1. Introduction
In the literature, nonlinear partial differential equations (NLPDEs) have an important role; therefore there has been much research into obtaining exact solutions for NLPDEs. As a result of these studies, many new methods for finding exact solutions of NLPDEs have been discovered. For example, Wazwaz applied the tanh technique to the Dodd–Bullough–Mikhailov equation [1], Hashemi et al utilized the Nucci reduction method with the (2+1)-dimensional Korteweg–de Vries equation [2], Biswas et al handled the method of undetermined coefficients [3], Zhou et al studied the Riccati equation mapping approach [4], Biswas et al applied the trial equation procedure to the Radhakrishnan–Kundu–Lakshmanan equation with a couple of integration schemes [5], Karakoc et al discussed the modified regularized long wave equation [6], Raza et al studied the extended trial approach [7], Hosseini et al applied the unified method to the generalized third-order nonlinear Schrödinger equation [8], Zafar et al utilized the modified extended tanh expansion method in the Biswas–Arshed model with full nonlinear form [9], Kaplan et al applied the exponential rational function procedure to the wave packet envelope [10], Rabie et al applied the extended F-expansion technique to the generalized derivative nonlinear Schrödinger equation with quintic nonlinearity [11], Hashemi et al handled the Nucci reduction method [12], Kudryashov and Lavrova employed the simplest equation method with the Chavy–Waddy–Kolokolnikov model [13] and Niwas and Kumar used the inverse (G'/G)-expansion method [14]. Finally, Kumar and colleagues used a generalized exponential rational function method [15], similarity reduction [16], the extended Jacobian elliptic function expansion method [17] and Lie symmetry analysis [18, 19].
The propagation pulse in the equation describing the propagation of pulses in optical fibers serves as a cornerstone for understanding, predicting and optimizing various phenomena in optical communication, nonlinear optics and other related fields, driving advances in diverse technological areas. This equation is important for the understanding signal transmission and exploring nonlinear phenomena. The equation describing the propagation of pulses in optical fibers, often referred to as the nonlinear Schrödinger equation or its variants, holds immense importance in several fields, particularly optical communications and nonlinear optics. Therefore, we have concluded that it is crucial to work on this model. In this paper, the propagation pulse in an optical fiber is explained using the following new PDE:1 ) , the Schrödinger equation is obtained as follows:1 ) is a generalization of the Schrödinger equation.
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{W}_{t}+{W}_{{xx}}+{\alpha }_{1}\,| W{| }^{2m-2n}W\\ \,+\,{\beta }_{1}\,| W{| }^{2m-n}W+{\gamma }_{1}\,| W{| }^{2m}W+{\delta }_{1}\,| W{| }^{2m+n}W\\ \,+\,{\lambda }_{1}\,| W{| }^{2n+2m}W=0,\end{array}\end{eqnarray}$
where W(x, t) is the complex function representing an optical wave and α1, β1, γ1, δ1 and λ1 are the parameters. If we substitute m = 0 in equation ( $\begin{eqnarray}\begin{array}{l}{\rm{i}}{W}_{t}+{W}_{{xx}}+{\alpha }_{1}\,| W{| }^{-2n}W+{\beta }_{1}\,| W{| }^{-n}W\\ \,+\,{\gamma }_{1}\,W+{\delta }_{1}\,| W{| }^{n}W+{\lambda }_{1}\,| W{| }^{2n}W=0.\end{array}\end{eqnarray}$
As a result, we can say that equation (The first equation
If we substitute m = n in equation (1 ), the following NLPDE is obtained:
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{W}_{t}+{W}_{{xx}}+{\alpha }_{1}\,W+{\beta }_{1}\,| W{| }^{n}W\\ \,+\,{\gamma }_{1}\,| W{| }^{2n}W+{\delta }_{1}\,| W{| }^{3n}W+{\lambda }_{1}\,| W{| }^{4n}W=0.\end{array}\end{eqnarray}$
The second equation
If we substitute m = 2n, in equation (1 ), the following NLPDE is obtained:
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{W}_{t}+{W}_{{xx}}+{\alpha }_{1}\,| W{| }^{2n}W+{\beta }_{1}\,| W{| }^{3n}W\\ \,+\,{\gamma }_{1}\,| W{| }^{4n}W+{\delta }_{1}\,| W{| }^{5n}W+{\lambda }_{1}\,| W{| }^{6n}W=0.\end{array}\end{eqnarray}$
The given models are handled in some studies in the literature, for example Kudryashov obtained the first integrals of the given models [20], Zayed and Alngar applied three different methods to obtain the optical soliton solutions [21] and Raza et al used the sine-Gordon expansion and modified auxiliary equation procedures [22].In this paper we will obtain the exact solutions of equations (3 ) and (4 ) via the modified simplest equation and the modified Kudryashov procedures. For this purpose, a description of the procedures will be given in section 2 . Then, the procedures will be applied in section 3 . Some plots of the results will be given in section 4 . Finally, conclusions will be given in section 5 .
2. Description of the procedures
We take a PDE of the form
$\begin{eqnarray}P(W,{W}_{x},{W}_{t},{W}_{{xx}},{W}_{{xt}},{W}_{{tt}},\ldots )=0,\end{eqnarray}$
where P is a polynomial of W and W is a complex and unknown function.Using the traveling wave transform5 ), resulting in a nonlinear ordinary differential equation (ODE)
$\begin{eqnarray}W(x,t)=V(\varepsilon ){{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\varepsilon =x-{c}_{1}t,{\rm{\Phi }}={k}_{1}x-{w}_{1}t,\end{eqnarray}$
we can modify equation ( $\begin{eqnarray}Q(V,V^{\prime} ,V^{\prime\prime} ,V\prime\prime\prime ,\ldots )=0,\end{eqnarray}$
where a prime denotes the derivative respect to ϵ.2.1. The modified simplest equation procedure
We extend the solution of equation (7 ) in a finite series form7 ). The function σ(ϵ) fulfills some ODEs. In this research, we use the Riccati equations as the simplest equation9 ) due to variations of δ.
$\begin{eqnarray}F(\varepsilon )=\displaystyle \sum _{i=1}^{m}{b}_{i}{\sigma }^{i}(\varepsilon ),\end{eqnarray}$
where bi (i = 1, 2, ..., m) are non-variables to be found and bm should be different from zero. m is a balance term and is calculated according to the principle of homogeneous balance. Let us determine the integer m by balancing the highest-order linear term with the highest-order nonlinear term in equation ( $\begin{eqnarray}\sigma ^{\prime} (\varepsilon )={\sigma }^{2}(\varepsilon )+\delta ,\end{eqnarray}$
where δ is a non-variable. After that we get family of solutions to equation (If δ < 0, solutions are given as follows:
$\begin{eqnarray}\sigma (\varepsilon )=-\sqrt{-\delta }\tanh \sqrt{-\delta }\varepsilon ,\end{eqnarray}$
$\begin{eqnarray}\sigma (\varepsilon )=-\sqrt{-\delta }\coth \sqrt{-\delta }\varepsilon ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=\sqrt{-\delta }\left(-\tanh \left(2\sqrt{-\delta }\,\varepsilon \right)\right.\\ \,\left.\pm \,\mathrm{isech}\left(2\sqrt{-\delta }\,\varepsilon \right)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=\sqrt{-\delta }\left(-\coth \left(2\sqrt{-\delta }\,\varepsilon \right)\right.\\ \,\left.\pm \,\mathrm{csch}\left(2\sqrt{-\delta }\,\varepsilon \right)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=-\displaystyle \frac{\sqrt{-\delta }}{2}\left(\tanh \left(\displaystyle \frac{\sqrt{-\delta }}{2}\,\varepsilon \right)\right.\\ \,\left.+\,\coth \left(\displaystyle \frac{\sqrt{-\delta }}{2}\,\varepsilon \right)\right).\end{array}\end{eqnarray}$
If δ > 0, the solutions are as follows: $\begin{eqnarray}\sigma (\varepsilon )=\sqrt{\delta }\tan \sqrt{\delta }\varepsilon ,\end{eqnarray}$
$\begin{eqnarray}\sigma (\varepsilon )=-\sqrt{\delta }\cot \sqrt{\delta }\varepsilon ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=\sqrt{\delta }\left(\tan \left(2\sqrt{\delta }\,\varepsilon \right)\right.\\ \,\left.\pm \,\sec \left(2\sqrt{\delta }\,\varepsilon \right)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=\sqrt{\delta }\left(-\cot \left(2\sqrt{\delta }\,\varepsilon \right)\right.\\ \,\left.\pm \,\csc \left(2\sqrt{\delta }\,\varepsilon \right)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\sigma (\varepsilon )=\displaystyle \frac{\sqrt{\delta }}{2}\left(\tan \left(\displaystyle \frac{\sqrt{\delta }}{2}\,\varepsilon \right)\right.\\ \,\left.-\,\cot \left(\displaystyle \frac{\sqrt{\delta }}{2}\,\varepsilon \right)\right).\end{array}\end{eqnarray}$
If δ = 0, the solution is as follows:
$\begin{eqnarray}\sigma (\varepsilon )=-\displaystyle \frac{1}{\varepsilon }.\end{eqnarray}$
Substituting equations (8 ) into (7 ) with (9 ) and setting all the coefficients of function σi to zero, the system of algebraic equations in the terms bi, c1, k1 and w1 is obtained; then we solve these algebraic equations. Finally, the obtained results are substituted into the auxiliary solutions [23, 24].
2.2. The modified Kudryashov procedure
We transform equation (7 ) using the wave transformation below22 ) satisfies the following ODE:23 ), equation (21 ) is substituted into equation (7 ) to produce a set of algebraic equations for bi, a, Γ, c1, k1 and w1. Eventually, after solving the resulting system, the exact solutions to equation (5 ) are determined [25, 26].
$\begin{eqnarray}F\left(\varepsilon \right)=\mathop{\displaystyle \sum _{i=0}}\limits^{m}{b}_{i}{\left(\sigma \left(\varepsilon \right)\right)}^{i},{b}_{m}\ne 0,\end{eqnarray}$
where ${b}_{i}\left(\,{,}(,i=0,1,\,\ldots ,\,m\right)$ are constants that will be determined later, m is calculated by the homogeneous balance principle and the function $\sigma \left(\varepsilon \right)$ is given by $\begin{eqnarray}\sigma \left(\varepsilon \right)=\displaystyle \frac{1}{1+{\rm{\Gamma }}{a}^{\varepsilon }},\end{eqnarray}$
where ( $\begin{eqnarray}{\sigma }^{{}^{{\prime} }}\left(\varepsilon \right)=\left({\sigma }^{2}\left(\varepsilon \right)-\sigma \left(\varepsilon \right)\right)\mathrm{ln}a.\end{eqnarray}$
Without ignoring equation (3. Application of the procedures
In this section, the given models will be reduced to an ODE by transformation (6 ). Then, two different procedures will be applied to the obtained reduced equations. Equations (3 ) and (4 ) can be reduced to the following nonlinear ODEs:3 ) and (4 ) give the speed of the traveling wave as24 ), we apply the following transformation:28 ), the constraint condition β1 = δ1 = 0 is imposed. Equation (28 ) becomes25 ), we apply the following transformation:25 ) takes the following form:31 ), the constraint condition α1 = γ1 = δ1 = 0 is been imposed. Equation (31 ) becomes29 ) and (32 ), we then derive m = 1.
$\begin{eqnarray}\begin{array}{l}V^{\prime\prime} -({w}_{1}-{k}_{1}^{2}+{\alpha }_{1})V+{\beta }_{1}\,{V}^{n+1}+{\gamma }_{1}\,{V}^{2n+1}\\ \quad +\,{\delta }_{1}\,{V}^{3n+1}+2\,{\lambda }_{1}\,{V}^{4n+1}=0,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}V^{\prime\prime} -({w}_{1}-{k}_{1}^{2})V+{\alpha }_{1}\,{V}^{2n+1}+{\beta }_{1}\,{V}^{3n+1}\\ \quad +\,{\gamma }_{1}\,{V}^{4n+1}+{\delta }_{1}\,{V}^{5n+1}+2\,{\lambda }_{1}\,{V}^{6n+1}=0.\end{array}\end{eqnarray}$
The imaginary parts of equations ( $\begin{eqnarray}{c}_{1}=2{k}_{1}.\end{eqnarray}$
In order to obtain the closed form solutions for equation ( $\begin{eqnarray}V={F}^{\displaystyle \frac{1}{2n}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2n}\left(\displaystyle \frac{1}{2n}-1\right){\left(F^{\prime} \right)}^{2}+\displaystyle \frac{1}{2n}\,F\,F^{\prime\prime} +({w}_{1}-{k}_{1}^{2}+{\alpha }_{1}){F}^{2}+{\beta }_{1}\,{F}^{\tfrac{5}{2}}\\ \quad +{\gamma }_{1}\,{F}^{3}+{\delta }_{1}\,{F}^{\tfrac{7}{2}}+2\,{\lambda }_{1}\,{F}^{4}=0.\end{array}\end{eqnarray}$
To obtain closed form solutions for equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2n}\left(\displaystyle \frac{1}{2n}-1\right){\left(F^{\prime} \right)}^{2}+\displaystyle \frac{1}{2n}\,F\,F^{\prime\prime} +({w}_{1}-{k}_{1}^{2}\\ \quad +\,{\alpha }_{1}){F}^{2}+{\gamma }_{1}\,{F}^{3}+2\,{\lambda }_{1}\,{F}^{4}=0.\end{array}\end{eqnarray}$
To obtain the closed form solutions for equation ( $\begin{eqnarray}V={F}^{\displaystyle \frac{1}{3n}}.\end{eqnarray}$
Equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{3n}\left(\displaystyle \frac{1}{3n}-1\right){\left(F^{\prime} \right)}^{2}+\displaystyle \frac{1}{3n}\,F\,F^{\prime\prime} +({w}_{1}-{k}_{1}^{2}){F}^{2}\\ \quad +\,{\alpha }_{1}\,{F}^{\tfrac{8}{3}}+{\beta }_{1}\,{F}^{3}+{\gamma }_{1}\,{F}^{\tfrac{10}{3}}\\ \quad +\,{\delta }_{1}\,{F}^{\tfrac{11}{3}}+{\lambda }_{1}\,{F}^{4}=0.\end{array}\end{eqnarray}$
To obtain closed form solutions for equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{3n}\left(\displaystyle \frac{1}{3n}-1\right){\left(F^{\prime} \right)}^{2}+\displaystyle \frac{1}{3n}\,F\,F^{\prime\prime} +({w}_{1}-{k}_{1}^{2}){F}^{2}\\ \quad +\,{\beta }_{1}\,{F}^{3}+{\lambda }_{1}\,{F}^{4}=0.\end{array}\end{eqnarray}$
Balancing ${\left(F^{\prime} \right)}^{2}$ and F4 in equations (3.1. The modified simplest equation procedure
We choose the auxiliary solution as follows for the equations because the balance number is 1:
$\begin{eqnarray}F(\varepsilon )={b}_{0}+{b}_{1}\sigma (\varepsilon ).\end{eqnarray}$
The first equation
By substituting equation (33 ) into (29 ) and using some mathematical operations, we obtain a nonlinear algebraic system. Solving this system for different values of b0, b1, λ1 and w1 results in Cases 1 and 2.
Case 1:
$\begin{eqnarray*}\begin{array}{l}\left\{{b}_{0}\to \displaystyle \frac{-n\delta -\delta }{{\gamma }_{1}{n}^{2}},{b}_{1}\to -\displaystyle \frac{{\rm{i}}(n+1)\sqrt{\delta }}{{\gamma }_{1}{n}^{2}},\right.\\ \left.{\lambda }_{1}\to \displaystyle \frac{{\gamma }_{1}^{2}{n}^{2}(2n+1)}{4{\left(n+1\right)}^{2}\delta },{w}_{1}\to \displaystyle \frac{{k}_{1}^{2}{n}^{2}-{\alpha }_{1}{n}^{2}+\delta }{{n}^{2}}\right\}.\end{array}\end{eqnarray*}$
When δ < 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}\,x-{w}_{1}\,t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\tanh \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}\,x-{w}_{1}\,t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\left(-\tanh \left(2\varepsilon \sqrt{-\delta }\right)\pm i{\rm{sech}} \left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\left(-\coth \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{csch}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\left(\delta -{\rm{i}}\sqrt{-{\delta }^{2}}\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}}.\end{array}\end{eqnarray}$
When δ > 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\tan \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)+{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\left(\tan \left(2\varepsilon \sqrt{\delta }\right)\pm \sec \left(2\varepsilon \sqrt{\delta }\right)\right)-{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\left(-\cot \left(2\varepsilon \sqrt{\delta }\right)\pm \csc \left(2\varepsilon \sqrt{\delta }\right)\right)-{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)+{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
where ϵ = x − c1 t.Case 2:
$\begin{eqnarray*}\begin{array}{l}\left\{{b}_{0}\to \displaystyle \frac{-n\delta -\delta }{{\gamma }_{1}{n}^{2}},{b}_{1}\to \displaystyle \frac{{\rm{i}}(n+1)\sqrt{\delta }}{{\gamma }_{1}{n}^{2}},\right.\\ \left.{\lambda }_{1}\to \displaystyle \frac{{\gamma }_{1}^{2}{n}^{2}(2n+1)}{4{\left(n+1\right)}^{2}\delta },{w}_{1}\to \displaystyle \frac{{k}_{1}^{2}{n}^{2}-{\alpha }_{1}{n}^{2}+\delta }{{n}^{2}}\right\}.\end{array}\end{eqnarray*}$
When δ < 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\tanh \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\left(-\tanh \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{isech}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\left(-\coth \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{csch}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{(n+1)\left(\delta +{\rm{i}}\sqrt{-{\delta }^{2}}\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}}.\end{array}\end{eqnarray}$
When δ > 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times {\left(\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\tan \left(\varepsilon \sqrt{\delta }\right)+{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\left(\tan \left(2\varepsilon \sqrt{\delta }\right)\pm \sec \left(2\varepsilon \sqrt{\delta }\right)\right)+{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\left(-\cot \left(2\varepsilon \sqrt{\delta }\right)\pm \csc \left(2\varepsilon \sqrt{\delta }\right)\right)+{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{w}_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{{\rm{i}}(n+1)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{{\gamma }_{1}{n}^{2}}\right)}^{\tfrac{1}{2n}},\end{array}\end{eqnarray}$
where ϵ = x − c1 t.The second equation
By substituting equation (33 ) into equation (32 ) and using some mathematical operations, we obtain a nonlinear algebraic system. Solving this system for different values of b0, b1, λ1 and ω1 results in Cases 1 and 2.
Case 1:
$\begin{eqnarray*}\begin{array}{l}\left\{{b}_{0}\to -\displaystyle \frac{2(3n\delta +2\delta )}{9{\beta }_{1}{n}^{2}},{b}_{1}\to -\displaystyle \frac{2{\rm{i}}(3n+2)\sqrt{\delta }}{9{\beta }_{1}{n}^{2}},\right.\\ \left.{\lambda }_{1}\to \displaystyle \frac{9{\beta }_{1}^{2}{n}^{2}(3n+1)}{4{\left(3n+2\right)}^{2}\delta },{\omega }_{1}\to \displaystyle \frac{9{k}^{2}{n}^{2}+4\delta }{9{n}^{2}}\right\}.\end{array}\end{eqnarray*}$
When δ < 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\tanh \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\left(-\tanh \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{isech}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\left(-\coth \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{csch}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}}.\end{array}\end{eqnarray}$
When δ > 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\tan \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)+{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\left(\tan \left(2\varepsilon \sqrt{\delta }\right)\pm \sec \left(2\varepsilon \sqrt{\delta }\right)\right)-{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\left(-\cot \left(2\varepsilon \sqrt{\delta }\right)\pm \csc \left(2\varepsilon \sqrt{\delta }\right)\right)-{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{2i(3n+2)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)+{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
where ϵ = x − c1 t.Case 2:
$\begin{eqnarray*}\begin{array}{l}\left\{{b}_{0}\to -\displaystyle \frac{2(3n\delta +2\delta )}{9{\beta }_{1}{n}^{2}},{b}_{1}\to \displaystyle \frac{2{\rm{i}}(3n+2)\sqrt{\delta }}{9{\beta }_{1}{n}^{2}},\right.\\ \left.{\lambda }_{1}\to \displaystyle \frac{9{\beta }_{1}^{2}{n}^{2}(3n+1)}{4{\left(3n+2\right)}^{2}\delta },{\omega }_{1}\to \displaystyle \frac{9{k}^{2}{n}^{2}+4\delta }{9{n}^{2}}\right\}.\end{array}\end{eqnarray*}$
When δ < 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\tanh \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\left(-\tanh \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{isech}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }-{\rm{i}}\sqrt{-\delta }\left(-\coth \left(2\varepsilon \sqrt{-\delta }\right)\pm \mathrm{csch}\left(2\varepsilon \sqrt{-\delta }\right)\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2(3n+2)\sqrt{\delta }\left(\sqrt{\delta }+{\rm{i}}\sqrt{-\delta }\coth \left(\varepsilon \sqrt{-\delta }\right)\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}}.\end{array}\end{eqnarray}$
When δ > 0, the following solutions are obtained: $\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\tan \left(\varepsilon \sqrt{\delta }\right)+i\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\left(\tan \left(2\varepsilon \sqrt{\delta }\right)\pm \sec \left(2\varepsilon \sqrt{\delta }\right)\right)+{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\left(-\cot \left(2\varepsilon \sqrt{\delta }\right)\pm \csc \left(2\varepsilon \sqrt{\delta }\right)\right)+{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}W(x,t)={{\rm{e}}}^{{\rm{i}}({k}_{1}x-{\omega }_{1}t)}\\ \,\times \,{\left(-\displaystyle \frac{2{\rm{i}}(3n+2)\delta \left(\cot \left(\varepsilon \sqrt{\delta }\right)-{\rm{i}}\right)}{9{\beta }_{1}{n}^{2}}\right)}^{\tfrac{1}{3n}},\end{array}\end{eqnarray}$
where ϵ = x − c1 t.3.2. The modified Kudryashov procedure
The first equation
If we take the following auxiliary solution for equation (29 ):74 ) into equation (29 ) and collect the coefficients of ${\sigma }^{i}\left(\varepsilon \right)$, we obtain the determining equation system. If we solve the obtained system, we get the following solution family (Cases 1 and 2).
$\begin{eqnarray}F\left(\varepsilon \right)={b}_{0}+{b}_{1}\sigma \left(\varepsilon \right),\end{eqnarray}$
then substitute equation (Case 1:
$\begin{eqnarray}\begin{array}{l}{b}_{0}\to 0,{b}_{1}\to \displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(n+1\right)}{2{\gamma }_{1}{n}^{2}},\\ {\lambda }_{1}\to -\displaystyle \frac{{\gamma }_{1}^{2}{n}^{2}\left(2n+1\right)}{{\left(\mathrm{ln}a\right)}^{2}\left({n}^{2}+2n+1\right)},\\ {\omega }_{1}\to -\displaystyle \frac{-4{k}^{2}{n}^{2}+4{\alpha }_{1}{n}^{2}+{\left(\mathrm{ln}a\right)}^{2}}{4{n}^{2}}.\end{array}\end{eqnarray}$
If we substitute the obtained values of the coefficients from equations (75 ) into equation (3 ) without ignoring equations (26 ), (27 ) and (74 ), we get the following solution:
$\begin{eqnarray}\begin{array}{l}W\left(x,t\right)={{\rm{e}}}^{{\rm{i}}\left({k}_{1}x-{\omega }_{1}t\right)}\\ \,\times \,{\left(\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(n+1\right)}{2{\gamma }_{1}{n}^{2}\left(1+{\rm{\Gamma }}{a}^{x-2{k}_{1}t}\right)}\right)}^{\tfrac{1}{2n}}.\end{array}\end{eqnarray}$
Case 2:77 ), we obtain the following exact solution:
$\begin{eqnarray}\begin{array}{l}{b}_{0}\to \displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(n+1\right)}{2{\gamma }_{1}{n}^{2}},\\ {b}_{1}\to -\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(n+1\right)}{2{\gamma }_{1}{n}^{2}},\\ {\lambda }_{1}\to -\displaystyle \frac{{\gamma }_{1}^{2}{n}^{2}\left(2n+1\right)}{{\left(\mathrm{ln}a\right)}^{2}\left({n}^{2}+2n+1\right)},\end{array}\end{eqnarray}$
$\begin{eqnarray*}{\omega }_{1}\to -\displaystyle \frac{-4{k}^{2}{n}^{2}+4{\alpha }_{1}{n}^{2}+{\left(\mathrm{ln}a\right)}^{2}}{4{n}^{2}}.\end{eqnarray*}$
If we use the values from ( $\begin{eqnarray}\begin{array}{l}W\left(x,t\right)={{\rm{e}}}^{{\rm{i}}\left({k}_{1}x-{\omega }_{1}t\right)}\\ \quad \times \,{\left(\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}{a}^{x-2{k}_{1}t}\left(n+1\right){\rm{\Gamma }}}{2{\gamma }_{1}{n}^{2}\left(1+{\rm{\Gamma }}{a}^{x-2{k}_{1}t}\right)}\right)}^{\tfrac{1}{2n}}.\end{array}\end{eqnarray}$
The second equation
If we take the auxiliary solution for equation (32 ) as (74 ), then substitute equation (74 ) in (32 ), then collect the coefficients of ${\sigma }^{i}\left(\varepsilon \right),$ we obtain the determining equation system. If we solve the obtained system, we get the following solution family (Cases 1 and 2):
Case 1:
$\begin{eqnarray}\begin{array}{l}{b}_{0}\to 0,{b}_{1}\to \displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(3n+2\right)}{9{\beta }_{1}{n}^{2}},\\ {\lambda }_{1}\to -\displaystyle \frac{9{\beta }_{1}^{2}{n}^{2}\left(3n+1\right)}{{\left(\mathrm{ln}a\right)}^{2}\left(9{n}^{2}+12n+4\right)},\\ {\omega }_{1}\to -\displaystyle \frac{-9{k}^{2}{n}^{2}+{\left(\mathrm{ln}a\right)}^{2}}{9{n}^{2}}.\end{array}\end{eqnarray}$
If we substitute the obtained values of the coefficients equations (79 ) in (4 ) without ignoring equations (26 ), (27 ) and (74 ), we get the following solution:
$\begin{eqnarray}\begin{array}{l}W\left(x,t\right)={{\rm{e}}}^{{\rm{i}}\left({k}_{1}x-{\omega }_{1}t\right)}\\ \quad \times \,{\left(\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(3n+2\right)}{9{\beta }_{1}{n}^{2}\left(1+{\rm{\Gamma }}{a}^{x-2{k}_{1}t}\right)}\right)}^{\tfrac{1}{3n}}.\end{array}\end{eqnarray}$
Case 2:81 ), the exact solution is given by
$\begin{eqnarray}\begin{array}{l}{b}_{0}\to \displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(3n+2\right)}{9{\beta }_{1}{n}^{2}},\\ {b}_{1}\to -\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}\left(3n+2\right)}{9{\beta }_{1}{n}^{2}},\\ {\lambda }_{1}\to -\displaystyle \frac{9{\beta }_{1}^{2}{n}^{2}\left(3n+1\right)}{{\left(\mathrm{ln}a\right)}^{2}\left(9{n}^{2}+12n+4\right)},\end{array}\end{eqnarray}$
$\begin{eqnarray*}{\omega }_{1}\to -\displaystyle \frac{-9{k}^{2}{n}^{2}+{\left(\mathrm{ln}a\right)}^{2}}{9{n}^{2}}.\end{eqnarray*}$
If we use values from ( $\begin{eqnarray}\begin{array}{l}W\left(x,t\right)={{\rm{e}}}^{{\rm{i}}\left({k}_{1}x-{\omega }_{1}t\right)}\\ \,\times \,{\left(\displaystyle \frac{{\left(\mathrm{ln}a\right)}^{2}{a}^{x-2{k}_{1}t}\left(3n+2\right){\rm{\Gamma }}}{9{\beta }_{1}{n}^{2}\left(1+{\rm{\Gamma }}{a}^{x-2{k}_{1}t}\right)}\right)}^{\tfrac{1}{3n}}.\end{array}\end{eqnarray}$
4. Plots of the results
Plots of some of the results will be given in this section. First, we give the plots for equation (34 ) in figure 1. When we set the parameters ω1 = − 0.8, n = 2, k1 = 0.5, α1 = 0.2 and γ1 = 0.7 in equation (34 ), the three-dimensional (3D) and contour plots shown in figure 1 are obtained. When we set t = 0, t = 0.5 and t = 1, red, green and blue lines, respectively, are obtained in the two-dimensional (2D) plot in figure 1; this figure represents the kink shape solution.
Figure 1. Plots of equation ( |
We show the plots for equation (59 ) in figure 2. When we set the parameters ω1 = 0.8, n = 2, k1 = 0.5 and β1 = 0.2 in equation (59 ), the 3D and contour plots obtained are shown in figure 2. When we set t = 0, t = 0.5 and t = 1, red, green and blue lines, respectively, are obtained in the 2D plot in figure 2; this figure also represents the periodic shape solution.
Figure 2. Plots of equation ( |
We give the plots for equation (76 ) in figure 3. When we set the parameters in equation (76 ) as a = 0.2, Γ = 0.1, n = 2, k1 = 0.8, γ1 = 1 and α1 = 0.3, the 3D and contour plots are obtained are shown in figure 3. When we set t = 0, t = 0.5 and t = 1, red, green and blue lines, respectively, are obtained in the 2D plot in figure 3; this figure represents the kink shape solution.
Figure 3. Plots of equation ( |
We show the plots for equation (82 ) in figure 4. We set the parameters in equation (82 ) as a = 0.2, Γ = 0.1, n = 2, k1 = 0.8 and β1 = 1, and the 3D and contour plots obtained are shown in figure 4. When we set t = 0, t = 0 and t = 1, red, green and blue lines, respectively, are obtained in 2D plot shown in figure 4; this figure represents kink shape solution.
Figure 4. Plots of equation ( |
5. Conclusion
In this study, a new form of propagation pulse in optical fibers is studied to obtain the exact solutions. We assumed different values for the arbitrary constants of the given model and obtained two different nonlinear differential equations. Then, we applied suitable wave transformations to the models. Thus, the given models were reduced to nonlinear ordinary differential equations. Then, we applied the two different procedures, the modified simplest equation and the modified Kudryashov procedures, to the reduced equations. The solutions are given by hyperbolic, trigonometric and rational functions. When we compared our methods with the other methods in the literature our methods gave simpler results: our solutions are expressed as different types of functions and our methods are easier to apply to equations. We concluded that the methods discussed in our study are effective and powerful. However, the methods have some disadvantages: while successful for specific types of equations, the modified Kudryashov method may not be applicable to all nonlinear equations. Its effectiveness depends heavily on the specific structure of the equation. We give the 3D, contour and 2D plots for some results. These figures are important for understanding the motion of the wave. The results can be used in various fields such as physics, engineering and optics. These methods can apply to the different equations or researchers could apply different methods to the given models.