Aly R Seadawy^,1, Muhammad Younis², Muhammad Z Baber³, Syed T R Rizvi⁴, Muhammad S Iqbal³¹Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
²Department of Computer Science, University of the Punjab, Lahore, Pakistan
³Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
⁴Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan

Received:2021-04-5Revised:2021-07-20Accepted:2021-07-29Online:2021-09-27


Abstract
In this study, the (3+1)-dimensional fractional time–space Kadomtsev–Petviashivili (FTSKP) equation is considered and analyzed analytically, which propagates the acoustic waves in an unmagnetized dusty plasma. The fractional derivatives are studied in a confirmable sense. The new modified extended direct algebraic (MEDA) approach is adopted to investigate the diverse nonlinear wave structures. A variety of new families of hyperbolic and trigonometric solutions are obtained in single and different combinations. The obtained results are also constructed graphically with the different parametric choices.
Keywords： exact solutions;fractional calculus;new MEDA technique;unmagnetized dusty plasma


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Cite this article
Aly R Seadawy, Muhammad Younis, Muhammad Z Baber, Syed T R Rizvi, Muhammad S Iqbal. Diverse acoustic wave propagation to confirmable time–space fractional KP equation arising in dusty plasma. Communications in Theoretical Physics, 2021, 73(11): 115004- doi:10.1088/1572-9494/ac18bb

1. Introduction

Fractional partial differential equations (FPDEs) describe the nonlinear behavior of many physical phenomena in different fields. Therefore, it is imperative to find exact solutions of such type of model. Kadomtsev and Petviashvili extracted solutions to a (2+1)-dimensional KP equation [1, 2].

FPDEs are commonly used to describe problems in fluid mechanics, geographies, plasma physics, and thermal and mechanical systems. Increasing attention has been given by different scientists to find the exact solution for the fractional order reasoning. Many methods are being developed and are gradually maturing [3–9].

In this paper, the (3+1)-dimensional FTSKP equation is under investigation. This equation is used to analyze the complex dust acoustic wave structures [10, 11]. Diverse families of hyperbolic, trigonometric and plane wave solutions are constructed with different arguments. The new MEDA method [12–15] is adopted to derive the exact solutions. The (3+1)-dimensional FTSKP equation is read as follow [16, 17];(1)$\begin{eqnarray}\begin{array}{l}{{\rm{D}}}_{t}^{\alpha }{{\rm{D}}}_{x}^{\alpha }u+{a}_{1}{\left({{\rm{D}}}_{x}^{\alpha }u\right)}^{2}+{a}_{1}u{{\rm{D}}}_{x}^{2\alpha }u\\ +{a}_{2}{{\rm{D}}}_{x}^{4\alpha }u+{a}_{3}{{\rm{D}}}_{y}^{2\alpha }u+{a}_{3}{{\rm{D}}}_{z}^{2\alpha }u=0.\end{array}\end{eqnarray}$In recent decades, exact solutions [18], analytical solutions [19] and numerical solutions [20] of many NPDEs have been successfully obtained. For example, the construction of bright-dark solitary waves and elliptic function solutions are observed in [21]. A fractional order model was used to find lump solutions in dusty plasma [22]. Dispersive shock wave solutions were also constructed in [23]. There are also many different methods for obtaining exact explicit solutions; the exp-function method [24, 25], the MEDA method [26], the extended auxiliary equation method [27] and modified method of simplest equation, the $(G^{\prime} /{G}^{2})$-expansion method [28], modified mapping method [29], extended homogeneous balance method [30] and extended Fan's sub-equation method [31], Lie algebraic discussion for affinity based information diffusion in social networks; analytical solution for an in-host viral infection model with time-inhomogeneous rates [32–34]; extended and modified direct algebraic method, extended mapping method, and Seadawy techniques [35–43].

2. Wave propagation

To find the exact solutions of equation (1) we convert it into an ordinary differential equation by using the following transformation(2)$\begin{eqnarray}\begin{array}{l}u(x,y,z,t)=U(\xi ),\quad {\rm{where}}\\ \xi =\displaystyle \frac{p}{\alpha }{x}^{\alpha }+\displaystyle \frac{q}{\alpha }{y}^{\alpha }+\displaystyle \frac{r}{\alpha }{z}^{\alpha }-\displaystyle \frac{c}{\alpha }{t}^{\alpha }.\end{array}\end{eqnarray}$Equation (1) changes into the fractional order ordinary deferential equation as(3)$\begin{eqnarray}\begin{array}{l}{{pcU}}^{{\prime\prime} }(\xi )+{\alpha }_{1}{\left({{pU}}^{{\prime} }(\xi )\right)}^{2}+{\alpha }_{1}{{pUU}}^{{\prime\prime} }(\xi )\\ +{\alpha }_{2}{p}^{4}{U}^{{\prime} \prime\prime\prime }(\xi )+{\alpha }_{3}{q}^{2}{U}^{{\prime\prime} }(\xi )+{\alpha }_{3}{r}^{2}{U}^{{\prime\prime} }(\xi )=0.\end{array}\end{eqnarray}$Suppose the solutions of equation (3) can be expressed as U(ξ) in the form of [15].(4)$\begin{eqnarray}U(\xi )=\sum _{i=0}^{N}{b}_{i}{Q}^{i}(\xi ),\quad {b}_{i}\ne 0,\end{eqnarray}$where b_i(0 ≤ i ≤ N) are constants and Q(ξ) is satisfy the equation (3). Here we take(5)$\begin{eqnarray}\begin{array}{l}{Q}^{{\prime} }(\xi )=\mathrm{ln}(B)(\mu +\lambda Q(\xi )+{{rQ}}^{2}(\xi ))\\ B\ne 0,1\end{array}\end{eqnarray}$The value of N for equation (4) is taken by homogeneous balancing principle from equation (3) by putting equal to highest derivative term and highest nonlinear term. It gives N = 1 and takes expression from the solutions of equation (4) as(6)$\begin{eqnarray}u(\xi )={b}_{0}+{b}_{1}Q(\xi ).\end{eqnarray}$Substituting equation (6) and its derivatives in equation (3) and equating the co-efficients of the same power of Q(ξ) equal to zero, we get the system of equations easily. We further solve this system of equations by using the mathematica or maple, and get the solutions set as follows:

Case 1: where free parameters are r and μ while along with b₀ = 0, ${b}_{1}=-\tfrac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}$,$\lambda =\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}$

Type 1: For λ² − μr < 0 and r ≠ 0, the mixed trigonometric solutions are found as$\begin{eqnarray*}\begin{array}{l}{u}_{1}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}+\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left.{\tan }_{B}\left(\displaystyle \frac{\sqrt{-(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r)}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solution u₁(x, t) are depicted in figure 1, for the different choices of parameters c = 100, p = 100.101, μ = 0.1, s = 0.0005, α₁ =20, α₂ = 0.9, α₃ = 1, r = 0.0002, q = 0.1, and B = 5.$\begin{eqnarray*}\begin{array}{l}{u}_{2}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}-\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left.{\cot }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solution u₁(x, t) are depicted in figure 2, for the values of parameters c = 120, p = 1, μ = 0.1, s = 0.0005, α₁ = 20, α₂ = 0.9, α₃ = 1, q = 0.0002, q = 0.1, and B = 5.$\begin{eqnarray*}\begin{array}{l}{u}_{3}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}+\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left({\tan }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right.\\ \left.\left.\pm \sqrt{({pq})}{\sec }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₃(x, t) are depicted in figure 3, for the values of parameters c = 20, p = 20, μ = 2.1, s = 10.5, α₁ = 20, α₂ = 0.9, α₃ = 1, q = 2.2, q = 1.1, and B = 2.$\begin{eqnarray*}\begin{array}{l}{u}_{4}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}-\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left({\cot }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right.\\ \left.\left.\pm \sqrt{({pq})}{\csc }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄(x, t) are depicted in figure 4, for the values of parameters c = 10, p = 20, μ = 21, s = 15, α₁ = 30, α₂ = 1.9, α₃ = 10, q = 22.2, q = 1.1, and B = 5.$\begin{eqnarray*}\begin{array}{l}{u}_{5}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}+\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left({\tan }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{4}\xi \right)\right.\\ \left.\left.-\sqrt{({pq})}{\cot }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{4}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₅(x, t) are depicted in figure 5, for the values of parameters c = 10, p = 20, μ = 21, s = 15, α₁ = 30, α₂ = 1.9, α₃ = 10, q = 22.2, q = 1.1, and B = 5.

Figure 1.

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Figure 1.Graphical representation and corresponding contour of u₁(x, t) for different values of parameters.

Figure 2.

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Figure 2.Graphical representation and corresponding contour of u₂(x, t) for different values of parameters.

Figure 3.

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Figure 3.Graphical representation and corresponding contour of u₃(x, t) for different values of parameters.

Figure 4.

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Figure 4.Graphical representation and corresponding contour of u₄(x, t) for different values of parameters.

Figure 5.

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Figure 5.Graphical representation and corresponding contour of u₅(x, t) for different values of parameters.


Type 2: For λ² − μr > 0 and r ≠ 0, different types of solutions are obtained.

The dark solutions are obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{6}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}+\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left.{\tanh }_{B}\left(\displaystyle \frac{\sqrt{-(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r)}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₆(x, t) are depicted in figure 6, for the values of parameters c = 10, p = 10.101, μ = 0.1, s = 11.5, α₁ = 20, α₂ = 0.9, α₃ = 1, q = 0.0002, q = 0.1, and B = 5.

Figure 6.

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Figure 6.Graphical representation and corresponding contour of u₆(x, t) for different values of parameters.


The singular solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{7}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}-\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left.{\coth }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The complex dark bright solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{8}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}-\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left({\tanh }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right.\\ \left.\left.\pm {\rm{i}}\sqrt{({pq})}{{\rm{{\rm{sech}} }}}_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₆(x, t) are depicted in figure 7, for the values of parameters c = 100, p = 100, μ = 21, s = 10.5, α₁ = 200, α₂ = 10.9, α₃ = 1, q = 20.2, q = 11, and B = 20.

Figure 7.

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Figure 7.Graphical representation and corresponding contour of u₈(x, t) for different values of parameters.


The mixed singular solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{9}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}-\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{2r}\right.\\ \left({\coth }_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right.\\ \left.\left.\pm \sqrt{({pq})}{{\rm{csch}}}_{B}\left(\sqrt{-\left(\displaystyle \frac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The dark solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{10}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \left(-\displaystyle \frac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{2\sqrt{5}\sqrt{{\alpha }_{2}}{{rp}}^{2}\mathrm{ln}(B)}+\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{4r}\right.\\ \left({\tanh }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{4}\xi \right)\right.\\ \left.\left.+{\coth }_{B}\left(\displaystyle \frac{\sqrt{-\left(\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}{\mathrm{ln}}^{2}(B)}-4\mu r\right)}}{4}\xi \right)\right)\right).\end{array}\end{eqnarray*}$

Type 3: For μr > 0 and λ = 0, we obtained trigonometric solutions as$\begin{eqnarray*}\begin{array}{rcl}{u}_{11}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\sqrt{\displaystyle \frac{\mu }{r}}{\tan }_{B}(\sqrt{\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₁(x, t) are depicted in figure 8, for the values of parameters c = 80, p = 1.01, μ = 0.1, s = 0.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.002, q = 0.1, and B = 3.$\begin{eqnarray*}\begin{array}{rcl}{u}_{12}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\sqrt{\displaystyle \frac{\mu }{r}}{\cot }_{B}(\sqrt{\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₂(x, t) are depicted in figure 9, for the values of parameters c = 80, p = 0.01, μ = 0.1, s = 0.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.002, q = 0.1, and B = 3.

Figure 8.

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Figure 8.Graphical representation and corresponding contour of u₁₁(x, t) for different values of parameters.

Figure 9.

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Figure 9.Graphical representation and corresponding contour of u₁₂(x, t) for different values of parameters.


The mixed trigonometric solutions are obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{13}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \sqrt{\displaystyle \frac{\mu }{r}}\left({\tan }_{B}(2\sqrt{\mu r}\xi )\pm \sqrt{{pq}}{\sec }_{B}(2\sqrt{\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₃(x, t) are depicted in figure 10, for the values of parameters c = 100, p = 100, μ = 1.1, s = 1.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.02, q = 0.1, and B = 3.$\begin{eqnarray*}\begin{array}{rcl}{u}_{14}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \sqrt{\displaystyle \frac{\mu }{r}}\left({\cot }_{B}\left(2\sqrt{\mu r}\xi \right)\pm \sqrt{{pq}}{\csc }_{B}\left(2\sqrt{\mu r}\xi \right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{15}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\mu }{r}}\left({\tan }_{B}(\displaystyle \frac{\sqrt{\mu r}}{2}\xi )-\sqrt{{pq}}{\cot }_{B}(\displaystyle \frac{\sqrt{\mu r}}{2}\xi )\right),\end{array}\end{eqnarray*}$

Figure 10.

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Figure 10.Graphical representation and corresponding contour of u₁₃(x, t) for different values of parameters.


Type 4: For μr < 0 and λ = 0, we obtained dark solutions as$\begin{eqnarray*}\begin{array}{rcl}{u}_{16}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\sqrt{-\displaystyle \frac{\mu }{r}}{\tanh }_{B}(\sqrt{-\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₆(x, t) are depicted in figure 11, for the values of parameters c = 80, p = 1.01, μ = 0.1, s = 0.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.02, q = 0.1, and B = 3. We get the singular solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{17}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\sqrt{-\displaystyle \frac{\mu }{r}}{\coth }_{B}(\sqrt{-\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₇(x, t) are depicted in figure 12, for the values of parameters c = 80, p = 0.01, μ = 0.1, s = 0.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.002, q = 0.1, and B = 3.

Figure 11.

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Figure 11.Graphical representation and corresponding contour of u₁₆(x, t) for different values of parameters.

Figure 12.

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Figure 12.Graphical representation and corresponding contour of u₁₇(x, t) for different values of parameters.


The different type of complex combo solutions are obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{18}(\xi )=\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \sqrt{-\displaystyle \frac{\mu }{r}}\left({\tanh }_{B}(2\sqrt{-\mu r}\xi )\pm {\rm{i}}\sqrt{{pq}}{{\rm{{\rm{sech}} }}}_{B}(2\sqrt{-\mu r}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₁₈(x, t) are depicted in figure 13, for the values of parameters c = 100, p = 100, μ = 1.1, s = 1.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.02, q = 0.1, and B = 3.$\begin{eqnarray*}\begin{array}{l}{u}_{19}(\xi )=\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \sqrt{-\displaystyle \frac{\mu }{r}}\left({\coth }_{B}(2\sqrt{-\mu r}\xi )\pm \sqrt{{pq}}{{csch}}_{B}(2\sqrt{-\mu r}\xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{20}(\xi )=-\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \times \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{-\mu }{r}}\left({\tanh }_{B}(\displaystyle \frac{\sqrt{-\mu r}}{2}\xi )-{\coth }_{B}(\displaystyle \frac{\sqrt{-\mu r}}{2}\xi )\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₂₀(x, t) are depicted in figure 14, for the values of parameters c = 100, p = 100, μ = 1.1, s = 1.05, α₁ = 10, α₂ = 0.9, α₃ = 1, q = 0.02, q = 0.1, and B = 3.

Figure 13.

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Figure 13.Graphical representation and corresponding contour of u₁₈(x, t) for different values of parameters.

Figure 14.

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Figure 14.Graphical representation and corresponding contour of u₂₀(x, t) for different values of parameters.


Type 5: For λ = 0 and μ = r, the periodic and mixed periodic solutions are obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{21}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\tan }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{22}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(-{\cot }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{23}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\tan }_{B}(2\mu \xi )\pm \sqrt{{pq}}{\sec }_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{24}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\cot }_{B}(2\mu \xi )\pm \sqrt{{pq}}{{\csc }}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{25}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \displaystyle \frac{1}{2}\left({\tan }_{B}(\displaystyle \frac{\sqrt{\mu }}{2}\xi )-{\cot }_{B}(\displaystyle \frac{\mu }{2}\xi )\right).\end{array}\end{eqnarray*}$Type 6: For λ = 0 and r = −μ, we obtained different types of solutions as$\begin{eqnarray*}\begin{array}{rcl}{u}_{26}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\tanh }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{27}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\coth }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{28}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\tanh }_{B}(2\mu \xi )\pm {\rm{i}}\sqrt{{pq}}{{{\rm{sech}} }}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{29}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({\coth }_{B}(2\mu \xi )\pm \sqrt{{pq}}{{csch}}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{30}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \displaystyle \frac{1}{2}\left({\tanh }_{B}(\displaystyle \frac{\sqrt{\mu }}{2}\xi )+{\coth }_{B}(\displaystyle \frac{\mu }{2}\xi )\right).\end{array}\end{eqnarray*}$Type 7: For λ² = 4μr we obtained only one solutions as$\begin{eqnarray*}\begin{array}{rcl}{u}_{31}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\displaystyle \frac{2\mu \left(\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}}\right)\xi +2\right)}{\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}\mathrm{ln}(B)}\xi }\right).\end{array}\end{eqnarray*}$Type 8: For λ = χ, μ = νχ(ν ≠ 0) and r = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{32}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left({B}^{\chi \xi }-\nu \right).\end{array}\end{eqnarray*}$Type 9: For λ = r = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{33}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \left(\mu \xi \mathrm{ln}(B)\right).\end{array}\end{eqnarray*}$Type 10: For λ = μ = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{34}(\xi ) & = & \displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\displaystyle \frac{1}{r\xi \mathrm{ln}(B)}\right).\end{array}\end{eqnarray*}$Type 11: For μ = 0, and λ ≠ 0 we obtained mixed hyperbolic solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{35}(\xi )=\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}}{r\left({\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)-{\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{36}(\xi )=\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)\right)}{r\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$Type 12: For λ = χ, r = νχ(ν ≠ 0) and μ = 0, we obtained a plane solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{37}(\xi ) & = & -\displaystyle \frac{6\sqrt{5}\sqrt{{\alpha }_{2}}{pr}\mathrm{ln}(B)\sqrt{{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{{\alpha }_{1}+{\alpha }_{1}p}\\ & & \times \left(\displaystyle \frac{{{pB}}^{\chi \xi }}{p-\nu {{qB}}^{\chi \xi }}\right).\end{array}\end{eqnarray*}$Case 2: where free parameters are μ, r and λ while along with ${b}_{0}=\tfrac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}$, ${b}_{1}=-\tfrac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)},\mu =-\tfrac{{\lambda }^{2}}{2r}$.

Type 1: For λ² − μr < 0 and r ≠ 0, the mixed trigonometric solutions are found as$\begin{eqnarray*}\begin{array}{rcl}{u}_{38}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}{\tan }_{B}\left(\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2}\xi \right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{39}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}{\cot }_{B}\left(\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2}\xi \right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{40}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}\left({\tan }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right.\right.\\ & & \left.\left.\pm \sqrt{({pq})}{\sec }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{41}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}\left({\cot }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right.\right.\\ & & \left.\left.\pm \sqrt{({pq})}{\csc }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{rcl}{u}_{42}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}\left({\tan }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right.\right.\\ & & \left.\left.-\sqrt{({pq})}{\cot }_{B}\left(\sqrt{-3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$Type 2: For λ² − μr > 0 and r ≠ 0, there are different types of solutions are obtained.

The dark solutions are obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{43}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{2r}{\tanh }_{B}\left(\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄₃(x, t) are depicted in figure 15, for the values of parameters c = 2, p = 2, μ = 3.1, s = 10.05, α₁ = 3, α₂ = 1.9, α₃ = 1, q = 0.2, q = 0.1, and B = 3.

Figure 15.

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Figure 15.Graphical representation and corresponding contour of u₄₃(x, t) for different values of parameters.


The singular solution is obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{44}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{2r}{\coth }_{B}\left(\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄₄(x, t) are depicted in figure 16, for the values of parameters c = 2, p = 2, μ = 3.1, s = 10.05, α₁ = 3, α₂ = 1.9, α₃ = 1, q = 0.2, q = 0.1, and B = 3.

Figure 16.

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Figure 16.Graphical representation and corresponding contour of u₄₄(x, t) for different values of parameters.


The complex dark bright solution is obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{45}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{2r}\left({\tanh }_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right.\right.\\ & & \left.\left.\pm {\rm{i}}\sqrt{({pq})}{{{\rm{sech}} }}_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄₅(x, t) are depicted in figure 17, for the values of parameters c = 20, p = 20, μ = 1.011, s = 1.05, α₁ = 10, α₂ = 10.9, α₃ = 2, q = 10.2, q = 10.1, and B = 5.

Figure 17.

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Figure 17.Graphical representation and corresponding contour of u₄₅(x, t) for different values of parameters.


The mixed singular solution is obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{46}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ & & \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-3{\lambda }^{2}}}{2r}\left({\coth }_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right.\right.\\ & & \left.\left.\pm \sqrt{({pq})}{{csch}}_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄₆(x, t) are depicted in figure 18, for the values of parameters c = 20, p = 20, μ = 1.011, s = 1.05, α₁ = 10, α₂ = 10.9, α₃ = 2, q = 10.2, q = 10.1, and B = 5.

Figure 18.

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Figure 18.Graphical representation and corresponding contour of u₄₆(x, t) for different values of parameters.


The dark solution is obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{47}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{4r}\right.\\ & & \left.\times \left({\tanh }_{B}\left(\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{4}\xi \right)+{\coth }_{B}\left(\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{4}\xi \right)\right)\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₄₇(x, t) are depicted in figure 19, for the values of parameters c = 20, p = 20, μ = 1.011, s = 1.05, α₁ = 10, α₂ = 10.9, α₃ = 2, q = 10.2, q = 10.1, and B = 5.

Figure 19.

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Figure 19.Graphical representation and corresponding contour of u₄₇(x, t) for different values of parameters.


Type 3: For λ² = μr we have trigonometric solutions obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{48}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & +\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(\displaystyle \frac{\lambda \mathrm{ln}(B)\xi +2}{\rho r\mathrm{ln}(B)}\right).\end{array}\end{eqnarray*}$Type 4: For λ = χ, μ = νχ(ν ≠ 0) and r = 0 we have trigonometric solutions obtained as$\begin{eqnarray*}\begin{array}{rcl}{u}_{49}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left({B}^{\chi \rho }-\nu \right).\end{array}\end{eqnarray*}$Type 5: For λ = χ, r = νχ(ν ≠ 0) and μ = 0 we have obtained a plane solution as$\begin{eqnarray*}\begin{array}{rcl}{u}_{50}(\xi ) & = & \displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ & & -\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\left(\displaystyle \frac{{{pB}}^{\chi \rho }}{p-\nu {{qB}}^{\chi \rho }}\right).\end{array}\end{eqnarray*}$Type 6: For μ = 0, and λ ≠ 0 we obtained mixed hyperbolic solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{51}(\xi )=\displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ +\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}}{r\left({\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)-{\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₅₁(x, t) are depicted in figure 20, for the values of parameters c = 200, p = 20, μ = 100, s = 1.05, α₁ = 100, α₂ = 10.9, α₃ = 2, q = 10.2, q = 10.1, and B = 5.$\begin{eqnarray*}\begin{array}{l}{u}_{52}(\xi )=\displaystyle \frac{{\alpha }_{2}\left(-{\lambda }^{2}\right){p}^{4}{\mathrm{ln}}^{2}(B)-{cp}-{\alpha }_{3}{q}^{2}-{\alpha }_{3}{s}^{2}}{{\alpha }_{1}p}\\ +\displaystyle \frac{12{\alpha }_{2}\lambda {p}^{3}r{\mathrm{ln}}^{2}(B)}{{\alpha }_{1}(2p+1)}\\ \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)\right)}{r\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$

Figure 20.

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Figure 20.Graphical representation and corresponding contour of u₅₁(x, t) for different values of parameters.


Case 3: Where free parameters are μ, r and λ while along with b₀ = b₀, ${b}_{1}=-\tfrac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}$

Type 1:For λ² − μr < 0 and r ≠ 0, the mixed trigonometric solutions are found as$\begin{eqnarray*}\begin{array}{l}{u}_{53}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}{\tan }_{B}\left(\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2}\xi \right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{54}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}{\cot }_{B}\left(\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2}\xi \right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{55}(\xi )={b}_{0}-\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}\left({\tan }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right.\right.\\ \left.\left.\pm \sqrt{({pq})}{\sec }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$The plot and its corresponding contour plot of the solutions u₅₅(x, t) are depicted in figure 21, for the values of parameters c = 2, p = 2, μ = 0.011, s = 1.05, α₁ = 1, α₂ = 0.9, α₃ = 2, q = 10.2, q = 10.1, and B = 5.$\begin{eqnarray*}\begin{array}{l}{u}_{56}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}\left({\cot }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right.\right.\\ \left.\left.\pm \sqrt{({pq})}{\csc }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{57}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}\left({\tan }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right.\right.\\ \left.\left.-\sqrt{({pq})}{\cot }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$

Figure 21.

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Figure 21.Graphical representation and corresponding contour of u₅₅(x, t) for different values of parameters.


Type 2 For λ² − μr > 0 and r ≠ 0, different types of solutions are obtained.

The dark solutions are obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{58}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}{\tanh }_{B}\left(\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2}\xi \right)\right),\end{array}\end{eqnarray*}$The singular solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{59}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}{\coth }_{B}\left(\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2}\xi \right)\right).\end{array}\end{eqnarray*}$The complex dark bright solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{60}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \,\left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}\left({\tanh }_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right.\right.\\ \left.\left.\pm {\rm{i}}\sqrt{({pq})}{{{\rm{sech}} }}_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$The mixed singular solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{61}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}-\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{2r}\left({\coth }_{B}\left(\sqrt{-({\lambda }^{2}-4\mu r)}\xi \right)\right.\right.\\ \left.\left.\pm \sqrt{({pq})}{{csch}}_{B}\left(\sqrt{3{\lambda }^{2}}\xi \right)\right)\Space{0ex}{3.5ex}{0ex}\right).\end{array}\end{eqnarray*}$The dark solution is obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{62}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-\displaystyle \frac{\lambda }{2r}+\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{4r}\left({\tanh }_{B}\left(\displaystyle \frac{\sqrt{3{\lambda }^{2}}}{4}\xi \right)\right.\right.\\ \left.\left.+{\coth }_{B}\left(\displaystyle \frac{\sqrt{-({\lambda }^{2}-4\mu r)}}{4}\xi \right)\right)\right).\end{array}\end{eqnarray*}$Type 3: For μr > 0 and λ = 0, we obtained trigonometric solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{63}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{\displaystyle \frac{\mu }{r}}{\tan }_{B}(\sqrt{\mu r}\xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{64}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{\displaystyle \frac{\mu }{r}}{\cot }_{B}(\sqrt{\mu r}\xi )\right),\end{array}\end{eqnarray*}$The mixed trigonometric solutions were obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{65}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{\displaystyle \frac{\mu }{r}}\left({\tan }_{B}(2\sqrt{\mu r}\xi )\pm \sqrt{{pq}}{\sec }_{B}(2\sqrt{\mu r}\xi )\right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{66}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{\displaystyle \frac{\mu }{r}}\left({\cot }_{B}(2\sqrt{\mu r}\xi )\pm \sqrt{{pq}}{\csc }_{B}(2\sqrt{\mu r}\xi )\right)\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{67}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\mu }{r}}\left({\tan }_{B}(\displaystyle \frac{\sqrt{\mu r}}{2}\xi )-\sqrt{{pq}}{\cot }_{B}(\displaystyle \frac{\sqrt{\mu r}}{2}\xi )\right).\end{array}\end{eqnarray*}$Type 4: For μr < 0 and λ = 0, we obtained dark solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{68}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{-\displaystyle \frac{\mu }{r}}{\tanh }_{B}(\sqrt{-\mu r}\xi )\right),\end{array}\end{eqnarray*}$We get the singular solution as$\begin{eqnarray*}\begin{array}{l}{u}_{69}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\sqrt{-\displaystyle \frac{\mu }{r}}{\coth }_{B}(\sqrt{-\mu r}\xi )\right),\end{array}\end{eqnarray*}$The different type of complex combo solutions are obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{70}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \sqrt{-\displaystyle \frac{\mu }{r}}\left({\tanh }_{B}(2\sqrt{-\mu r}\xi )\right.\\ \left.\pm {\rm{i}}\sqrt{{pq}}{{\rm{{\rm{sech}} }}}_{B}(2\sqrt{-\mu r}\xi \right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{71}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \sqrt{-\displaystyle \frac{\mu }{r}}\left({\coth }_{B}(2\sqrt{-\mu r}\xi )\right.\\ \left.\pm \sqrt{{pq}}{{\rm{csch}}}_{B}(2\sqrt{-\mu r}\xi \right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{72}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{-\mu }{r}}\left({\tanh }_{B}(\displaystyle \frac{\sqrt{-\mu r}}{2}\xi )-{\coth }_{B}(\displaystyle \frac{\sqrt{-\mu r}}{2}\xi )\right).\end{array}\end{eqnarray*}$Type 5: For λ = 0 and μ = r, the periodic and mixed periodic solutions are obtained as$\begin{eqnarray*}\begin{array}{l}{u}_{73}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\tan }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{74}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(-{\cot }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{75}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\tan }_{B}(2\mu \xi )\pm \sqrt{{pq}}{\sec }_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{76}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\cot }_{B}(2\mu \xi )\pm \sqrt{{pq}}{{\rm{\csc }}}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{77}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \displaystyle \frac{1}{2}\left({\tan }_{B}(\displaystyle \frac{\sqrt{\mu }}{2}\xi )-{\cot }_{B}(\displaystyle \frac{\mu }{2}\xi )\right).\end{array}\end{eqnarray*}$Type 6: For λ = 0 and r = −μ, we obtained different types of solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{78}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\tanh }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{79}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\coth }_{B}(\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{80}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\tanh }_{B}(2\mu \xi )\pm {\rm{i}}\sqrt{{pq}}{{\rm{{\rm{sech}} }}}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{81}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left({\coth }_{B}(2\mu \xi )\pm \sqrt{{pq}}{{\rm{csch}}}_{B}(2\mu \xi )\right),\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{82}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \displaystyle \frac{1}{2}\left({\tanh }_{B}(\displaystyle \frac{\sqrt{\mu }}{2}\xi )+{\coth }_{B}(\displaystyle \frac{\mu }{2}\xi )\right).\end{array}\end{eqnarray*}$Type 7: For λ² = 4μr we obtained only one solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{83}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\displaystyle \frac{2\mu \left(\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}}\right)\xi +2\right)}{\tfrac{{\alpha }_{3}({q}^{2}+{s}^{2})}{5{\alpha }_{2}{p}^{4}\mathrm{ln}(B)}\xi }\right).\end{array}\end{eqnarray*}$Type 8: For λ = χ, μ = νχ(ν ≠ 0) and r = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{l}{u}_{84}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \left({B}^{\chi \xi }-\nu \right).\end{array}\end{eqnarray*}$Type 9: For λ = r = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{l}{u}_{85}(\xi )=-{b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \left(\mu \xi \mathrm{ln}(B)\right).\end{array}\end{eqnarray*}$Type 10: For λ = μ = 0, we obtained only one solution as$\begin{eqnarray*}\begin{array}{l}{u}_{86}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \left(\displaystyle \frac{1}{r\xi \mathrm{ln}(B)}\right).\end{array}\end{eqnarray*}$Type 11: For μ = 0, and λ ≠ 0 we obtained mixed hyperbolic solutions as$\begin{eqnarray*}\begin{array}{l}{u}_{87}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}}{r\left({\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)-{\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{88}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \times \left(\displaystyle \frac{p\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)\right)}{r\left({\sinh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+{\cosh }_{B}\left(\tfrac{\sqrt{{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}}}{\sqrt{5}\sqrt{{\alpha }_{2}}{p}^{2}\mathrm{ln}(B)}\xi \right)+p\right)}\right).\end{array}\end{eqnarray*}$Type 12: For λ = χ, r = νχ(ν ≠ 0) and μ = 0, we obtained a plane solution as$\begin{eqnarray*}\begin{array}{l}{u}_{89}(\xi )={b}_{0}\\ -\displaystyle \frac{\lambda \left({\alpha }_{1}{b}_{0}p+{\alpha }_{2}{\lambda }^{2}{p}^{4}{\mathrm{ln}}^{2}(B)+8{\alpha }_{2}\mu {p}^{4}r{\mathrm{ln}}^{2}(B)+{cp}+{\alpha }_{3}{q}^{2}+{\alpha }_{3}{s}^{2}\right)}{{\alpha }_{1}\mu {p}^{2}}\\ \left(\displaystyle \frac{{{pB}}^{\chi \xi }}{p-\nu {{qB}}^{\chi \xi }}\right).\end{array}\end{eqnarray*}$In the all above solutions, the generalized hyperbolic and trigonometric functions are defined as

${\sinh }_{B}(\rho )$	$\tfrac{{{pB}}^{\rho }-{{qB}}^{-\rho }}{2}$	${\cosh }_{B}(\rho )$	$\tfrac{{{pB}}^{\rho }+{{qB}}^{-\rho }}{2}$
${\tanh }_{B}(\rho )$	$\tfrac{{{pB}}^{\rho }-{{qB}}^{-\rho }}{{{pB}}^{\rho }+{{qB}}^{-\rho }}$	${\coth }_{B}(\rho )$	$\tfrac{{{pB}}^{\rho }+{{qB}}^{-\rho }}{{{pB}}^{\rho }-{{qB}}^{-\rho }}$
${\sin }_{B}(\rho )$	$\tfrac{{{pB}}^{{\rm{i}}\rho }-{{qB}}^{-{\rm{i}}\rho }}{2}$	${\cos }_{B}(\rho )$	$\tfrac{{{pB}}^{{\rm{i}}\rho }+{{qB}}^{-{\rm{i}}\rho }}{2}$
${\cot }_{B}(\rho )$	$-{\rm{i}}\tfrac{{{pB}}^{{\rm{i}}\rho }-{{qB}}^{-{\rm{i}}\rho }}{{{pB}}^{{\rm{i}}\rho }+{{qB}}^{-{\rm{i}}\rho }}$	${\cot }_{B}(\rho )$	$i\tfrac{{{pB}}^{{\rm{i}}\rho }+{{qB}}^{-{\rm{i}}\rho }}{{{pB}}^{{\rm{i}}\rho }-{{qB}}^{-{\rm{i}}\rho }}$

where ρ = ρ(x, y, z, t) and p, q > 0.

3. Conclusion

In this work, the (3 + 1)-dimensional fractional time-space KP (FTSKP) equation is under investigation, which propagates the acoustic waves in an unmagnetized dusty plasma. This fractional model is defined using the confirmable fractional derivatives. The diverse new families of hyperbolic, trigonometric, rational, and plane wave solutions are obtained in single and different combinations using the new modified extended direct algebraic (MEDA) technique. Graphical representations of the obtained results are also depicted with different choices of parameters.

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[1]Xu Z Chen H Dai Z 2014 Rogue wave for the (2+1)-dimensional Kadomtsev-Petviashvili equation
Appl. Math. Lett. 37 34 38
DOI:10.1016/j.aml.2014.05.005 [Cited within: 1]

[2]Seadawy A R 2017 Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma, Mathematical methods and applied
Sciences 40 1598 1607
[Cited within: 1]

[3]Kalim Ul-Haq TariqSeadawy Aly R 2019 Soliton solutions of (3+1)-dimensional Korteweg-de Vries Benjamin-Bona-Mahony, Kadomtsev-Petviashvili Benjamin-Bona-Mahony and modified Korteweg de Vries-Zakharov-Kuznetsovequations and their applications in water waves
J. King Saud University—Sci. 31 8 13
DOI:10.1016/j.jksus.2017.02.004 [Cited within: 1]

[4]Seadawy Aly R 2017 Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves
Eur. Phys. J Plus 132 1 1329
DOI:10.1140/epjp/i2017-11313-4

[5]Aly R 2017 Seadawy, Solitary wave solutions of tow-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in a dust acoustic plasmas
The Pramana—J. Phys. 89 1 1149


[6]Seadawy A El-Rashidy K 2018 Dispersive Solitary wave solutions of Kadomtsev-Petviashivili and modified Kadomtsev-Petviashivili dynamical equations in unmagnetized dust plasma
Results Phys. 8 1216 1222
DOI:10.1016/j.rinp.2018.01.053

[7]Ul-Haq Tariq K Seadawy A R 2018 Computational soliton solutions to (3+1)-dimensional generalized Kadomtsev-Petviashvili and (2+1)-dimensional Gardner-Kadomtsev-Petviashvili models and their applications
The Pramana—J. Phys. 91 1 13


[8]Ahmed I Seadawy A R Lu D 2019 Mixed lump-solitons, periodic lump and breather soliton solutions for (2+1)-dimensional extended Kadomtsev-Petviashvili dynamical equation
Int. J. Mod. Phys. B 33 1950019
DOI:10.1142/S021797921950019X

[9]Seadawy A R Ali A Albarakati W A 2019 Analytical wave solutions of the (2+1)-dimensional first integro-differential Kadomtsev-Petviashivili hierarchy equation by using modified mathematical methods
Results Phys. 15 102775
DOI:10.1016/j.rinp.2019.102775 [Cited within: 1]

[10]Guo M et al. 2018 Study of ion-acoustic solitary waves in a magnetized plasma using the three-dimensional time-space fractional Schamel-KdV equation
Complexity 2018 6852548
DOI:10.1155/2018/6852548 [Cited within: 1]

[11]Fu L Yang H 2019 An application of (3+1)-dimensional time-space fractional ZK model to analyze the complex dust acoustic waves
Complexity 2019 2806724
DOI:10.1155/2019/2806724 [Cited within: 1]

[12]Ma S H Fang J P Zheng C L 2009 New exact solutions of the (2+1)-dimensional breaking soliton system via an extended mapping method
Chaos Solitons and Fractals 40 210 214
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[13]Soliman A A Abdo H A 2012 New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method
11 45 57arXiv:1207.5127


[14]Younas U Younis M Seadawy A R Rizvi S T R Althobaiti S Sayed S 2021 Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative
Results Phys. 20 103766
DOI:10.1016/j.rinp.2020.103766

[15]Younas U Seadawy A R Younis M Rizvi S T R 2020 Dispersive of propagation wave structures to the Dullin-Gottwald-Holm dynamical equation in a shallow water waves
Chin. J. Phys. 68 348 364
DOI:10.1016/j.cjph.2020.09.021 [Cited within: 2]

[16]Sun J Fu L Yang H 2019 Analytical study of (3+1)-dimensional fractional ultralow-frequency dust acoustic waves in a dual-temperature plasma
J. Low Freq. Noise Vib. Active Control 38 928 952
DOI:10.1177/1461348418817991 [Cited within: 1]

[17]Fu L Yang H 2019 An application of (3+1)-dimensional time-space fractional ZK model to analyze the complex dust acoustic waves
Complexity2019
DOI:10.1155/2019/2806724 [Cited within: 1]

[18]Obaidullah U Jamal S 2020 A computational procedure for exact solutions of Burgers' hierarchy of nonlinear partial differential equations
J. Appl. Math. Comput.1 11
DOI:10.1007/s12190-020-01403-x [Cited within: 1]

[19]Zhang R F Bilige S 2019 Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation
Nonlinear Dyn. 95 3041 3048
DOI:10.1007/s11071-018-04739-z [Cited within: 1]

[20]Tu K M Yih K A Chou F I Chou J H 2020 Numerical solution and Taguchi experimental method for variable viscosity and non-Newtonian fluids effects on heat and mass transfer by natural convection in porous media
Int. J. Comput. Sci. Eng. 22 252 261
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[21]Sarwar A Gang T Arshad M Ahmed I 2020 Construction of bright-dark solitary waves and elliptic function solutions of space-time fractional partial differential equations and their applications
Phys. Scr. 95 045227
DOI:10.1088/1402-4896/ab6d46 [Cited within: 1]

[22]Sun J C Zhang Z G Dong H H Yang H W 2019 Fractional Order Model and Lump Solution in Dusty Plasma 68 210201
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[23]Bettelheim E Abanov A G Wiegmann P 2006 Nonlinear quantum shock waves in fractional quantum Hall edge states
Phys. Rev. Lett. 97 246401
DOI:10.1103/PhysRevLett.97.246401 [Cited within: 1]

[24]Younas U Younis M Seadawy A R Rizvi S T R Althobaiti S Sayed S 2020 Diverse exact solutions for modified nonlinear Schrödinger equation with conformable fractional derivative
Results Phys. 20 103766
[Cited within: 1]

[25]Manafian J Zamanpour I 2013 Analytical treatment of the coupled Higgs equation and the Maccari system via exp-function method
Acta Univ. Apulensis 33 276 287
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[26]Elwakil S A El-Labany S K Zahran M A Sabry R 2002 Modified extended tanh-function method for solving nonlinear partial differential equations
Phys. Lett. A 299 179 188
DOI:10.1016/S0375-9601(02)00669-2 [Cited within: 1]

[27]Rizvi S T R Seadawy A R Ashraf F Younis M Iqbal H 2020 Dumitru Baleanu, Lump and Interaction solutions of a geophysical Korteweg-de Vries equation
Results Phys. 19 103661
DOI:10.1016/j.rinp.2020.103661 [Cited within: 1]

[28]Lu D Seadawy A R Yaro D 2019 Analytical wave solutions for the nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov and two-dimensional Kadomtsev-Petviashvili-Burgers equations
Results Phys. 12 2164 2168
DOI:10.1016/j.rinp.2019.02.049 [Cited within: 1]

[29]Zhang Z Y Liu Z H Miao X J Chen Y Z 2010 New exact solutions to the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity
Appl. Math. Comput. 216 3064 3072
DOI:10.1016/j.amc.2010.04.026 [Cited within: 1]

[30]Mirzazadeh M 2015 The extended homogeneous balance method and exact 1- soliton solutions of Maccari system
Comput. Methods Differ. Equ. 2 1523 1536
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[31]Yomba E 2005 The extended Fan's sub-equation method and its application to (2.1)-dimensional dispersive long wave and Whitham-Broer-Kaup equations
Chinese J Phys 43 789 805
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[32]Shang Y 2015 Analytical solution for an in-host viral infection model with time-inhomogeneous rates
Acta Phys. Pol. B 46 1567 1577
DOI:10.5506/APhysPolB.46.1567 [Cited within: 1]

[33]Shang Y 2017 Lie algebraic discussion for affinity based information diffusion in social networks
Open Physics 15 705 711
DOI:10.1515/phys-2017-0083

[34]Shang Yilun 2013 Lie algebra method for solving biological population model
J. Theor. Appl. Phys. 7 67
DOI:10.1186/2251-7235-7-67 [Cited within: 1]

[35]Ali A Seadawy A R Lu D 2018 Computational methods and traveling wave solutions for the fourth-Order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications
Open Physics 16 219 226
DOI:10.1515/phys-2018-0032 [Cited within: 1]

[36]Asghar A R Seadawy A Lu D 2018 New solitary wave solutions of some nonlinear models and their Applications
Advances in Difference Equations 2018 1 12232


[37]Arshad M Seadawy A Lu D 2017 Bright-Dark Solitary Wave Solutions of generalized higher-order nonlinear Schrodinger equation and its applications in optics
J. Electromagn. Waves Appl. 31 1711 1721
DOI:10.1080/09205071.2017.1362361

[38]Ahmed I Seadawy A R Lu D 2019 M-shaped rational solitons and their interaction with kink waves in the Fokas-lenells equation
Phys. Scr. 94 055205
DOI:10.1088/1402-4896/ab0455

[39]Farah N Seadawy A R Ahmad S Rizvi S T R Younis M 2020 Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model
Opt. Quantum Electron. 52 1 15
DOI:10.1007/s11082-020-02443-0

[40]Seadawy A R Tariq K U Liu J-G 2019 Symbolic computations: Dispersive Soliton solutions for (3+1)-dimensional Boussinesqand Kadomtsev-Petviashvili dynamical equations and its applications
Int. J. Mod. Phys. B33 1950342
DOI:10.1142/S0217979219503429

[41]Çelik N Seadawy A R Özkan Y S Yasar E 2021 A model of solitary waves in a nonlinear elastic circular rod: abundant different type exact solutions and conservation laws
Chaos Solitons Fractals 143 110486
DOI:10.1016/j.chaos.2020.110486

[42]Younas U Younis M Seadawy A R Rizvi S T R 2020 Optical solitons and closed form solutions to (3+1)-dimensional resonant Schrodinger equation
Int. J. Mod. Phys. B 34 2050291
DOI:10.1142/S0217979220502914

[43]Ali I Seadawy A R Rizvi S T R Younis M 2021 Painlevé analysis for various nonlinear Schrödinger dynamical equations
Int. J. Mod. Phys. B 35 2150038
DOI:10.1142/S0217979221500387 [Cited within: 1]