\xi \right)}{\psi \left(\xi \right)}\right)}^{2}.\end{eqnarray}$Inserting solution (9) 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:$\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 (11), (12), (14), (15), (17) and (18) give the subsequent results:
$\dot{p}=0$ 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 obtain$\begin{eqnarray*}{a}_{1}\left(t\right)=\pm \displaystyle \frac{12{\rm{i}}\sqrt{\dot{r}\,\left(h+k\right)}\,\sqrt{{h}^{2}\beta -hk\beta +{k}^{2}\beta +hk\gamma }}{h\alpha }.\end{eqnarray*}$Since the values of $p,$ $q,$ ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi $ satisfy equation (16), the values of $p,$ $q,$ ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi $ are admissible to determine the soliton solutions.
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 be$\begin{eqnarray}V\left(\xi \right)=\pm \displaystyle \frac{144\,\dot{r}}{h\alpha \,}\,{\left[\pm 11\,\cosh \left(\displaystyle \frac{G\xi }{2}\right)\pm 13\,\sinh \left(\displaystyle \frac{G\xi }{2}\right)\right]}^{-2}.\end{eqnarray}$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 (8)$\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), yields$\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 (9), we attain the ensuing solution$\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 as$\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 (33), we extract$\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 obtain$\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}$$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1+\displaystyle \frac{3}{2}{\mathrm{csch}}^{2}\left(\displaystyle \frac{H}{2}\left(hx+ky+r\left(t\right)\right)\right)\right].\end{eqnarray}$The periodic soliton for solutions (38) and (39) is considered to be:$\begin{eqnarray}V\left(x,y,t\right)=-\displaystyle \frac{2\dot{r}}{h\alpha }\left[1-\displaystyle \frac{3}{2}{\text{sec}}^{2}\left(\displaystyle \frac{G}{2}\left(hx+ky+r\left(t\right)\right)\right)\right],\end{eqnarray}$$\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 (28)–(31) and (38)–(41) represent bright, parabolic soliton, bell-shape soliton, singular periodic soliton, and compacton.
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 variable$\begin{eqnarray}\begin{array}{l}v\left(x,y,z,\,t\right)=V(\xi ),\,\xi =p\left(t\right)x+q\left(t\right)y\\ \,+r\left(t\right)z+s(t),\end{array}\end{eqnarray}$reduces the (3+1)-dimensional ZK equation (2) into the ensuing 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 (43) with zero integrating constant, we obtain$\begin{eqnarray}\begin{array}{l}2\left(\dot{p}x+\dot{q}y+\dot{r}z+\dot{s}\right)V+\alpha r{V}^{2}\\ \,+2(\beta {r}^{3}+\gamma {p}^{2}r+\delta {q}^{2}r)V^{\prime\prime} =0.\end{array}\end{eqnarray}$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:$\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 (47)–(49), (51)–(53) and (55)–(57), it is found$\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 acquire$\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 (54) 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.
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 attain$\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 (62) turns out to be$\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 (44) in the ensuing$\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}$$\begin{eqnarray}V\left(x,y,z,t\right)=\displaystyle \frac{3\dot{s}}{n\alpha }{\mathrm{csch}}^{2}\left(M\left(lx+my+nz+s\left(t\right)\right)/2\right).\end{eqnarray}$In relation to (66) and (67), the trigonometric function solutions can be simulated as:$\begin{eqnarray}V\left(x,y,z,t\right)=-\displaystyle \frac{3\dot{s}}{n\alpha }{\text{sec}}^{2}\left(N\left(lx+my+nz+s\left(t\right)\right)/2\right),\end{eqnarray}$$\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 ${a}_{1},$ yield$\begin{eqnarray*}{a}_{1}=\pm \displaystyle \frac{12\,\sqrt{{s}_{1}}\,\sqrt{{r}^{2}\beta +{p}^{2}\gamma +{q}^{2}\delta }}{\sqrt{r}\alpha }.\end{eqnarray*}$For these values of ${a}_{0},$ ${a}_{1},$ ${a}_{2}$ and $\psi ,$ from solution (45), we attain$\begin{eqnarray}V\left(\xi \right)=-\dot{\displaystyle \frac{2s}{n\alpha }}\displaystyle \frac{\left({c}_{3}^{2}\,{{\rm{e}}}^{\pm 2T\xi }\,\mp 4\,T\,{c}_{3}{c}_{4}\,{{\rm{e}}}^{\pm T\xi }+{c}_{4\,}^{2}{T}^{2}\right)}{{\left({c}_{3}\,{{\rm{e}}}^{\pm T\xi }+{c}_{4}T\right)}^{2}},\end{eqnarray}$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 as$\begin{eqnarray}\begin{array}{l}V\left(\xi \right)\\ \,=\,-\displaystyle \frac{2\dot{s}}{n\alpha }\left(1\mp \displaystyle \frac{6T{{c}}_{3}{{c}}_{4}}{{\left\{\left({{c}}_{3}\pm {{c}}_{4}{\rm{T}}\right)\cosh \left(\displaystyle \frac{T\xi }{2}\right)\pm \left({{c}}_{3}\mp {{c}}_{4}T\right)\sinh \left(\displaystyle \frac{T\xi }{2}\right)\,\right\}}^{2}}\right).\end{array}\,\end{eqnarray}$For the values ${c}_{3}=\pm 6$ and ${c}_{4}=\pm \tfrac{1}{T},$ solution (71) turns to the soliton solutions:$\begin{eqnarray}V\left(\xi \right)=-\displaystyle \frac{2\dot{s}}{n\alpha }\left(1\mp \displaystyle \frac{36}{{\left\{\pm 7\,\cosh \left(\displaystyle \frac{T\xi }{2}\right)\pm 5\,\sinh \left(\displaystyle \frac{T\xi }{2}\right)\,\right\}}^{2}}\right).\end{eqnarray}$Furthermore, if we set ${c}_{3}=\pm T,$ ${c}_{4}=\pm 1$ or ${c}_{3}=\pm 1,$ ${c}_{4}=\pm \tfrac{1}{T},$ solution (72) turns into:$\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}$Relating to (75) and (76), the trigonometric function solutions can be constructed as:$\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 (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.
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.
New window|Download| PPT slide Figure 1.The 3D and 2D graphics for $r\left(t\right)=\lambda t$ of the solution (24) with the parametric values $h=2,\,k=1,\,\alpha =6,\,\beta =2,\,\gamma =2$ and $\lambda =2.$
Figure 2.
New window|Download| PPT slide Figure 2.The 3D and 2D graphics for r (t)=λt of the solution (28) with the parametric values $h=0.5,$ $k=0.2,$ $\alpha =0.3,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =0.5.$
Figure 3.
New window|Download| PPT slide Figure 3.The 3D and 2D graphics for $r\left(t\right)=\,\sinh \left(\lambda t\right)$ of the solution (28) with parametric values $h=0.7,$ $k=-1,$ $\alpha =-1,$ $\beta =2,$ $\gamma =3$ and $\lambda =0.2.$
Figure 4.
New window|Download| PPT slide Figure 4.The 3D and 2D graphics for $r\left(t\right)=\lambda t$ of the solution (29) with particular values $h=4,$ $k=4,$ $\alpha =3,$ $\beta =-0.1,$ $\gamma =0.3$ and $\lambda =4.$
Figure 5.
New window|Download| PPT slide Figure 5.The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution (30) with the particular values $h=2.8,$ $k=3,$ $\alpha =3,$ $\beta =2,$ $\gamma =1.9$ and $\lambda =2.$
Figure 6.
New window|Download| PPT slide Figure 6.The 3D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution (34) with the particular values $h=0.7,$ $k=1,$ $\alpha =0.4,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =-1.$
Figure 7.
New window|Download| PPT slide Figure 7.The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the solution (38) with the particular values $h=0.7,$ $k=1,$ $\alpha =0.4,$ $\beta =0.4,$ $\gamma =0.6$ and $\lambda =1.5.$
Figure 8.
New window|Download| PPT slide Figure 8.The 3D and 2D graphics for $r\left(t\right)=\lambda {\rm{t}}$ of the particular solution (40) with the parametric values $h=9,$ $k=-0.8,$ $\alpha =2,$ $\beta =4,$ $\gamma =6$ and $\lambda =-0.1.$
Figure 9.
New window|Download| PPT slide Figure 9.The 3D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the solution (63) with the parametric values $l=-0.1,$ $m=0.3,$ $n=1,$ $\delta =0.8,$ $\gamma =0.6,$ $\beta =0.9,$ $\alpha =0.4$ and $\lambda =0.2.$
Figure 10.
New window|Download| PPT slide Figure 10.The 3D and 2D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the general solution (66) with the parametric values $l=0.6,$ $m=0.8,$ $n=1,$ $\alpha =2,$ $\beta =4,$ $\gamma =2,$ $\delta =1$ and λ=1.
Figure 11.
New window|Download| PPT slide Figure 11.The 3D and 2D graphics for $s\left(t\right)={{\rm{e}}}^{\lambda {\rm{t}}}$ of the solution (66) with the particular values $l=5,$ $m=0.5,$ $n=2,$ $\alpha =3,$ $\beta =1,$ $\gamma =2,$ $\delta =2$ and $\lambda =-0.4.$
Figure 12.
New window|Download| PPT slide Figure 12.The 3D and 2D graphics for $s\left(t\right)=\,\sin \left(\lambda {\rm{t}}\right)$ of the particular solution (67) with the particular values $l=1,$ $m=0.2,$ $n=0.9,$ $\delta =1,$ $\gamma =1.5,$ $\beta =2.5,$ $\alpha =1.5$ and $\lambda =0.1.$
Figure 13.
New window|Download| PPT slide Figure 13.The 3D and 2D graphics for $s\left(t\right)=\,\tanh \left(\lambda {\rm{t}}\right)$ of the solution (68) with the particular values $l=5,$ $m=5,$ $n=2,$ $\delta =1,$ $\gamma =2,$ $\beta =1,$ $\alpha =3$ and $\lambda =0.4.$
Figure 14.
New window|Download| PPT slide Figure 14.The 3D and 2D graphics for $s\left(t\right)=\,\cos \left(\lambda {\rm{t}}\right)$ of the particular solution (75) with the particular values $l=4,$ $m=7,$ $n=2.5,$ $\alpha =3.5,$ $\beta =2.5,$ $\gamma =3.5,$ $\delta =8$ and $\lambda =0.6.$
Figure 15.
New window|Download| PPT slide Figure 15.The 3D and 2D graphics for $s\left(t\right)=\lambda {\rm{t}}$ of the particular solution (76) with the particular values $l=1,$ $m=7,$ $n=5,$ $\alpha =3,$ $\beta =0.5,$ $\gamma =1.5,$ $\delta =1$ and $\lambda =1.$
Figure 16.
New window|Download| PPT slide Figure 16.The 3D and 2 graphics for $s\left(t\right)=\,\cos (\lambda {\rm{t}})$ of the solution (77) with the parametric values $l=-0.9,$ $m=7,$ $n=2.5,$ $\alpha =3.5,$ $\beta =2.5,$ $\gamma =3.5,$ $\delta =8$ and $\lambda =0.1.$
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.
Acknowledgments
The authors would like to express their gratitude to the anonymous referees for their insightful remarks and ideas to improve the quality of the article.
LuDSeadawyA RWangJArshadMFarooqU2019 Soliton solutions of the generalized third-order nonlinear Schrödinger equation by two mathematical methods and their stability Pramana 93 19 DOI:10.1007/s12043-019-1804-5 [Cited within: 1]
KayumM AAkbarM AOsmanM S2020 Competent closed form soliton solutions to the nonlinear transmission and the low-pass electrical transmission lines Eur. Phys. J. Plus 135 120 DOI:10.1140/epjp/s13360-020-00573-8 [Cited within: 1]
LuDSeadawyAArshadM2017 Applications of extended simple equation method on unstable nonlinear Schrödinger equations Optik 140 136144 DOI:10.1016/j.ijleo.2017.04.032 [Cited within: 1]
KaplanMBekirAAkbulutA2016 A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics Nonlinear Dyn. 85 28432850 DOI:10.1007/s11071-016-2867-1 [Cited within: 1]
RezazadehHKorkmazAEslamiMMirhosseini-AlizaminiS M2019 A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method Opt. Quantum Electron. 51 112 DOI:10.1007/s11082-019-1801-4 [Cited within: 1]
BaskonusH MKoçD ABulutH2016 New travelling wave prototypes to the nonlinear Zakharov-Kuznetsov equation with power law nonlinearity Nonlinear Sci. Lett. A 7 6776 [Cited within: 1]
IncMAliyuA IYusufA2017 Dark optical, singular solitons and conservation laws to the nonlinear Schrödinger's equation with spatio-temporal dispersion Mod. Phys. Lett. B 31 1750163 DOI:10.1142/S0217984917501639 [Cited within: 1]
Al QurashiM MYusufAAliyuA IIncM2017 Optical and other solitons for the fourth-order dispersive nonlinear Schrödinger equation with dual-power law nonlinearity Superlattices Microstruct. 105 183197 DOI:10.1016/j.spmi.2017.03.022
KayumM AAraSOsmanM SAkbarM AGepreelK A2021 Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics Results Phys. 20 103762 DOI:10.1016/j.rinp.2020.103762 [Cited within: 1]
Abdul KayumMAli AkbarMOsmanM S2020 Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma Wave Random Complex122 DOI:10.1080/17455030.2020.1831711 [Cited within: 1]
OsmanM S2019 One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation Nonlinear Dyn. 96 14911496 DOI:10.1007/s11071-019-04866-1 [Cited within: 1]
KhaterM MAnwarSTariqK UMohamedM S2021 Some optical soliton solutions to the perturbed nonlinear Schrödinger equation by modified Khater method AIP Adv. 11 025130 DOI:10.1063/5.0038671 [Cited within: 1]
KumarSJiwariRMittalR CAwrejcewiczJ2021 Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model Nonlinear Dyn. 104 661682 [Cited within: 1]
KhanY2021 A novel type of soliton solutions for the Fokas-Lenells equation arising in the application of optical fibers Mod. Phys. Lett. B 35 2150058 DOI:10.1142/S0217984921500585 [Cited within: 1]
KilicBIncM2017 Optical solitons for the Schrödinger-Hirota equation with power law nonlinearity by the Bäcklund transformation Optik 138 6467 DOI:10.1016/j.ijleo.2017.03.017 [Cited within: 1]
ZayedE MAlngarM E2021 Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms Math. Met. Appl. Sci. 44 315324 DOI:10.1002/mma.6736 [Cited within: 1]
GepreelK AZayedE M EAlngarM E M2021 New optical solitons perturbation in the birefringent fibers for the CGL equation with Kerr law nonlinearity using two integral schemes methods Optik 227 166099 DOI:10.1016/j.ijleo.2020.166099 [Cited within: 1]
AkbarM AAliN H MZayedE M E2012 Abundant exact traveling wave solutions of generalized bretherton equation via improved (G′/G)-expansion method Commun. Theor. Phys. 57 173 DOI:10.1088/0253-6102/57/2/01 [Cited within: 1]
KhanKAkbarM A2013 Application of exp-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation World Appl. Sci. J. 24 13731377 [Cited within: 1]
Ali AkbarMAliN H M2017 The improved F-expansion method with Riccati equation and its applications in mathematical physics Cogent Math. 4 1282577 DOI:10.1080/23311835.2017.1282577 [Cited within: 1]
AkbarM AKayumM AOsmanM SAbdel-AtyA HEleuchH2021 Analysis of voltage and current flow of electrical transmission lines through mZK equation Results Phys. 20 103696 DOI:10.1016/j.rinp.2020.103696 [Cited within: 1]
KayumM AAraSBarmanH KAkbarM A2020 Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines Results Phys. 18 103269 DOI:10.1016/j.rinp.2020.103269
KayumM ABarmanH KAkbarM A2021 Exact soliton solutions to the nano-bioscience and biophysics equations through the modified simple equation method Proc. of the 6th Int. Conf. on Mathematics and Computing Singapore Springer 469482
KayumM ASeadawyA RAkbarA MSugatiT G2020 Stable solutions to the nonlinear RLC transmission line equation and the Sinh-Poisson equation arising in mathematical physics Open Phys. 18 710725 DOI:10.1515/phys-2020-0183 [Cited within: 1]
ZayedE M EArnousA H2013 The enhanced modified simple equation method for solving nonlinear evolution equations with variable coefficients AIP Conf. Proc. 1558 19992004 [Cited within: 1]
IncMAtesETchierF2016 Optical solitons of the coupled nonlinear Schrödinger's equation with spatiotemporal dispersion Nonlinear Dyn. 85 13191329 DOI:10.1007/s11071-016-2762-9 [Cited within: 1]
AslanE CIncM,2017 Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis Waves Random Complex Media 27 594601 DOI:10.1080/17455030.2017.1286060
AslanE CTchierFIncM2017 On optical solitons of the Schrödinger-Hirota equation with power law nonlinearity in optical fibers Superlattices Microstruct. 105 4855 DOI:10.1016/j.spmi.2017.03.014
KayumM ARoyRAkbarM AOsmanM S2021 Study of W-shaped, V-shaped, and other type of surfaces of the ZK-BBM and GZD-BBM equations Opt. Quantum Electron. 53 387 DOI:10.1007/s11082-021-03031-6 [Cited within: 1]
BarmanH KAktarM SUddinM HAkbarM ABaleanuDOsmanM S2021 Physically significant wave solutions to the Riemann wave equations and the Landau-Ginsburg-Higgs equation Results Phys. 27 104517 DOI:10.1016/j.rinp.2021.104517
AlmusawaHNur AlamMFayz-Al-AsadMOsmanM S2021 New soliton configurations for two different models related to the nonlinear Schrödinger equation through a graded-index waveguide AIP Adv. 11 065320 DOI:10.1063/5.0053565
LiuJ GOsmanM SWazwazA M2019 A variety of nonautonomous complex wave solutions for the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients in nonlinear optical fibers Optik 180 917923 DOI:10.1016/j.ijleo.2018.12.002
DingYOsmanM SWazwazA M2019 Abundant complex wave solutions for the nonautonomous Fokas–Lenells equation in presence of perturbation terms Optik 181 503513 DOI:10.1016/j.ijleo.2018.12.064 [Cited within: 1]
SeadawyA R2016 Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma Comput. Math. Appl. 71 201212 DOI:10.1016/j.camwa.2015.11.006 [Cited within: 1]
NuruddeenR IAboodhK SAliK K2018 Analytical investigation of soliton solutions to three quantum Zakharov-Kuznetsov equations Commun. Theor. Phys. 70 405 DOI:10.1088/0253-6102/70/4/405 [Cited within: 1]
MoslemW MAliSShuklaP KTangX YRowlandsG2007 Solitary, explosive, and periodic solutions of the quantum Zakharov-Kuznetsov equation and its transverse instability Phys. Plasmas 14 082308 DOI:10.1063/1.2757612 [Cited within: 2]
WangG WXuT ZJohnsonSBiswasA2014 Solitons and Lie group analysis to an extended quantum Zakharov-Kuznetsov equation Astrophys. Space Sci. 349 317327 DOI:10.1007/s10509-013-1659-z [Cited within: 1]
JiangY LLuYChenC2016 Conservation Laws and optimal system of extended quantum Zakharov-Kuznetsov equation J. Nonlinear Math. Phys. 23 157166 DOI:10.1080/14029251.2016.1161258 [Cited within: 1]
El-GanainiSAkbarM A2016 Conservation laws and multiple simplest equation methods to an extended quantum Zakharov-Kuznetsov equation Nonlinear Anal. Differ. Equ. 4 405426 DOI:10.12988/nade.2016.6633 [Cited within: 1]
BaskonusH MBulutHSulaimanT A2017 Investigation of various travelling wave solutions to the extended (2+1)-dimensional quantum ZK equation Eur. Phys. J. Plus 132 18 DOI:10.1140/epjp/i2017-11778-y [Cited within: 1]
ZhangB GLiuZ RXiaoQ2010 New exact solitary wave and multiple soliton solutions of quantum Zakharov-Kuznetsov equation Appl. Math. Comput. 217 392402 DOI:10.1016/j.amc.2010.05.074 [Cited within: 1]
EbadiGMojaverAMilovicDJohnsonSBiswasA2012 Solitons and other solutions to the quantum Zakharov-Kuznetsov equation Astrophys. Space Sci. 341 507513 DOI:10.1007/s10509-012-1072-z [Cited within: 1]
BhrawyA HAbdelkawyM AKumarSJohnsonSBiswasA2013 Solitons and other solutions to quantum Zakharov-Kuznetsov equation in quantum magneto-plasmas Indian J. Phys. 87 455463 DOI:10.1007/s12648-013-0248-x [Cited within: 1]
ZayedE M EAlurrfiK A E2016 Extended generalized (G′/G) -expansion method for solving the nonlinear quantum Zakharov-Kuznetsov equation Ric. di Mat. 65 235254 DOI:10.1007/s11587-016-0276-x [Cited within: 1]
LuDSeadawyA RArshadMWangJ2017 New solitary wave solutions of (3+1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications Results Phys. 7 899909 DOI:10.1016/j.rinp.2017.02.002 [Cited within: 1]
SindiC TManafianJ2017 Soliton solutions of the quantum Zakharov-Kuznetsov equation which arises in quantum magneto-plasmas Eur. Phys. J. Plus 132 123 DOI:10.1140/epjp/i2017-11354-7 [Cited within: 1]
ZayedE MShahootA MAlurrfiK A2018 The (G′/G,1/G) -expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrödinger equation and the quantum Zakharov-Kuznetsov equation Opt. Quantum Electron. 50 96 DOI:10.1007/s11082-018-1337-z [Cited within: 1]
VinitaRayS S2021 Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3+1)-dimensional extended quantum Zakharov-Kuznetsov equation in quantum physics Mod. Phys. Lett. B 35 2150163 DOI:10.1142/S0217984921501633 [Cited within: 1]
RoshidM MAliM ZRezazadehH2020 Kinky periodic pulse and interaction of bell wave with kink pulse wave propagation in nerve fibers and wall motion in liquid crystals Partial Differ. Equ. Appl. Math. 2 100012 DOI:10.1016/j.padiff.2020.100012 [Cited within: 2]
ZhangCZhangZ2017 Application of the enhanced modified simple equation method for Burger-Fisher and modified Volterra equations Adv. Differ. Equ. 2017 18 DOI:10.1186/s13662-017-1198-y [Cited within: 1]
RachedZ S2018 On exact solutions of Phi-4 partial differential equation using the enhanced modified simple equation method Asian J. Appl. Sci. 6 14 DOI:10.24203/ajas.v6i4.5433 [Cited within: 1]
RachedZ2019 On exact solutions of Chafee-Infante differential equation using enhanced modified simple equation method J. Interdis. Math. 22 969974 DOI:10.1080/09720502.2019.1696922 [Cited within: 1]
IslamM SRoshidM MRahmanA LAkbarM A2019 Solitary wave solutions in plasma physics and acoustic gravity waves of some nonlinear evolution equations through enhanced MSE method J. Phys. Commun. 3 125011 DOI:10.1088/2399-6528/ab5eac [Cited within: 2]
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