Md Nur Alam¹, M S Osman^,2,3,∗1Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh
²Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
³Department of Mathematics, Faculty of Applied Science, Umm Alqura University, Makkah 21955, Saudi Arabia

First author contact: ∗Author to whom any correspondence should be addressed.
Received:2020-09-11Revised:2020-12-7Accepted:2020-12-7Online:2021-02-04


Abstract
This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves. The modified $(G^{\prime} /G)$-expansion procedure is utilized to raise new closed-form wave solutions. Those solutions are investigated through hyperbolic, trigonometric and rational function. The graphical design makes the dynamics of the equations noticeable. It provides the mathematical foundation in diverse sectors of underwater acoustics, electromagnetic wave propagation, design of specific optoelectronic devices and physics quantum mechanics. Herein, we concluded that the studied approach is advanced, meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.
Keywords： the modified (G′/G)-expansion method;ion sound wave;Langmuir wave;closed-form wave solutions


PDF (2878KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
Md Nur Alam, M S Osman. New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves. Communications in Theoretical Physics, 2021, 73(3): 035001- doi:10.1088/1572-9494/abd849

1. Introduction

The nonlinear model of the wave equations has evolved to form the physical characteristics of science and engineering [1, 2]. The nonlinear partial differential equations (NPDEs) have importance in many categories of physical sciences, such as ocean engineering, solid state, geophysics, optics, chemistry, plasma physics, biology, mechanics, material sciences, mathematical physics and mathematics [3–10]. The closed-form wave solutions of the NPDEs are a crucial task necessary for understanding the processes and physical phenomena in the diverse applied sciences sectors. Numerous new approaches have been exploited to study this kind of solution for the NPDEs. Some crucial processes include the extended direct algebraic method [11], the sech-tanh technique [12], the sine-Gordon expansion scheme [13, 14], the finite series Jacobi elliptic cosine function ansatz [15], the extended tanh expansion method [16], the generalized exponential rational function method [17, 18], the improved tan (φ(ξ)/2)-expansion scheme [19, 20], the novel $(G^{\prime} /G)$-expansion approach [21, 22], the extended trial equation method (ETEM) [23], the improved tan (φ(ξ)/2)-expansion method (ITEM) [24], optimal homotopy and the differential transform method [25], the homogeneous balance method [26], the modified Kudryashov method [27], Riccati–Bernoulli sub-ODE method [28], the exp (− φ(ξ))-expansion method [29, 30], and the unified method and its generalized form [31–37].

This research deals with the model for Langmuir waves (LWs) and ion sound waves (ISWs) [24, 38, 39] as below(1a)$\begin{eqnarray}{\rm{i}}{E}_{t}+\displaystyle \frac{1}{2}{E}_{{xx}}-{nE}=0,\end{eqnarray}$(1b)$\begin{eqnarray}{n}_{{tt}}-{n}_{{xx}}-2{\left(| E{| }^{2}\right)}_{{xx}}=0,\end{eqnarray}$where the electric field of the Langmuir oscillation and the density perturbation are represented by $E\,{{\rm{e}}}^{{\rm{i}}{\omega }_{p}t}$ and n in the normal form, respectively. Here, we supervise the method of dynamical models concerning the ISW with the effect of the ponderomotive force owing to the high-frequency area, as well as for the LW, which is one variety of nonlinear wave models. The implementation of novel types of soliton solutions for the LW and ISWs has an extremely prominent status through the contributors. A few studies have converged on the Langmuir solitons. Musher et al [40] give support to the application of the weak turbulence hypothesis to LW turbulence, and consideration was given to plasma under relative unmagnetized and magnetized electron and ion temperatures. Zakharov [41] expressed the method of dynamical models for the LW. Benilov [42] illustrated the stability of solitons through the Zakharov model, that recognizes the interaction between ISWs and LWs. In [38, 40], a system of equations for the ISW under the action of the ponderomotive force due to the high-frequency field and the LW was discussed.

The main motivation for this work is to study the model for the LWs and ISWs via the modified $\left(\tfrac{G^{\prime} }{G}\right)$-expansion method [22, 43]. The essential advantage of this technique over the other methods in the literature is that it provides novel explicit analytical wave solutions, including many real free parameters. The closed-form wave solutions of nonlinear NPDEs have their significant meaning to reveal the interior device of the physical phenomena. Furthermore, the calculations in this method are very simple and vital to providing new solutions compared to the steps in other approaches.

The synopsis of this paper is shown below. We mention the algorithm of the new method in section 2. The implementation of our approach for solving the solitary wave phenomena of the studied model is manifested and some discussions are provided in section 3. Finally, some outcomes of the research are presented in section 4.

2. A brief description of the modified $\left(\tfrac{G^{\prime} }{G}\right)$-expansion scheme

The following strides summarize the modified $\left(\tfrac{G^{\prime} }{G}\right)$-expansion method [22, 43].

Suppose an NPDE has the form:(2)$\begin{eqnarray}H(h,{h}_{x},{h}_{{xx}},{h}_{t},{h}_{{tt}},{h}_{{xt}},\cdots )=0,\end{eqnarray}$where h = h(x, t) and H is a polynomial in its arguments. Assume that:(3)$\begin{eqnarray}h=h(x,t)=h(\xi ),\xi =k(x-{Vt}+{\xi }_{0}),\end{eqnarray}$where k, ξ₀ and V are constants. From equation (2) and equation (3), we get(4)$\begin{eqnarray}R(h,{{kh}}^{{\prime} },{k}^{2}{h}^{{\prime\prime} },-{{kVh}}^{{\prime} },{k}^{2}{V}^{2}{h}^{{\prime\prime} },-{k}^{2}{V}^{2}{h}^{{\prime\prime} },\cdots )=0.\end{eqnarray}$•Postulation 1: calculate m using the rule of the homogeneous analysis in (4).
•Postulation 2: the modified $\left(\tfrac{G^{\prime} }{G}\right)$-expansion technique suggested that(5)$\begin{eqnarray}h(\xi )=\displaystyle \sum _{i=-m}^{m}{A}_{i}{{\rm{\Theta }}}^{i},\end{eqnarray}$where ${\rm{\Theta }}=\left(\tfrac{{G}^{{\prime} }}{G}+\tfrac{\lambda }{2}\right)$, ∣A_−m∣ + ∣A_m∣ ≠ 0 and G = G(ξ) is given by(6)$\begin{eqnarray}{G}^{{\prime\prime} }+\lambda {G}^{{\prime} }+\mu G=0,\end{eqnarray}$where A_i, $\delta$ and $\mu$ are free parameters. From (6), we have(7)$\begin{eqnarray}{\rm{\Theta }}^{\prime} =b-{{\rm{\Theta }}}^{2},\end{eqnarray}$where $b=\tfrac{{\lambda }^{2}-4\mu }{4}$. So, Θ now satisfies the Riccati-like equation (7) that admits the following solutions [44]•If b > 0, then(8a)$\begin{eqnarray}{\rm{\Theta }}=\sqrt{b}\tanh (\sqrt{b}\xi ),\end{eqnarray}$(8b)$\begin{eqnarray}{\rm{\Theta }}=\sqrt{b}\coth (\sqrt{b}\xi ),\end{eqnarray}$
•If b = 0, then(9)$\begin{eqnarray}{\rm{\Theta }}=\displaystyle \frac{1}{\xi },\end{eqnarray}$
•If b < 0, then(10a)$\begin{eqnarray}{\rm{\Theta }}=-\sqrt{-b}\tan (\sqrt{-b}\xi ),\end{eqnarray}$(10b)$\begin{eqnarray}{\rm{\Theta }}=\sqrt{-b}\cot (\sqrt{-b}\xi ).\end{eqnarray}$

•Postulation 3: inserting (5) and (7) into (4), and collecting all terms with the same order of Θ together, the left-hand side of equation (4) is converted into the polynomial in Θ. Equating each coefficient of this polynomial to zero yields a set of algebraic equations which can be solved to find the values of the unknown parameters. Thus, the general solutions of equation (2) are obtained.

3. Soliton solutions for the model system related to ISWs and LWs

Here, we study the coupled nonlinear system of the model related to ISWs and LWs [24, 38, 39]:(11a)$\begin{eqnarray}{\rm{i}}{E}_{t}+\displaystyle \frac{1}{2}{E}_{{xx}}-{nE}=0,\end{eqnarray}$(11b)$\begin{eqnarray}{n}_{{tt}}-{n}_{{xx}}-2{\left(| E{| }^{2}\right)}_{{xx}}=0.\end{eqnarray}$Let us consider the transformations:(12)$\begin{eqnarray}\begin{array}{rcl}E(x,t) & = & U(\xi ){{\rm{e}}}^{{\rm{i}}\theta },n(x,t)=V(\xi ),\\ \xi & = & {rx}+{st},\theta ={px}+{qt},\end{array}\end{eqnarray}$where r, s, p and q are constants. We obtain:(13)$\begin{eqnarray}{\rm{i}}(s+{rp})U^{\prime} =0,\end{eqnarray}$(14)$\begin{eqnarray}{r}^{2}U^{\prime\prime} -(2q+{p}^{2})U-2{UV}=0,\end{eqnarray}$(15)$\begin{eqnarray}({s}^{2}-{r}^{2})V^{\prime\prime} -2{r}^{2}({U}^{2})^{\prime\prime} =0.\end{eqnarray}$From (13) and by integrating (15) twice w.r.t. ξ, we get:(16)$\begin{eqnarray}V(\xi )=\displaystyle \frac{2}{{p}^{2}-1}{U}^{2}(\xi ),s=-{rp},\end{eqnarray}$where the integration's constants are identical zero. Substituting equation (16) into equation (14), we reach:(17)$\begin{eqnarray}{r}^{2}({p}^{2}-1)U^{\prime\prime} -({p}^{2}-1)(2q+{p}^{2})U-4{U}^{3}=0,\end{eqnarray}$where $U^{\prime} =\tfrac{{\rm{d}}U}{{\rm{d}}\xi }$ and $U^{\prime\prime} =\tfrac{{{\rm{d}}}^{2}U}{{\rm{d}}{\xi }^{2}}$. Applying the homogeneous balance condition on 17 yields the positive integer M = 1. So, the general solution takes the form:(18)$\begin{eqnarray}U(\xi )={A}_{1}{\rm{\Theta }}(\xi )+{A}_{0}+{A}_{-1}{{\rm{\Theta }}}^{-1}(\xi ).\end{eqnarray}$Plugging (18) into (17) and by applying postulation 3, we find that:•Result 1:$q=\pm \tfrac{\sqrt{4\mu {r}^{2}-2{p}^{2}-{\lambda }^{2}{r}^{2}}}{2}$, ${A}_{0}=0,{A}_{1}=0,{f}_{1}\,=\pm \sqrt{\tfrac{{p}^{2}-1}{2}}$, and ${A}_{-1}=\tfrac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}$. We get the following solutions:$\begin{eqnarray*}\begin{array}{rcl}{E}_{11}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{11}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{12}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{12}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{13}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{13}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[-\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{14}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl} & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{14}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{15}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{15}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$
•Result 2:$q=\pm \tfrac{\sqrt{4\mu {r}^{2}-2{p}^{2}-{\lambda }^{2}{r}^{2}}}{2}$, A₀ = 0, A₁ = f₁r, and A₋₁ = 0. We have:$\begin{eqnarray*}\begin{array}{rcl}{E}_{21}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{21}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{22}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{22}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{23}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{23}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[-\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{E}_{24}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{24}(x,t) & = & -\displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{25}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{25}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$
•Result 3:$q=\pm \sqrt{\tfrac{{\lambda }^{2}{r}^{2}-4\mu {r}^{2}-{p}^{2}}{2}}$, A₀ = 0, A₁ = f₁r, and ${A}_{-1}=-\tfrac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}$. We obtain:$\begin{eqnarray*}\begin{array}{rcl}{E}_{31}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \left.\displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{31}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{32}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{32}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{33}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{33}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{34}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl} & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{34}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{E}_{35}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{35}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$
•Result 4:$q=\pm \sqrt{\tfrac{8\mu {r}^{2}-2{\lambda }^{2}{r}^{2}-{p}^{2}}{2}}$, A₀ = 0, A₁ = f₁r, and ${A}_{-1}=\tfrac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}$. We get:$\begin{eqnarray*}\begin{array}{rcl}{E}_{41}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{41}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{42}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{42}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{43}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{43}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{44}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{n}_{44}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{45}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{45}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$

3.1. Physical and graphical explanation of constructed solutions

In this section, we have explained the results of the coupled nonlinear system of the model for ISWs and LWs by drawing some 3D, 2D and the contour figures of the succeeded solutions with the support of the symbolic calculation software Maple. The graphical illustrations of 3D, 2D and contour plots for some of the obtained solutions are given in figures 1–12. Figure 1 depicts the 3D plot, 2D plot and the corresponding contour plot of solution E₁₁(x, t) that behaves like the spike shape solution. Figures 2, 11 and 12 depict the 3D plot, 2D plot and the corresponding contour plot of solutions n₁₁(x, t), E₄₅(x, t) and n₄₅(x, t), respectively, that represent the singular soliton solutions. Singular solitons are other varieties of solitary waves that give up under a singularity, typically countless discontinuity. Moreover, they can be connected under solitary waves when the midpoint location of the extraordinary wave is missing. Simultaneously, with these lines it is not necessary to address the problem of singular solitons. This answer has spiked and, in this manner, can likely clarify the improvement of rogue waves. Figures 3–6, 8 and 10 depict the 3D plot, 2D plot and the corresponding contour plots of solutions E₁₃(x, t), n₁₃(x, t) E₁₄(x, t), n₁₄(x, t), n₂₃(x, t) and n₂₄(x, t), respectively, that represent the periodic wave solution. Herein, the special kind of periodic wave solutions of the solutions E₂₃(x, t) and E₂₄(x, t) are shown in figures 7 and 9.

Figure 1.

New window|Download| PPT slide
Figure 1.Graphical descriptions of the solution E₁₁(x, t) under the values p = 2, q = 0.5, r = 2, $\mu$ = 1, $\delta$ = 3 and t = 0.01 for 2D graphics.

Figure 2.

New window|Download| PPT slide
Figure 2.Graphical descriptions of the solution n₁₁(x, t) under the values p = 2, q = 0.5, r = 2, $\mu$ = 1, $\delta$ = 3 and t = 0.01 for 2D graphics.

Figure 3.

New window|Download| PPT slide
Figure 3.Graphical descriptions of the solution E₁₃(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 4.

New window|Download| PPT slide
Figure 4.Graphical descriptions of the solution n₁₃(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 5.

New window|Download| PPT slide
Figure 5.Graphical descriptions of the solution E₁₄(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 6.

New window|Download| PPT slide
Figure 6.Graphical descriptions of the solution n₁₄(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 7.

New window|Download| PPT slide
Figure 7.Graphical descriptions of the solution E₂₃(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 8.

New window|Download| PPT slide
Figure 8.Graphical descriptions of the solution n₂₃(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 9.

New window|Download| PPT slide
Figure 9.Graphical descriptions of the solution E₂₄(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 10.

New window|Download| PPT slide
Figure 10.Graphical descriptions of the solution n₂₄(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 11.

New window|Download| PPT slide
Figure 11.Graphical descriptions of the solution E₄₅(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

Figure 12.

New window|Download| PPT slide
Figure 12.Graphical descriptions of the solution n₄₅(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01.

4. Conclusion

In this work, we successfully executed the modified $(G^{\prime} /G)$-expansion approach to obtain the closed-form wave structures of the studied equation. Furthermore, this approach concerned new closed-form wave structures like rational, trigonometrical and hyperbolic function solutions. The techniques prescribed the well-designed systems for supervising various nonlinear wave equations with integer-order and also with fraction order arising in several applications of applied sciences. As can be seen from the above solution process, the modified $(G^{\prime} /G)$-expansion method is very effective for solving different types of NPDEs. Our results show that the structures of the obtained wave solutions are multifarious in nonlinear dynamic systems. In the near future, we will modify the algorithm presented here to deal with different NPDEs when their coefficients are variables, for exhaling nonautonomous wave solutions.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1]Durur H Ilhan E Bulut H 2020Novel complex wave solutions of the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation
Fractal Fract. 4 41
DOI:10.3390/fractalfract4030041 [Cited within: 1]

[2]Gao W Baskonus H M Shi L 2020New investigation of bats-hosts-reservoir-people coronavirus model and application to 2019-nCoV system
Adv. Differ. Equ. 2020 391
DOI:10.1186/s13662-020-02831-6 [Cited within: 1]

[3]Manafian J Mohammed S A Alizadeh A A Baskonus H M Gao, W 2020Investigating lump and its interaction for the third-order evolution equation arising propagation of long waves over shallow water
Eur. J. Mech. B Fluids 84 289301
DOI:10.1016/j.euromechflu.2020.04.013 [Cited within: 1]

[4]Gao W Senel M Yel G Baskonus H M Senel B 2020New complex wave patterns to the electrical transmission line model arising in network system
AIMS Math. 5 18811892
DOI:10.3934/math.2020125

[5]Goyal M Baskonus H M Prakash A 2020Regarding new positive, bounded and convergent numerical solution of nonlinear time fractional HIV/AIDS transmission model
Chaos Soliton Fract. 139110096
DOI:10.1016/j.chaos.2020.110096

[6]Liu J G Zhu W H Osman M S Ma W X 2020An explicit plethora of different classes of interactive lump solutions for an extension form of 3D-Jimbo-Miwa model
Eur. Phys. J. Plus 135 412
DOI:10.1140/epjp/s13360-020-00405-9

[7]Ali K K Osman M S Abdel-Aty M 2020New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via Sine-Gordon expansion method
Alex. Eng. J. 59 11911196
DOI:10.1016/j.aej.2020.01.037

[8]Gao W Veeresha P Baskonus H M Prakasha D G Kumar P 2019A new study of unreported cases of 2019-nCOV epidemic outbreaks
Chaos Soliton 109929
DOI:10.1016/j.chaos.2020.109929

[9]Garca Guirao J L Baskonus H M Kumar A Rawat M S Yel G 2020Complex patterns to the (3 + 1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation
Symmetry 12 17
DOI:10.3390/sym12010017

[10]Lu D Osman M S Khater M M A Attia R A M Baleanu D 2020Analytical and numerical simulations for the kinetics of phase separation in iron (Fe-Cr-X (X = Mo, Cu)) based on ternary alloys
Physica 537122634
DOI:10.1016/j.physa.2019.122634 [Cited within: 1]

[11]Kurt A Tozar A Tasbozan O 2020Applying the new extended direct algebraic method to solve the equation of obliquely interacting waves in shallow waters
J. Ocean U. China 19 772780
DOI:10.1007/s11802-020-4135-8 [Cited within: 1]

[12]Guo M Dong H Liu J Yang H 2019The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method
Nonlinear Anal. Model. 24 119
DOI:10.15388/NA.2019.1.1 [Cited within: 1]

[13]Abdul Kayum M Ali Akbar M Osman M S 2020Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma
Wave Random Complex
DOI:10.1080/17455030.2020.1831711 [Cited within: 1]

[14]Baskonus H M Bulut H 2016New wave behaviors of the system of equations for the ion sound and Langmuir waves
Waves Random Complex Media 26 613625
DOI:10.1080/17455030.2016.1181811 [Cited within: 1]

[15]Kumar V S Rezazadeh H Eslami M Izadi F Osman M S 2019Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity
Int. J. Appl. Comput. Math. 5 127
DOI:10.1007/s40819-019-0710-3 [Cited within: 1]

[16]Osman M S Ali K K Gómez-Aguilar J F 2020A variety of new optical soliton solutions related to the nonlinear Schrödinger equation with time-dependent coefficients
Optik 222165389
DOI:10.1016/j.ijleo.2020.165389 [Cited within: 1]

[17]Ghanbari B Osman M S Baleanu D 2019Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative
Mod. Phys. Lett. A 341950155
DOI:10.1142/S0217732319501554 [Cited within: 1]

[18]Osman M S Ghanbari B Machado J A T 2019New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity
Eur. Phys. J. Plus 134 20
DOI:10.1140/epjp/i2019-12442-4 [Cited within: 1]

[19]Mohyud-Din S T Irshad A Ahmed N Khan U 2017Exact solutions of (3 + 1)-dimensional generalized KP equation arising in physics
Results Phys. 7 39013909
DOI:10.1016/j.rinp.2017.10.007 [Cited within: 1]

[20]Raza N Osman M S Abdel-Aty A H Abdel-Khalek S Besbes H R 2020Optical solitons of space-time fractional Fokas-Lenells equation with two versatile integration architectures
Adv. Differ. Equ. 2020 517
DOI:10.1186/s13662-020-02973-7 [Cited within: 1]

[21]Alam M N Akbar M A Mohyud-Din S T 2014A novel $(G^{\prime} /G)$-expansion method and its application to the Boussinesq equation
Chin. Phys. B 23020203020203020210
DOI:10.1088/1674-1056/23/2/020203 [Cited within: 1]

[22]Akbar M A Alam M N Hafez M G 2016Application of the Novel $({G}^{{\prime} }/G)$-expansion method to traveling wave solutions for the positive Gardner-KP equation
Indian J. Pure Appl. Math. 47 8596
DOI:10.1007/s13226-016-0171-x [Cited within: 3]

[23]Demiray S T Bulut H 2016New exact solutions of the system of equations for the ion sound and Langmuir waves by ETEM
Math. Comput. Appl. 21 11
DOI:10.3390/mca21020011 [Cited within: 1]

[24]Manafian J 2017Application of the ITEM for the system of equations for the ion sound and Langmuir waves
Opt. Quant. Electron. 49 17
DOI:10.1007/s11082-016-0860-z [Cited within: 3]

[25]Parsa A B Rashidi M M Anwar O B Sadri S M 2013Semi-computational simulation of magnetohemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods
Comput. Biol. Med. 43 11421153
DOI:10.1016/j.compbiomed.2013.05.019 [Cited within: 1]

[26]Zhao X Wang L Sun W 2006The repeated homogeneous balance method and its applications to nonlinear partial differential equations
Chaos Soliton. Frac. 28 448453
DOI:10.1016/j.chaos.2005.06.001 [Cited within: 1]

[27]Hosseini K Ansari R 2017New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method
Waves Random Complex Media 27 628636
DOI:10.1080/17455030.2017.1296983 [Cited within: 1]

[28]Mirzazadeh M Alqahtani R T Biswas A 2017Optical soliton perturbation with quadratic-cubic nonlinearity by Riccati-Bernoulli sub-ODE method and Kudryashov's scheme
Optik 145 7478
DOI:10.1016/j.ijleo.2017.07.011 [Cited within: 1]

[29]Alam M N Alam M M 2017An analytical method for solving exact solutions of a nonlinear evolution equation describing the dynamics of ionic currents along microtubules
J. Taibah Univ. Sci. 11 939948
DOI:10.1016/j.jtusci.2016.11.004 [Cited within: 1]

[30]Alam M N Belgacem F B M 2016Microtubules nonlinear models dynamics investigations through the exp-φ(ξ)-expansion method implementation
Mathematics 4 6
DOI:10.3390/math4010006 [Cited within: 1]

[31]Abdel-Gawad H I Osman M S 2013On the variational approach for analyzing the stability of solutions of evolution equations
Kyungpook Math. J. 53 661680
DOI:10.5666/KMJ.2013.53.4.680 [Cited within: 1]

[32]Osman M S Rezazadeh H Eslami M 2019Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
Nonlinear Eng. 8 559567
DOI:10.1515/nleng-2018-0163

[33]Osman M S Rezazadeh H Eslami M Neirameh A Mirzazadeh M 2018Analytical study of solitons to Benjamin-Bona-Mahony-Peregrine equation with power law nonlinearity by using three methods
U.P.B. Sci. Bull. Series A 80 267278


[34]Abdel-Gawad H I Osman M 2014Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method
Indian J. Pure Appl. Math. 45 112
DOI:10.1007/s13226-014-0047-x

[35]Osman M S 2017Analytical study of rational and double-soliton rational solutions governed by the KdV-Sawada-Kotera-Ramani equation with variable coefficients
Nonlinear Dyn. 89 22832289
DOI:10.1007/s11071-017-3586-y

[36]Osman M S Abdel-Gawad H I 2015Multi-wave solutions of the (2 + 1)-dimensional Nizhnik-Novikov-Veselov equations with variable coefficients
Eur. Phys. J. Plus 130 215
DOI:10.1140/epjp/i2015-15215-1

[37]Abdel-Gawad H I Elazab N S Osman M 2013Exact solutions of space dependent Korteweg-de Vries equation by the extended unified method
J. Phys. Soc. Jpn. 82044004
DOI:10.7566/JPSJ.82.044004 [Cited within: 1]

[38]Baskonus H M Bulut H 2016New wave behaviors of the system of equations for the ion sound and Langmuir waves
Waves Random Complex Media 26 613625
DOI:10.1080/17455030.2016.1181811 [Cited within: 3]

[39]Khater M M A Attia R A M Park C Lu D 2020On the numerical investigation of the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves
Results Phys. 18103317
DOI:10.1016/j.rinp.2020.103317 [Cited within: 2]

[40]Musher S L Rubenchik A M Zakharov V E 1995Weak Langmuir turbulence
Phys. Rep. 252 178274
DOI:10.1016/0370-1573(94)00071-A [Cited within: 2]

[41]Zakharov V E 1972Collapse of Langmuir waves
Sov. Phys. JETP 35 908914
[Cited within: 1]

[42]Benilov E S 1985Stability of plasma solitons
Zhurnal. Eksp. Theor. Fiz. 88 120128
[Cited within: 1]

[43]Alam M N Tunc C 2020Constructions of the optical solitons and other solitons to the conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
J. Taibah Univ. Sci. 14 94100
DOI:10.1080/16583655.2019.1708542 [Cited within: 2]

[44]Engui F 2000Extended tanh-function method and its applications to nonlinear equations
Phys. Lett. A 277 212218
DOI:10.1016/S0375-9601(00)00725-8 [Cited within: 1]