Open Journal of Mathematical Sciences
Vol. 6 (2022), Issue 1, pp. 269 – 280
ISSN: 2523-0212 (Online) 2616-4906 (Print)
DOI: 10.30538/oms2022.0192

Application of the newly \(\varphi^{6}\)-model expansion approach to the nonlinear reaction-diffusion equation

Muhammad Abubakar Isah\(^{1,*}\) and Asif Yokus\(^{1}\)
\(^{1}\) Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey.
Correspondence should be addressed to Muhammad Abubakar Isah at myphysics_09@hotmail.com

Copyright © 2022 Muhammad Abubakar Isah and Asif Yokus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: June 30, 2022 – Accepted: October 21, 2022 – Published: December 31, 2022

Abstract

In this paper, we use the \(\varphi ^{6}\)-model expansion method to construct the traveling wave solutions for the reaction-diffusion equation. The method of \(\varphi ^{6}\)-model expansion enables the explicit retrieval of a wide variety of solution types, such as bright, singular, periodic, and combined singular soliton solutions. Kink-type solitons, also known as topological solitons in the context of water waves, are another type of solution that can be explicitly retrieved. This study’s results might enhance the equation’s nonlinear dynamical properties. The method proposes a practical and efficient method for solving a sizable class of nonlinear partial differential equations. The dynamical features of the data are explained and highlighted using exciting graphs.

Keywords:

Reaction-diffusion equation; \(\varphi ^{6}\)-model expansion method; Soliton solutions; Kink soliton.

1. Introduction

The study of surfaces in geometry (see [1,2,3,4,5]) and a variety of mechanical issues are where partial differential equations (PDEs) first showed up. The study of various issues brought on by partial differential equations attracted the attention of eminent mathematicians worldwide in the late 19th century. This work was primarily motivated by the fact that partial differential equations commonly appear in the mathematical analysis of a wide range of problems in science and engineering and describe many fundamental natural laws [6]. It is now incredibly beneficial to look for precise answers to nonlinear evolution equations and partial differential equations NLEEs using various techniques. There are numerous effective techniques, such as the inverse scattering transform approach [7], the Homoclinic technique [8], the sinh-Gordon function method [9], the generalized exponential rational function method [10], the auxiliary equation method [11], An alternative method [12], the Bernoulli sub-equation function method [13,14], the sub-equation analytical method [15], the modified sub-equation method [16], the auto-Backlund transformation method [17] and so on.

This study focuses on the reaction-diffusion equation, and the equation has been investigated via many direct methods. Among these are; The sine-Gordon expansion method [18], the rational \((G'/G)\)-expansion method [19], the \((G'/G)\)-expansion method [20], the projective Riccati equation method [21]. The reaction-diffusion model is studied in this research using the newly developed \(\varphi ^{6}\)--model expansion method [22,23,24,25]), which results in the restoration of optical solitary wave solutions.

The plan for this work is provided below. In §2, a presentation of the \(\varphi ^{6}\)-model expansion method will be provided. Next, the reaction-diffusion model will be developed using the \(\varphi 6\) approach in §3 to provide new traveling wave solutions to the reaction-diffusion equation. Moreover, the associated 3D, 2D, and density graphs clearly illustrate the physical structure of the traveling wave solution. Finally, in §4, conclusions are reached.

2. The \(\varphi ^{6}\)-model expansion technique

According to [22,23,24,25], the steps involves for the \(\varphi ^{6}\)-model expansion technique are given as:

Step-1: Assuming the nonlinear evolution equation (NLEE) for \(W=W(x,t)\) is in the form.

\begin{equation} H(W,W_{x},W_{t},W_{xx},W_{xt},...)=0, \label{9} \end{equation}

(1)

here \(H\) is a polynomial of \(W(x,t)\) which involves highest order partial derivatives and its nonlinear terms.

Step-2: By using the wave transformation

\begin{equation} W(x,t)=W(\zeta ), \zeta =x - vt, \label{10} \end{equation}

(2)

where \(v\) represents wave speed and Eq. (1) can be converted into the nonlinear ordinary differential equation shown below.

\begin{equation} \Omega (W,W^{^{\prime }},WW^{^{\prime }},W^{^{\prime \prime }},...)=0, \label{11} \end{equation}

(3)

where the derivatives with respect to \(\zeta \) are shown by prime.

Step-3: Suppose that the formal solution to Eq. (3) exists:

\begin{equation} W\left( \zeta \right) =\sum\limits_{j=0}^{2M}\alpha _{j}Q^{j}(\zeta ), \label{12} \end{equation}

(4)

\(M\) can be gotten using the balancing rule, \(\alpha _{j}(j=0,1,2, \ldots ,M)\) are to be determined constants and \(Q(\zeta )\) satisfies the auxiliary NLODE;

\begin{equation}\label{13} \begin{cases} Q^{\prime 2}(\zeta )=h_{0}+h_{2}Q^{2}(\zeta )+h_{4}Q^{4}(\zeta )+h_{6}Q^{6}(\zeta ), \\ Q^{^{\prime \prime }}(\zeta ) =h^{2}Q(\zeta )+2h_{4}Q^{3}(\zeta )+3h_{6}Q^{5}(\zeta ), \end{cases}\end{equation}

(5)

here \(h_{j}(j=0,2,4,6)\) are real constants that will be found later.

Step-4: It is known that the solution to Eq. (5) is given as;

\begin{equation} Q(\zeta )=\frac{P(\zeta )}{\sqrt{fP^{2}(\zeta )+g}}, \label{14} \end{equation}

(6)

\(P(\zeta )\) is the Jacobi elliptic equation solution, provided that \(0< fP^{2}(\zeta )+g\)

\begin{equation} P^{\prime 2}(\zeta )=l_{0}+l_{2}P^{2}(\zeta )+l_{4}P^{4}(\zeta ), \label{15} \end{equation}

(7)

where \(l_{j}(j=0,2,4)\) are unknown constants to be determined, \(g\) and \(f\) are given by

\begin{equation}\label{16} \begin{cases} f =\frac{h_{4}(l_{2}-h_{2})}{% (l_{2}-h_{2})^{2}+3l_{0}l_{4}-2l_{2}(l_{2}-h_{2})}, \\ g =\frac{3l_{0}h_{4}}{(l_{2}-h_{2})^{2}+3l_{0}l_{4}-2l_{2}(l_{2}-h_{2})}, \end{cases}\end{equation}

(8)

under the restricted condition

\begin{equation} h_{4}^{2}(l_{2}-h_{2})[9l_{0}l_{4}-(l_{2}-h_{2})(2l_{2}+h_{2})]+3h_{6}[-l_{2}^{2}+h_{2}^{2}+3l_{0}l_{4}]^{2}=0. \label{17} \end{equation}

(9)

Step-5: The Jacobi elliptic solutions of Eq. (7) can be calculated when \(0< m< 1\), the exact solutions of Eq. (1) can be derived by substituting Eq. (6) and Eq. (7) into Eq. (4).

Function	\(m\rightarrow 1\)	\(m\rightarrow 0\)	Function	\(m\rightarrow 1\)	\(m\rightarrow 0\)
\(sn(\zeta ,m)\)	\(tanh(\zeta )\)	\(sin(\zeta )\)	\(ds(\zeta ,m)\)	\(csch(\zeta )\)	\(csc(\zeta )\)
\(cn(\zeta ,m)\)	\(sech(\zeta )\)	\(cos(\zeta )\)	\(sc(\zeta ,m)\)	\(sinh(\zeta )\)	\(tan(\zeta )\)
\(dn(\zeta ,m)\)	\(sech(\zeta )\)	\(1\)	\(sd(\zeta ,m)\)	\(sinh(\zeta )\)	\(sin(\zeta )\)
\(ns(\zeta ,m)\)	\(coth(\zeta )\)	\(csc(\zeta )\)	\(nc(\zeta ,m)\)	\(cosh(\zeta )\)	\(sec(\zeta )\)
\(cs(\zeta ,m)\)	\(csch(\zeta )\)	\(cot(\zeta )\)	\(cd(\zeta ,m)\)	\(1\)	\(cos(\zeta )\)

3. Application to the reaction-diffusion equation

The \(\varphi ^{6}\)-model expansion method, which was explained in the previous part, we take into account the reaction-diffusion equation's following form.

\begin{equation} {W_{tt}} + d W + n{W^3} + \beta {W_{xx}} = 0, \label{18} \end{equation}

(10)

here real constants \(\beta\) , \(d\) and \(n\) are the real constants, Eq. (10) is reduced to the following ODE using the traveling wave transformation \(W(x, t) = W(\zeta) = G(x- vt)\):

\begin{equation} {(v^2 + \beta)}{W^{''}} + n{W^3} + d W = 0, \label{19} \end{equation}

(11)

where \(M+2=3M\), therefore, \(M= 1\) is obtained as a result of the balance principle between \({W^{''}}\) and \({W^3}\); so, the solution form can be expressed as

\begin{equation} W(\zeta )=\alpha _{0}+\alpha _{1}Q(\zeta )+\alpha _{2}Q^{2}(\zeta ), \label{21} \end{equation}

(12)

where \(\alpha _{0},\alpha _{1}\) and \(\alpha _{2}\) are constants to be determined.

We obtain the following algebraic equations by substituting Eq. (12) along with Eq. (5) into Eq. (11) and setting the coefficients of all powers of \(Q^{j}(\zeta )\), \(j=0,1,\ldots ,6\) to be equal to zero;

\begin{equation} \begin{cases} {Q^0}(\zeta ): \alpha _0 d+2 \alpha _2 \beta h_0+2 \alpha _2 h_0 v^2+\alpha _0^3 n= 0,\\ {Q^1}(\zeta ): \alpha _1 d+\alpha _1 \beta h_2+\alpha _1 h_2 v^2+3 \alpha _0^2 \alpha _1 n= 0,\\ {Q^2}(\zeta ): \alpha _2 d+4 \alpha _2 \beta h_2+4 \alpha _2 h_2 v^2+3 \alpha _0 \alpha _1^2 n+3 \alpha _0^2 \alpha _2 n= 0,\\ {Q^3}(\zeta ): 2 \alpha _1 \beta h_4+2 \alpha _1 h_4 v^2+\alpha _1^3 n+6 \alpha _0 \alpha _2 \alpha _1 n= 0,\\ {Q^4}(\zeta ): 6 \alpha _2 \beta h_4+6 \alpha _2 h_4 v^2+3 \alpha _0 \alpha _2^2 n+3 \alpha _1^2 \alpha _2 n= 0,\\ {Q^5}(\zeta ): 3 \alpha _1 \beta h_6+3 \alpha _1 h_6 v^2+3 \alpha _1 \alpha _2^2 n= 0,\\ {Q^6}(\zeta ): 8 \alpha _2 \beta h_6+8 \alpha _2 h_6 v^2+\alpha _2^3 n=0. \end{cases} \end{equation}

(13)

The following solutions are obtained after solving the above system of equations:%

\begin{equation}\label{23} \begin{cases} \alpha _0 = 0,\quad {\alpha _1} = \frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2}}{\sqrt{n}},\quad \alpha _2=0, \\ d=h_2 \left(-v^2\right)-\beta h_2,\quad {h_6} = 0.\end{cases} \end{equation}

(14)

The following solutions of Eq. (10) can be obtained with the help of Eqs. (6), (12) and (14) along with the Jacobi elliptic functions in the table above.

1. If \(l_{0}=1\), \(l_{2}=-(1+m^{2})\), \(l_{4}=m^{2}\), \(0< m< 1\), then \(% P(\zeta )=sn(\zeta ,m)\) or \(P(\zeta )=cd(\zeta ,m)\), and we have

\begin{equation} W_{_{1,0}}\left( x,t\right) =\alpha _1 \left( \frac{sn(\zeta ,m)}{\sqrt{f\left( sn(\zeta ,m)\right) ^{2}+g}}\right) , \label{24} \end{equation}

(15)

\begin{equation} W_{_{1,1}}\left( x,t\right) =\alpha _1 \left( \frac{cd(\zeta ,m)}{\sqrt{f\left( cd(\zeta ,m)\right) ^{2}+g}}\right), \label{25} \end{equation}

(16)

such that \(\zeta =x-vt\) and \(f\) and \(g\) in Eq. (8) are given by \begin{align*} f& =\frac{h_4 \left(h_2+m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4}{-h_2^2+m^4-m^2+1}, \end{align*} under the restriction condition \begin{equation*} h_4^2 \left(-h_2-m^2-1\right)\left(9 m^2-\left(-h_2-m^2-1\right) \left(h_2+2 \left(-m^2-1\right)\right)\right) =0. \end{equation*} If \(m\rightarrow 1\), then the kink soliton is obtained

\begin{equation} W_{_{1,2}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \tanh (t v-x)}{\sqrt{n} \sqrt{\frac{h_4 \left(3-\left(h_2+2\right) \tanh ^2(t v-x)\right)}{h_2^2-1}}} , \label{26} \end{equation}

(17)

such that \begin{equation*} h_4^2 \left(-h_2-2\right)\left(9-\left(-h_2-2\right) \left(h_2-4\right)\right) =0. \end{equation*}

Figure 1. The numerical simulations corresponding to \(\left| {{W}_{1,2}} \right|\) given by Eq. (17), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.1, v =0.6, n = 0.5, h_4=0.3, h_2=0.1 .\)

If \(m\rightarrow 0\), then the periodic solution is obtained

\begin{equation} W_{_{1,3}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \sin (t v-x)}{\sqrt{n} \sqrt{h_4 \left(\frac{3}{h_2^2-1}-\frac{\sin ^2(t v-x)}{h_2-1}\right)}}, \label{27} \end{equation}

(18)

such that \begin{equation*} h_4^2 \left(-h_2-1\right)\left(-\left(-h_2-1\right) \left(h_2-2\right)\right)=0. \end{equation*}

Figure 2. The numerical simulations corresponding to \(\left| {{W}_{1,3}} \right|\) given by Eq. (18), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.2, v =0.12, n = 0.15, h_4=0.03, h_2=0.1 .\)

2. If \(l_{0}=1-m^{2}\), \(l_{2}=2m^{2}-1\), \(% l_{4}=-m^{2} \), \(0< m< 1\), then \(P(\zeta )=cn(\zeta ,m)\) therefore

\begin{equation} W_{_{2,1}}\left( x,t\right) =\alpha _1 \left( \frac{cn(\zeta ,m)}{\sqrt{f\left( cn(\zeta ,m)\right) ^{2}+g}}\right), \label{28} \end{equation}

(19)

where \(f\) and \(g\) are determined by \begin{align*} f& =\frac{h_4 \left(h_2-2 m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =\frac{3 h_4 \left(m^2-1\right)}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2+2 m^2-1\right)\left(-\left(-h_2+2 m^2-1\right) \left(h_2+2 \left(2 m^2-1\right)\right)-9 \left(1-m^2\right) m^2\right)=0 \end{equation*} If \(m\rightarrow 1\), then the bright soliton is retrieved

\begin{equation} {{W}_{2,2}}\left( x,t \right)=\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \text{sech}(t v-x)}{\sqrt{n} \sqrt{-\frac{h_4 \text{sech}^2(t v-x)}{h_2+1}}} \label{29} \end{equation}

(20)

provided that \begin{equation*} h_4^2 \left(1-h_2\right)\left(-\left(1-h_2\right) \left(h_2+2\right)\right) =0. \end{equation*} If \(m\rightarrow 0\), then the periodic solution is obtained

\begin{equation} W_{_{2,3}}\left( x,t\right) =\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \cos (t v-x)}{\sqrt{n} \sqrt{h_4 \left(\frac{3}{h_2^2-1}-\frac{\cos ^2(t v-x)}{h_2-1}\right)}}, \label{30} \end{equation}

(21)

such that \begin{equation*} h_4^2 \left(-h_2-1\right)\left(-\left(-h_2-1\right) \left(h_2-2\right)\right)=0. \end{equation*}

Figure 3. The numerical simulations corresponding to \(\left| {{W}_{2,3}} \right|\) given by Eq. (21), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.3, v =1.2, n = 1.2, h_4=0.4, h_2=0.3 \).

3. If \(l_{0}=m^{2}-1\), \(l_{2}=2-m^{2}\), \(l_{4}=-1\), \(0< m< 1\), then \(% P(\zeta )=dn(\zeta ,m)\) which gives

\begin{equation} W_{_{3,1}}\left( x,t\right) =\alpha _1 \left( \frac{dn(\zeta ,m)}{\sqrt{f\left( dn(\zeta ,m)\right) ^{2}+g}}\right), \label{31} \end{equation}

(22)

where \(f\) and \(g\) are determined by \begin{align*} f& =\frac{h_4 \left(h_2+m^2-2\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4 \left(m^2-1\right)}{-h_2^2+m^4-m^2+1}, \end{align*} under the restriction condition \begin{equation*} h_4^2 \left(-h_2-m^2+2\right)\left(-\left(-h_2-m^2+2\right) \left(h_2+2 \left(2-m^2\right)\right)-9 \left(m^2-1\right)\right)=0 \end{equation*} If \(m\rightarrow 0\), then the rational solution is obtained

\begin{equation} W_{_{3,3}}\left( x,t\right) =\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2}}{\sqrt{-\frac{\left(h_2-2\right) h_4}{h_2^2-1}-\frac{3 h_4}{h_2^2-1}} \sqrt{n}}, \label{33} \end{equation}

(23)

such that \begin{equation*} h_4^2 \left(2-h_2\right)\left(9-\left(2-h_2\right) \left(h_2+4\right)\right) =0. \end{equation*}

4. If \(l_{0}=m^{2}\), \(l_{2}=-\left( 1+m^{2}\right) \), \(l_{4}=1\), \(% 0< m< 1\), then \(P(\zeta )=ns(\zeta ,m)\) or \(P(\zeta )=dc(\zeta ,m)\) then

\begin{equation} W_{_{4,0}}\left( x,t\right) =\alpha _1 \left( \frac{ns(\zeta ,m)}{\sqrt{f\left( ns(\zeta ,m)\right) ^{2}+g}}\right), \label{34} \end{equation}

(24)

\begin{equation} W_{_{4,1}}\left( x,t\right) =\alpha _1 \left( \frac{dc(\zeta ,m)}{\sqrt{f\left( dc(\zeta ,m)\right) ^{2}+g}}\right), \label{35} \end{equation}

(25)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2+m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4 m^2}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2-m^2-1\right)\left(9 m^2-\left(-h_2-m^2-1\right) \left(h_2+2 \left(-m^2-1\right)\right)\right) =0. \end{equation*} If \(m\rightarrow 1\), then the dark singular solution is obtained

\begin{equation} {{W}_{4,2}}\left( x,t \right)=-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \coth (t v-x)}{\sqrt{n} \sqrt{-\frac{h_4 \left(\left(h_2+2\right) \text{csch}^2(t v-x)+h_2-1\right)}{h_2^2-1}}} \label{36} \end{equation}

(26)

such that \begin{equation*} h_4^2 \left(-h_2-2\right)\left(9-\left(-h_2-2\right) \left(h_2-4\right)\right) =0 \end{equation*}

Figure 4. The numerical simulations corresponding to \(\left| {{W}_{4,2}} \right|\) given by Eq. (26), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=3.5, v =1.001, n = 2.1, h_4=0.6, h_2=0.1 \).

If \(m\rightarrow 0\), then the periodic solution is obtained

\begin{equation} W_{_{4,3}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \csc (t v-x)}{\sqrt{n} \sqrt{-\frac{h_4 \csc ^2(t v-x)}{h_2-1}}}, \label{37} \end{equation}

(27)

such that \begin{equation*} h_4^2 \left(-h_2-1\right)\left(-\left(-h_2-1\right) \left(h_2-2\right)\right) =0. \end{equation*}

5. If \(l_{0}=-m^{2}\), \(l_{2}=2m^{2}-1\), \(l_{4}=1-m^{2}\), \(0< m< 1\), then \(P(\zeta )=nc(\zeta ,m)\) and we have

\begin{equation} W_{_{5,1}}\left( x,t\right) =\alpha _1 \left( \frac{nc(\zeta ,m)}{\sqrt{f\left( nc(\zeta ,m)\right) ^{2}+g}}\right), \label{38} \end{equation}

(28)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2-2 m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =\frac{3 h_4 m^2}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2+2 m^2-1\right)\left(-\left(-h_2+2 m^2-1\right) \left(h_2+2 \left(2 m^2-1\right)\right)-9 \left(1-m^2\right) m^2\right) =0. \end{equation*} If \(m\rightarrow 1\), then the singular solitary wave solution is obtained

\begin{equation} W_{_{5,2}}\left( x,t\right) =\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \cosh (t v-x)}{\sqrt{n} \sqrt{h_4 \left(-\frac{\cosh ^2(t v-x)}{h_2+1}-\frac{3}{h_2^2-1}\right)}}, \label{39} \end{equation}

(29)

such that \begin{equation*} h_4^2 \left(1-h_2\right)\left(-\left(1-h_2\right) \left(h_2+2\right)\right)=0. \end{equation*}

Figure 5. The numerical simulations corresponding to \(\left| {{W}_{5,2}} \right|\) given by Eq. (29), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.9, v =0.75, n = 0.2, h_4=1.8, h_2=0.3 \).

If \(m\rightarrow 0\), then the periodic solution is obtained

\begin{equation} W_{_{5,3}}\left( x,t\right) =\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \sec (t v-x)}{\sqrt{n} \sqrt{-\frac{h_4 \sec ^2(t v-x)}{h_2-1}}}, \label{40} \end{equation}

(30)

such that \begin{equation*} h_4^2 \left(-h_2-1\right)\left(-\left(-h_2-1\right) \left(h_2-2\right)\right) =0. \end{equation*}

6. If \(l_{0}=-1\), \(l_{2}=2-m^{2}\), \(l_{4}=-\left( 1-m^{2}\right) \), \(0< m< 1\), then \(P(\zeta )=nd(\zeta ,m)\) and we have

\begin{equation} W_{_{6}}\left( x,t\right) =\alpha _1 \left( \frac{nd(\zeta ,m)}{\sqrt{f\left( nd(\zeta ,m)\right) ^{2}+g}}\right), \label{41} \end{equation}

(31)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2+m^2-2\right)}{-h_2^2+m^4-m^2+1}, \\ g& =\frac{3 h_4}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2-m^2+2\right)\left(-\left(-h_2-m^2+2\right) \left(h_2+2 \left(2-m^2\right)\right)-9 \left(m^2-1\right)\right) =0. \end{equation*}

7. If \(l_{0}=1\), \(l_{2}=2-m^{2}\), \(l_{4}=1-m^{2}\),\(0< m< 1\), then \(P(\zeta )=sc(\zeta ,m)\), and we have

\begin{equation} W_{_{7,1}}\left( x,t\right) =\alpha _1 \left( \frac{sc(\zeta ,m)}{\sqrt{f\left( sc(\zeta ,m)\right) ^{2}+g}}\right), \label{42} \end{equation}

(32)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2+m^2-2\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2-m^2+2\right)\left(9 \left(1-m^2\right)-\left(-h_2-m^2+2\right) \left(h_2+2 \left(2-m^2\right)\right)\right) =0. \end{equation*} If \(m\rightarrow 1\), then the soliton solution is retrieved

\begin{equation} W_{_{7,2}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \sinh (t v-x)}{\sqrt{n} \sqrt{h_4 \left(\frac{3}{h_2^2-1}-\frac{\sinh ^2(t v-x)}{h_2+1}\right)}}, \label{43} \end{equation}

(33)

such that \begin{equation*} h_4^2 \left(1-h_2\right)\left(-\left(1-h_2\right) \left(h_2+2\right)\right)=0. \end{equation*}

Figure 6. The numerical simulations corresponding to \(\left| {{W}_{7,2}} \right|\) given by Eq. (33), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.1, v =0.9, n = 0.2, h_4=0.24, h_2=0.1 \).

If \(m\rightarrow 0\), then the periodic wave solution is obtained

\begin{equation} W_{_{7,3}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \tan (t v-x)}{\sqrt{n} \sqrt{\frac{h_4 \left(3-\left(h_2-2\right) \tan ^2(t v-x)\right)}{h_2^2-1}}}, \label{44} \end{equation}

(34)

such that \begin{equation*} h_4^2 \left(2-h_2\right)\left(9-\left(2-h_2\right) \left(h_2+4\right)\right)=0. \end{equation*}

Figure 7. The numerical simulations corresponding to \(\left| {{W}_{7,3}} \right|\) given by Eq. (34), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=0.5, v =1.8, n = 0.3, h_4=0.06, h_2=0.1 \).

8. If \(l_{0}=1\), \(l_{2}=2m^{2}-1\), \(l_{4}=-m^{2}\left( 1-m^{2}\right) \), \(0< m< 1\), then \(P(\zeta )=sd(\zeta ,m)\) and we have

\begin{equation} W_{_{8}}\left( x,t\right) =\alpha _1 \left( \frac{sd(\zeta ,m)}{\sqrt{f\left( sd(\zeta ,m)\right) ^{2}+g}}\right), \label{45} \end{equation}

(35)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2-2 m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2+2 m^2-1\right)\left(-\left(-h_2+2 m^2-1\right) \left(h_2+2 \left(2 m^2-1\right)\right)-9 \left(1-m^2\right) m^2\right) =0. \end{equation*}

9. If \(l_{0}=1-m^{2}\), \(l_{2}=2-m^{2}\), \(l_{4}=1\), \(0< m< 1\), then \(% P(\zeta )=cs(\zeta ,m)\) and we have

\begin{equation} W_{_{9,1}}\left( x,t\right) =\alpha _1 \left( \frac{cs(\zeta ,m)}{\sqrt{f\left( cs(\zeta ,m)\right) ^{2}+g}}\right), \label{46} \end{equation}

(36)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2+m^2-2\right)}{-h_2^2+m^4-m^2+1}, \\ g& =\frac{3 h_4 \left(m^2-1\right)}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2-m^2+2\right)\left(9 \left(1-m^2\right)-\left(-h_2-m^2+2\right) \left(h_2+2 \left(2-m^2\right)\right)\right) =0. \end{equation*} If \(m\rightarrow 1\), then the singular soliton solution is obtained

\begin{equation} {{W}_{9,2}}\left( x,t \right)=-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \text{csch}(t v-x)}{\sqrt{n} \sqrt{-\frac{h_4 \text{csch}^2(t v-x)}{h_2+1}}}, \label{47} \end{equation}

(37)

such that \begin{equation*} h_4^2 \left(1-h_2\right)\left(-\left(1-h_2\right) \left(h_2+2\right)\right)=0. \end{equation*} If \(m\rightarrow 0\), then the periodic wave solution is obtained

\begin{equation} W_{_{9,3}}\left( x,t\right) =-\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} \cot (t v-x)}{\sqrt{n} \sqrt{\frac{h_4 \left(-\left(h_2-2\right) \csc ^2(t v-x)+h_2+1\right)}{h_2^2-1}}}, \label{48} \end{equation}

(38)

such that \begin{equation*} h_4^2 \left(2-h_2\right)\left(9-\left(2-h_2\right) \left(h_2+4\right)\right) =0. \end{equation*}

10. If \(l_{0}=-m^{2}\left( 1-m^{2}\right) \), \(l_{2}=2m^{2}-1\), \(% l_{4}=1\), \(0< m< 1\), then \(P(\zeta )=ds(\zeta ,m)\) and we have

\begin{equation} W_{_{10}}\left( x,t\right) =\alpha _1 \left( \frac{ds(\zeta ,m)}{\sqrt{f\left( ds(\zeta ,m)\right) ^{2}+g}}\right), \label{49} \end{equation}

(39)

where \(f\) and \(g\) are given by \begin{align*} f& =\frac{h_4 \left(h_2-2 m^2+1\right)}{-h_2^2+m^4-m^2+1}, \\ g& =-\frac{3 h_4 m^2 \left(m^2-1\right)}{-h_2^2+m^4-m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(-h_2+2 m^2-1\right)\left(-\left(-h_2+2 m^2-1\right) \left(h_2+2 \left(2 m^2-1\right)\right)-9 \left(1-m^2\right) m^2\right) =0. \end{equation*}

11. If \(l_{0}=\frac{1-m^{2}}{4}\), \(l_{2}=\frac{1+m^{2}}{2}\), \(l_{4}=% \frac{1-m^{2}}{4}\), \(0< m< 1\), then \(P(\zeta )=nc(\zeta ,m)\pm sc(\zeta ,m)\) or \(P(\zeta )=\frac{cn(\zeta ,m)}{1\pm sn(\zeta ,m)}\) and we have

\begin{equation} W_{_{11,0}}\left( x,t\right) =\alpha _1 \left( \frac{nc(\zeta ,m)\pm sc(\zeta ,m)% }{\sqrt{f\left( nc(\zeta ,m)\pm sc(\zeta ,m)\right) ^{2}+g}}\right), \label{50} \end{equation}

(40)

\begin{equation} W_{_{11,1}}\left( x,t\right) =\alpha _1 \left( \frac{\frac{cn(\zeta ,m)}{1\pm sn(\zeta ,m)}}{\sqrt{f\left( \frac{cn(\zeta ,m)}{1\pm sn(\zeta ,m)}\right) ^{2}+g}}\right), \label{51} \end{equation}

(41)

where \(f\) and \(g\) are given by \begin{align*} f& =-\frac{8 h_4 \left(-2 h_2+m^2+1\right)}{-16 h_2^2+m^4+14 m^2+1}, \\ g& =\frac{12 h_4 \left(m^2-1\right)}{-16 h_2^2+m^4+14 m^2+1}, \end{align*} under the constraint condition \begin{equation*} h_4^2 \left(\frac{1}{2} \left(m^2+1\right)-h_2\right)\left(\frac{9}{16} \left(1-m^2\right)^2-\left(\frac{1}{2} \left(m^2+1\right)-h_2\right) \left(h_2+m^2+1\right)\right) =0. \end{equation*} If \(m\rightarrow 1\), then the exponential solution is obtained

\begin{equation} W_{_{11,2}}\left( x,t\right) =\frac{i \sqrt{2} \sqrt{h_4} \sqrt{\beta +v^2} e^{x-t v}}{\sqrt{n} \sqrt{-\frac{h_4 e^{2 x-2 t v}}{h_2+1}}}, \label{52} \end{equation}

(42)

such that \begin{equation*} h_4^2 \left(1-h_2\right)\left(-\left(1-h_2\right) \left(h_2+2\right)\right)=0. \end{equation*} If \(m\rightarrow 0\), then the combined periodic wave solutions are retrieved

\begin{equation} W_{_{11,3}}\left( x,t\right) =\frac{i \sqrt{h_4} \sqrt{\beta +v^2} (\sec (t v-x)-\tan (t v-x))}{\sqrt{2} \sqrt{n} \sqrt{\frac{h_4 \left(4 h_2 (\sin (t v-x)-1)+\sin (t v-x)+5\right)}{\left(16 h_2^2-1\right) (\sin (t v-x)+1)}}}, \label{54} \end{equation}

(43)

\begin{equation} W_{_{11,4}}\left( x,t\right) =-\frac{i \sqrt{h_4} \sqrt{\beta +v^2} \cos (t v-x)}{\sqrt{2} \sqrt{n} (\sin (t v-x)-1) \sqrt{\frac{h_4 \left(\frac{2 \left(1-2 h_2\right) \cos ^2(t v-x)}{(\sin (t v-x)-1)^2}+3\right)}{16 h_2^2-1}}}, \label{55} \end{equation}

(44)

are obtained, such that \begin{equation*} h_4^2 \left(\frac{1}{2}-h_2\right)\left(\frac{9}{16}-\left(\frac{1}{2}-h_2\right) \left(h_2+1\right)\right)=0. \end{equation*}

Figure 8. The numerical simulations corresponding to \(\left| {{W}_{11,4}} \right|\) given by Eq. (44), for \(m=1\) ; (a) is the 3D graphic, (b) is the 2D-contour graphic while (c) is the 2D graphic for \( \beta=2.9, v =0.968, n = 0.4, h_4=0.3, h_2=0.1 \).

4. Conclusion

The reaction-diffusion equation is examined in this study. Using the \(\varphi^6\)-model expansion technique, bright, kink, periodic, and combined periodic soliton solutions are retrieved. Furthermore, singular soliton solutions are seen favorably. The soliton solutions at any given time are shown in Figures \(1-8\), which is important for the movement of energy from one location to another. It is the internal dynamics of the traveling wave for various parameter values. We might conclude that the traveling wave behavior varies for different values of each. The study's findings are hoped to boost the equation's nonlinear dynamical features. The method suggests a promising and efficient strategy for solving a large class of nonlinear partial differential equations.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

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