Exact Optical Solitons for Generalized Kudryashov’s Equation by Lie Symmetry Method

Abstract

In this article, we use Lie point symmetry analysis to extract some new optical soliton solutions for the generalized Kudryashov’s equation (GKE) with an arbitrary power nonlinearity. Using a traveling wave transformation, the GKE is transformed into a nonlinear second order ordinary differential equation (ODE). Using Lie point symmetry analysis, the nonlinear second-order ODE is reduced to a first-order ODE. This first-order ODE is solved in two cases to retrieve some new bright, dark, and kink soliton solutions of the GKE. These soliton solutions for the GKE are obtained here for the first time.

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.

1. Introduction

Nonlinear partial differential equations (NLPDEs) play an important role in different branches of mathematical and physical sciences such as fluid mechanics [1–4], particle physics [5, 6], quantum mechanics [7, 8], chemistry [9], optical fibers [10–12], and other areas of engineering [13–15]. Exact solutions and the study of dynamical behaviors of optical solitons in nonlocal nonlinear media of NLPDEs are very important for understanding various nonlinear physical phenomena. There are many methods used to obtain the exact solutions of NLPDEs such as Hirota’s method [16], complete polynomial discriminant system method [17], extended generalized Kudryashov’s method [18], extended trial function scheme [19], exponential expansion method [20, 21], sine-Gordon method [22, 23], Jacobi elliptic ansatz method [24], extended variational approach [25], direct similarity technique [26], and Lie symmetry method [27, 28].

Optical solitons play a very important role in fiber optics communication systems. Therefore, many researchers are interested in seeking this type of solutions for many NLPDEs describing pulse propagation inside optical fibers [18–20]. One of these models is the following GKE with an arbitrary power nonlinearity [18]:$$\begin{array}{}\text{(1)}& i{q}_t+a{q}_{xx}+\frac{c_1}{q^{4n}}+\frac{c_2}{q^{3n}}+\frac{c_3}{q^{2n}}+\frac{c_4}{q^n}+c_5{q}^n+c_6{q}^{2n}+c_7{q}^{3n}+c_8{q}^{4n}q=0,\end{array}$$where $a,c_j,1\le j\le 8$ are coefficients of chromatic dispersion and self-phase modulation (SPM), respectively. $n$ represents power nonlinearity parameter, while $i=\sqrt{-1}$, $q=qx,t$ is a complex valued function that describes the pulse profile, where $x$ and $t$ represent spatial and temporal variables, respectively.

Equation (1) was proposed and solved in reference [18]. When $c_2=c_4=c_5=c_7=0,$ some bright, dark, and singular soliton solutions are obtained in reference [18]. In reference [29], equation (1) is solved when $n=1,c_2=c_3=c_4=c_5=c_7=0$ using Jacobi’s elliptic function expansion method. In reference [30], equation (1) is studied when $c_1=c_2=c_7=c_8=0$ using the exponential rational function method and Painlevé approach. In this paper, we find some new soliton solutions for equation (1). Some bright, dark, and kink soliton solutions will be obtained when $c_1=c_2=0,$ whereas some bright and dark soliton solutions will be obtained when $c_7=c_8=0.$

Lie point Symmetry analysis is considered as one of the most important methods that can be used for seeking new exact solutions for PDEs and ODEs [31–35]. Details of this method can be found in reference [31]. This method will be used in this paper to find some new optical soliton solutions for equation (1).

This paper is organized as follows: In Section 2, equation (1) will be transformed into a second ODE using a traveling wave transformation. Then, this second-order ODE will be reduced to a first-order ODE using the Lie point symmetry method. New dark, bright, and kink soliton solutions for equation (1) will be obtained in Section 3. Finally, Section 4 concludes the paper.

2. Lie Point Symmetry Method

For solving equation (1), we use the traveling wave transformation as follows:$$\begin{array}{}\text{(2)}& qx,t=yz{e}^{i\phi },\hspace{1em}z=kx-\mu t,\phi =-kx+\omega t+\theta ,\end{array}$$where $yz$ is the amplitude of the soliton, $k$ and $\mu$ are constants, $\omega$ represents the frequency of soliton, $k$ represents the wave number, and $\theta$ is the phase constant.

Substituting equation (2) into equation (1) and putting the imaginary and real parts to zero, we obtain the real part as follows:$$\begin{array}{}\text{(3)}& -a{k}^{2}yz-wyz+c_{1}y{z}^{1-4n}+c_{2}y{z}^{1-3n}+c_{3}y{z}^{1-2n}+c_{4}y{z}^{1-n}+c_{5}y{z}^{1+n}+c_{6}y{z}^{1+2n}+c_{7}y{z}^{1+3n}+c_{8}y{z}^{1+4n}+a{k}^2{y}^{\prime \prime }z=0,\end{array}$$and the imaginary part gives$$\begin{array}{}\text{(4)}& \mu =-2a{k}^2.\end{array}$$

Here, we use Lie symmetry analysis [31–35] to reduce equation (3) to a first-order ODE with well-known solutions. The autonomous ODE in equation (3) admits the Lie symmetry generator [31].$$\begin{array}{}\text{(5)}& \Gamma =\frac{\partial }{\partial z}.\end{array}$$

The canonical coordinates $r,Sr$ associated with the infinitesimal generator (5) are given by$$\begin{array}{}\text{(6)}& r=y,Sr=z,\end{array}$$which are prolonged to$$\begin{array}{c}\text{(7)}& \frac{\text{d}S}{\text{d}r}=\frac{1}{y″},\frac{{\text{d}}^{2}S}{\text{d}{r}^2}=\frac{-y″}{y{\prime }^3}.\end{array}$$

Substituting equations (6) and (7) into equation (3), we get$$\begin{array}{}\text{(8)}& -a{k}^{2}r-wr+c_1{r}^{1-4n}+c_2{r}^{1-3n}+c_3{r}^{1-2n}+c_4{r}^{1-n}+c_5{r}^{1+n}+c_6{r}^{1+2n}+c_7{r}^{1+3n}+c_8{r}^{1+4n}{S}^{\prime 3}-a{k}^2\frac{d^{2}S}{\text{d}{r}^2}=0.\end{array}$$

Let$$\begin{array}{}\text{(9)}& Qr=S^{\prime }r,\end{array}$$to get$$\begin{array}{}\text{(10)}& Q^{\prime }r=\frac{1}{a{k}^2}-a{k}^{2}r-wr+c_1{r}^{1-4n}+c_2{r}^{1-3n}+c_3{r}^{1-2n}+c_4{r}^{1-n}+c_5{r}^{1+n}+c_6{r}^{1+2n}+c_7{r}^{1+3n}+c_8{r}^{1+4n}Q{r}^3.\end{array}$$

Equation (10) can be solved easily to obtain$$\begin{array}{}\text{(11)}& \frac{1}{Q{r}^2}=r^2+\frac{r^{2}w}{a{k}^2}-\frac{r^{2-4n}{c}_1}{k^{2}a-2an}+\frac{2r^{2-3n}{c}_2}{a{k}^2-2+3n}+\frac{r^{2-2n}{c}_3}{a{k}^2-1+n}+\frac{2r^{2-n}{c}_4}{a{k}^2-2+n}-\frac{2r^{2+n}{c}_5}{a{k}^{2}2+n}-\frac{r^{2+2n}{c}_6}{k^{2}a+an}-\frac{2r^{2+3n}{c}_7}{k^{2}2a+3an}-\frac{r^{2+4n}{c}_8}{k^{2}a+2an}.\end{array}$$

Using equation (6) and (9), equation (11) becomes$$\begin{array}{}\text{(12)}& y^{\prime }{z}^2=y{z}^2+\frac{wy{z}^2}{a{k}^2}-\frac{c_{1}y{z}^{2-4n}}{k^{2}a-2an}+\frac{2c_{2}y{z}^{2-3n}}{a{k}^2-2+3n}+\frac{c_{3}y{z}^{2-2n}}{a{k}^2-1+n}+\frac{2c_{4}y{z}^{2-n}}{a{k}^2-2+n}-\frac{2c_{5}y{z}^{2+n}}{a{k}^{2}2+n}-\frac{c_{6}y{z}^{2+2n}}{k^{2}a+an}-\frac{2c_{7}y{z}^{2+3n}}{k^{2}2a+3an}-\frac{c_{8}y{z}^{2+4n}}{k^{2}a+2an}.\end{array}$$

Equation (12) has many solutions, but we are interested in soliton solutions only which will appear in the next section.

3. Soliton Solutions

(i) In order to obtain the soliton solutions, assume that$$\begin{array}{}\text{(13)}& yz=G{z}^{1/n}.\end{array}$$

  Substituting equation (13) into equation (12), we obtain$$\begin{array}{}\text{(14)}& G^{\prime 2}=G^2{n}^2+\frac{n^{2}w}{a{k}^2}-\frac{n^2{c}_1}{k^{2}a-2an}{G}^{-2}+\frac{2n^2{c}_2}{a{k}^2-2+3n}{G}^{-1}+\frac{n^2{c}_3}{a{k}^2-1+n}+\frac{2n^2{c}_4}{a{k}^2-2+n}G-\frac{2n^2{c}_5}{a{k}^{2}2+n}{G}^3-\frac{n^2{c}_6}{k^{2}a+an}{G}^4-\frac{2n^2{c}_7}{k^{2}2a+3an}{G}^5-\frac{n^2{c}_8}{k^{2}a+2an}{G}^6.\end{array}$$

  Let $c_1=c_2=0,$ to get$$\begin{array}{}\text{(15)}& G^{\prime 2}=\frac{n^2{c}_3}{a{k}^2-1+n}+\frac{2n^2{c}_4}{a{k}^2-2+n}G+n^2+\frac{n^{2}w}{a{k}^2}{G}^2-\frac{2n^2{c}_5}{a{k}^{2}2+n}{G}^3-\frac{n^2{c}_6}{k^{2}a+an}{G}^4-\frac{2n^2{c}_7}{k^{2}2a+3an}{G}^5-\frac{n^2{c}_8}{k^{2}a+2an}{G}^6.\end{array}$$

  Assume that$$\begin{array}{}\text{(16)}& Gz=A+Vz.\end{array}$$

  Hence, equation (15) can be reduced to the following:$$\begin{array}{c}\text{(17)}& V^{\prime 2}=A^2{n}^2+\frac{A^2{n}^{2}w}{a{k}^2}+\frac{n^2{c}_3}{a{k}^2-2+n}+\frac{2A{n}^2{c}_4}{a{k}^2-2+n}+\frac{2A^3{n}^2{c}_5}{a{k}^{2}2+n}-\frac{A^2{n}^2{c}_6}{k^{2}a1+n}-\frac{2A^5{n}^2{c}_7}{k^{2}a2+3n}-\frac{A^2{n}^2{c}_8}{k^{2}a1+2n}+2A{n}^2+\frac{2A{n}^{2}w}{a{k}^2}+\frac{2n^2{c}_4}{a{k}^2-2+n}-\frac{6A^2{n}^2{c}_5}{a{k}^{2}2+n}-\frac{4A^3{n}^2{c}_6}{k^{2}a1+n}-\frac{10A^4{n}^2{c}_7}{k^{2}a2+3n}-\frac{6A^5{n}^2{c}_8}{k^{2}a1+2n}V\\ +n^2+\frac{n^{2}w}{a{k}^2}-\frac{6A{n}^2{c}_5}{a{k}^{2}2+n}-\frac{6A^2{n}^2{c}_6}{k^{2}a1+n}-\frac{20A^3{n}^2{c}_7}{k^{2}a2+3n}-\frac{15A^4{n}^2{c}_8}{k^{2}a1+2n}{V}^2\\ +-\frac{2n^2{c}_5}{a{k}^{5}2+n}-\frac{4A{n}^2{c}_6}{k^{2}a1+n}-\frac{20A^2{n}^2{c}_7}{k^{2}a2+3n}-\frac{20A^3{n}^2{c}_8}{k^{2}a1\_2n}{V}^3\\ +-\frac{n^2{c}_6}{k^{2}a1+n}-\frac{10A{n}^2{c}_7}{k^{2}a2+3n}-\frac{15A^2{n}^2{c}_8}{k^{2}a1+2n}{V}^4\\ +-\frac{2n^2{c}_7}{k^{2}a2+3n}-\frac{6A{n}^2{c}_8}{k^{2}a1+2n}{V}^5-\frac{n^2{c}_8}{k^{2}a1+2n}{V}^6.\end{array}$$

  It is well known that equation (17) has many solutions as mentioned in reference [7]. We choose the following three solutions from them:

  Case (1). The bright soliton solution.

  Equation (17) admits a bright soliton solution in the following form:$$\begin{array}{}\text{(18)}& Vz=\sqrt{\frac{2{\lambda }_1}{\sqrt{{\lambda }_2^2-4{\lambda }_1{\lambda }_3}\text{Cosh}2\sqrt{{\lambda }_1}z-{\lambda }_2}}.\end{array}$$

  Solution (18) satisfies equation (17) when$$\begin{array}{c}\text{(19)}& w=-a{k}^2+\frac{16{1+2n}^3{c}_7^4}{27{2+3n}^4{c}_8^3}-\frac{4{1+2n}^2{c}_6{c}_7^2}{91+n{2+3n}^2{c}_8^2}+\frac{32+3n{c}_4{c}_8}{-2+n1+2n{c}_7},{\lambda }_1=-\frac{n^2{1+2n}^3{c}_7^4}{3a{k}^2{2+3n}^4{c}_8^3}+\frac{2n^2{1+2n}^2{c}_6{c}_7^2}{9a{k}^{2}1+n{2+3n}^2{c}_8^2}+\frac{3n^{2}2+3n{c}_4{c}_8}{a{k}^2-2+n1+2n{c}_7},\\ c_3=\frac{-13-1+n{1+2n}^5{c}_7^6}{729{2+3n}^6{c}_8^5}+\frac{-1+n{1+2n}^4{c}_6{c}_7^4}{81{2+3n}^{4}1+n{c}_8^4}+\frac{-1+n1+2n{c}_7{c}_4}{32+3n-2+n{c}_8},\\ c_5=\frac{22+n1+2n{c}_7-101+n1+2n{c}_7^2+9{2+3n}^2{c}_6{c}_8}{271+n2+3n{c}_8^2},\\ {\lambda }_2=\frac{n^2}{3a{k}^2}-\frac{3c_6}{1+n}+\frac{51+2n{c}_7^2}{{2+3n}^2{c}_8},\\ {\lambda }_3=-\frac{n^2{c}_8}{k^{2}a1+2n},\\ A=-\frac{1+2n{c}_7}{32+3n{c}_8},\end{array}$$

  provided that ${\lambda }_1>0,$${\lambda }_2^2-4{\lambda }_1{\lambda }_3>0$, $c_4,c_6,c_7$, and $c_8$ are arbitrary constants.

  Substituting equations (16) and (18) into equation (13), we obtain$$\begin{array}{}\text{(20)}& yz={A+\sqrt{\frac{2{\lambda }_1}{\sqrt{{\lambda }_2^2-4{\lambda }_1{\lambda }_3}\text{Cosh}2\sqrt{{\lambda }_1}z-{\lambda }_2}}}^{1/n}.\end{array}$$

  Substitute equation (20) into equation (2), to get the following bright soliton solution of equation (1):$$\begin{array}{c}\text{(21)}& qx,t={A+\sqrt{\frac{2{\lambda }_1}{\sqrt{{\lambda }_2^2-4{\lambda }_1{\lambda }_3}\text{Cosh}2\sqrt{{\lambda }_1}kx+2a{k}^{2}t-{\lambda }_2}}}^{1/n}\exp i-kx+-a{k}^2+\frac{16{1+2n}^3{c}_7^4}{27{2+3n}^4{c}_8^3}-\frac{4{1+2n}^2{c}_6{c}_7^2}{91+n{2+3n}^2{c}_8^2}+\frac{32+3n{c}_4{c}_8}{-2+n1+2n{c}_7}t+\theta .\end{array}$$

  Figure 1 represents the 3D plot of the bright soliton solution (21), while Figure 2 represents the propagation of the bright soliton solution (21).

  Case 2. Kink soliton solution.

  Equation (17) admits a kink soliton solution in the following form:$$\begin{array}{}\text{(22)}& Vz=\sqrt{\frac{5{1+2n}^2{c}_7^2}{12c_8^2{2+3n}^2}-\frac{3c_{6}1+2n}{12c_{8}1+n}1+\tanh \sqrt{{\lambda }_4}z},\end{array}$$

  where$$\begin{array}{}\text{(23)}& {\lambda }_4=-\frac{1+2n{5n1+n1+2n{c}_7^2-3n{2+3n}^2{c}_6{c}_8}^2}{36a{k}^2{1+n}^2{2+3n}^4{c}_8^3}.\end{array}$$

  Solution (22) satisfies equation (17) when$$\begin{array}{c}\text{(24)}& w=-a{k}^2+\frac{1+2n}{108c_8^3}\frac{25{1+2n}^2{c}_7^4}{{2+3n}^4}+\frac{181+2n{c}_6{c}_7^2{c}_8}{1+n{2+3n}^2}-\frac{27c_6^2{c}_8^2}{{1+n}^2},c_3=-\frac{-1+n{1+2n}^3{131+n1+2n{c}_7^3-9{2+3n}^2{c}_6{c}_7{c}_8}^2}{2916{1+n}^2{2+3n}^6{c}_8^5},\\ c_4=-\frac{13-2+n{1+2n}^4{c}_7^5}{108{2+3n}^5{c}_8^4}+\frac{11-2+n{1+2n}^3{c}_6{c}_7^3}{541+n{2+3n}^3{c}_8^3}-\frac{-2+n{1+2n}^2{c}_6^2{c}_7}{12{1+n}^{2}2+3n{c}_8^2},\\ c_5=\frac{-202+n{1+2n}^2{c}_7^3}{27{2+3n}^3{c}_8^2}+\frac{22+n1+2n{c}_7{c}_6}{31+n2+3n{c}_8}.\end{array}$$

  Substituting equations (16) and (22) into equation (13), we obtain$$\begin{array}{}\text{(25)}& yz={-\frac{1+2n{c}_7}{32+3n{c}_8}+\sqrt{\frac{5{1+2n}^2{c}_7^2}{12c_8^2{2+3n}^2}-\frac{3c_{6}1+2n}{12c_{8}1+n}1+\tanh \sqrt{{\lambda }_4}z}}^{1/n}.\end{array}$$

  Substituting equation (25) into equation (2), to get the following kink soliton solution of equation (1):$$\begin{array}{c}\text{(26)}& qx,t={-\frac{1+2n{c}_7}{32+3n{c}_8}+\sqrt{\frac{5{1+2n}^2{c}_7^2}{12c_8^2{2+3n}^2}-\frac{3c_{6}1+2n}{12c_{8}1+n}1+\tanh \sqrt{{\lambda }_4}kx+2n^{2}t}}^{1/n}\exp i-kx+-a{k}^2+\frac{1+2n}{108c_8^3}\frac{25{1+2n}^2{c}_7^4}{{2+3n}^4}+\frac{181+2n{c}_6{c}_7^2{c}_8}{1+n{2+3n}^2}-\frac{27c_6^2{c}_8^2}{{1+n}^2}t+\theta .\end{array}$$

  Figure 3 represents the 3D plot of the kink soliton solution of equation (26) and Figure 4 represents the propagation of the kink soliton solution of equation (26).

  Case (3). Dark soliton solution.

  Equation (17) admits a dark soliton solution in the following form:$$\begin{array}{}\text{(27)}& Vz=\sqrt{\frac{-8{\lambda }_5{\tanh }^2\sqrt{-{\lambda }_5/3}z}{n^2/a{k}^2-3c_6/1+n+51+2n{c}_7^2/{2+3n}^2{c}_{8}3+{\tanh }^2\sqrt{\sqrt{-{\lambda }_5/3}z}z}},\end{array}$$

  where$$\begin{array}{}\text{(28)}& {\lambda }_5=-\frac{1+2n{5n1+n1+2n{c}_7^2-3n{2+3n}^2{c}_6{c}_8}^2}{36a{k}^2{1+n}^2{2+3n}^4{c}_8^3}.\end{array}$$

  Solution (27) satisfies equation (17) when$$\begin{array}{c}\text{(29)}& w=-a{k}^2+\frac{1+2n}{108c_8^3}\frac{25{1+2n}^2{c}_7^4}{{2+3n}^4}+\frac{181+2n{c}_6{c}_7^2{c}_8}{1+n{2+3n}^2}-\frac{27c_6^2{c}_8^2}{{1+n}^2},c_3=\frac{1}{108{1+n}^3{2+3n}^6{c}_8^5}-1+n{1+2n}^{2}31+n1+2n{c}_7^2-2{2+3n}^2{c}_6{c}_8{1+n1+2n{c}_7^2-{2+3n}^2{c}_6{c}_8}^2,\\ c_4=-\frac{1}{108{1+n}^2{2+3n}^5{c}_8^4}-2+n{1+2n}^2{c}_{7}13{1+n}^2{1+2n}^2{c}_7^4-221+n1+2n{2+3n}^2{c}_6{c}_7^2{c}_8+9{2+3n}^4{c}_6^2{c}_8^2,\\ c_5=\frac{22+n1+2n{c}_7}{271+n{2+3n}^3{c}_8^2}-101+n1+2n{c}_7^2+9{2+3n}^2{c}_6{c}_8.\end{array}$$

  Substituting equations (16) and (27) into equation (13), we obtain$$\begin{array}{}\text{(30)}& yz={A+\sqrt{\frac{-8{\lambda }_5{\tanh }^2\sqrt{-{\lambda }_5/3}z}{n^2/a{k}^2-3c_6/1+n+51+2n{c}_7^2/{2+3n}^2{c}_{8}3+{\tanh }^2\sqrt{-{\lambda }_5/3}z}}}^{1/n},\end{array}$$

  where $A=-1+2n{c}_7/32+3n{c}_8$. Substituting equation (30) into equation (2), we obtain the following dark soliton solution of equation (1):$$\begin{array}{c}\text{(31)}& qx,t={A+\sqrt{\frac{-8{\lambda }_5{\tanh }^2\sqrt{-{\lambda }_5/3}kx+2n^{2}t}{n^2/--3c_6/1+n+51+2n{c}_7^2/{2+3n}^2{c}_{8}3+{\tanh }^2\sqrt{-{\lambda }_5/3}kx+2n^{2}t}}}^{1/n}\exp i-kx+-a{k}^2+\frac{1+2n}{108c_8^3}\frac{25{1+2n}^2{c}_7^4}{{2+3n}^4}+\frac{181+2n{c}_6{c}_7^2{c}_8}{1+n{2+3n}^2}-\frac{27c_6^2{c}_8^2}{{1+n}^2}t+\theta .\end{array}$$

  Figure 5 represents the 3D plot of the dark soliton solution of equation (31), while Figure 6 represents the propagation of the dark soliton solution of equation (31).

(ii) To get another soliton solution for equation (12), we use the following substitution:$$\begin{array}{}\text{(32)}& yz=G{z}^{-1/n}.\end{array}$$

  Substituting equation (32) into equation (12), we obtain$$\begin{array}{}\text{(33)}& G^{\prime 2}=n^2+\frac{n^{2}w}{a{k}^2}{G}^2-\frac{n^2{c}_1}{k^{2}a-2an}{G}^6+\frac{2n^2{c}_2}{a{k}^2-2+3n}{G}^5+\frac{n^2{c}_3}{a{k}^2-1+n}{G}^4+\frac{2n^2{c}_4}{a{k}^2-2+n}{G}^3-\frac{2n^2{c}_5}{a{k}^{2}2+n}G-\frac{n^2{c}_6}{k^{2}a+an}-\frac{2n^2{c}_7}{k^{2}2a+3anG}-\frac{n^2{c}_8}{k^{2}a+2an{G}^2}.\end{array}$$

  Let $c_7=c_8=0,$ to get$$\begin{array}{}\text{(34)}& G^{\prime 2}=n^2+\frac{n^{2}w}{a{k}^2}{G}^2-\frac{n^2{c}_1}{k^{2}a-2an}{G}^6+\frac{2n^2{c}_2}{a{k}^2-2+3n}{G}^5+\frac{n^2{c}_3}{a{k}^2-1+n}{G}^4+\frac{2n^2{c}_4}{a{k}^2-2+n}{G}^3-\frac{2n^2{c}_5}{a{k}^{2}2+n}G-\frac{n^2{c}_6}{k^{2}a+an}.\end{array}$$

  Assume that$$\begin{array}{}\text{(35)}& Gz=h_1+h_{2}Vz.\end{array}$$

  Hence, equation (34) can be reduced to the following:$$\begin{array}{c}\text{(36)}& V^{\prime 2}=\frac{h_1^2{n}^2}{h_2^2}+\frac{h_1^2{n}^{2}w}{a{h}_2^2{k}^2}-\frac{h_1^2{n}^2{c}_1}{h_2^2{k}^{2}1-2n}+\frac{2h_1^5{n}^2{c}_2}{a{h}_2^2{k}^2-2+3n}+\frac{h_1^4{n}^2{c}_3}{a{h}_2^2{k}^2-1+n}+\frac{2h_1^3{n}^2{c}_4}{a{h}_2^2{k}^2-2+n}-\frac{2h_1{n}^2{c}_5}{a{h}_2^2{k}^{2}2+n}-\frac{n^2{c}_6}{h_2^2{k}^{2}a1+n}\\ +\frac{2h_1{n}^2}{h_2}+\frac{2h_1{n}^{2}w}{a{h}_2{k}^2}-\frac{6h_1^5{n}^2{c}_1}{h_2{k}^{2}a1-2n}+\frac{10h_1^4{n}^2{c}_2}{a{h}_2{k}^2-2+3n}+\frac{a{h}_1^3{n}^2{c}_3}{a{h}_2{k}^2-1\_n}+\frac{6h_1^2{n}^2{c}_4}{a{h}_2{k}^2-2+n}-\frac{2n^2{c}_5}{a{h}_2{k}^{2}2+n}V\\ +n^2+|\frac{n^{2}w}{a{k}^2}-\frac{15h_1^4{n}^2{c}_1}{k^{2}a1-2n}+\frac{20h_1^3{n}^2{c}_2}{a{k}^2-2+3n}+\frac{6h_1^2{n}^2{c}_3}{a{k}^2-1+n}+\frac{6h_1{n}^2{c}_4}{a{k}^2-2+n}{V}^2\\ +-\frac{20h_1^3{h}_2{n}^2{c}_1}{k^{2}a1-|2n}+\frac{20h_1^2{h}_2{n}^2{c}_2}{a{k}^2-2+3n}+\frac{4h_1{h}_2{n}^2{c}_3}{a{k}^2-1+n}+\frac{2h_2{n}^2{c}_4}{a{k}^2-2+n}{V}^3\\ +-\frac{15h_1^2{h}_2^2{n}^2{c}_1}{k^{2}a1-2n}+\frac{10h_1{h}_2{n}^2{c}_2}{a{k}^2-2+3n}+\frac{h_2^2{n}^2{c}_3}{a{k}^2-1+n}{V}^4\\ +-\frac{6h_1{h}_2^3{n}^4{c}_1}{k^{2}a1-2n}+\frac{2h_2^3{n}^2{c}_2}{a{k}^2-2+3n}{V}^5-\frac{h_2^4{n}^2{c}_1}{k^{2}a1-2n}{V}^6.\end{array}$$

  It is well known that equation (36) has many solutions mentioned in reference [7], we choose from them the following solution:$$\begin{array}{}\text{(37)}& Vz=\sqrt{\frac{2{\lambda }_6}{\sqrt{{\lambda }_8}\text{Cosh}2\sqrt{{\lambda }_6}z-{\lambda }_7}},\end{array}$$

  where$$\begin{array}{c}\text{(38)}& {\lambda }_6=\frac{14n^2{-1+2n}^3{c}_2^4}{81a{k}^2{2-3n}^4{c}_1^3}-\frac{{1-2n}^2{n}^2{c}_2^2{c}_3}{9a{k}^2{2-3n}^2-1+n{c}_1^2}-\frac{9{2-3n}^2{n}^2{c}_1^2{c}_6}{a{k}^2{1-2n}^{2}1+n{c}_2^2},{\lambda }_7=\frac{h_2^2{n}^2}{3a{k}^2}\frac{51-2n{c}_2^2}{{2-3n}^2{c}_1}+\frac{3c_3}{-1+n},\\ {\lambda }_8=\frac{h_2^4{n}^4}{81a^2{k}^4}\frac{169{1-2n}^2{c}_2^4}{{2-3n}^4{c}_1^2}+\frac{2341-2n{c}_2^2{c}_3}{{2-3n}^2-1+n{c}_1}+\frac{81c_3^2}{{-1+n}^2}+\frac{2916{2-3n}^2{c}_1^3{c}_6}{1+n{-1+2n}^3{c}_2^2}.\end{array}$$

  Solution (37) satisfies equation (36) when$$\begin{array}{c}\text{(39)}& w=-a{k}^2-\frac{61{-1+2n}^3{c}_2^4}{81{2-3n}^4{c}_1^3}+\frac{5{1-2n}^2{c}_2^2{c}_3}{9{2-3n}^2-1+n{c}_1^2}-\frac{9{2-3n}^2{c}_1^2{c}_6}{{1-2n}^{2}1+n{c}_2^2},h_1=\frac{1-2n{c}_2}{3-2+3n{c}_1},\\ c_4=-\frac{20{1-2n}^2-2+n{c}_2^3}{27{-2+3n}^3{c}_1^2}+\frac{2-2+n-1+2n{c}_2{c}_3}{3-1+n-2+3n{c}_1},\\ c_5=\frac{13{1-2n}^{4}2+n{c}_2^5}{243{-2+3n}^5{c}_1^4}-\frac{2+n{-1+2n}^3{c}_2^3{c}_3}{27-1+n{-2+3n}^3{c}_1^3}+\frac{32+n-2+3n{c}_1{c}_6}{-1+n+2n^2{c}_2},\end{array}$$

  where $h_2,c_1,c_2,c_3$, and $c_6$ are arbitrary constants. Substituting equations (35) and (37) into equation (32), we obtain$$\begin{array}{}\text{(40)}& yz={\frac{1-2n{c}_2}{3-2+3n{c}_1}+h_2\sqrt{\frac{2{\lambda }_6}{\sqrt{{\lambda }_8}\text{Cosh}2\sqrt{{\lambda }_6}z-{\lambda }_7}}}^{-1/n}.\end{array}$$

  Substituting equation (40) into equation (2), we obtain$$\begin{array}{}\text{(41)}& qx,t={\frac{1-2n{c}_2}{3-2+3n{c}_1}+h_2\sqrt{\frac{2{\lambda }_6}{\sqrt{{\lambda }_8}\text{Cosh}2\sqrt{{\lambda }_6}z-{\lambda }_7}}}^{-1/n}\exp ikx+-a{k}^2-\frac{61{-1+2n}^3{c}_2^4}{81{2-3n}^4{c}_1^3}+\frac{5{1-2n}^2{c}_2^2{c}_3}{9{2-3n}^2-1+n{c}_1^2}-\frac{9{2-3n}^2{c}_1^2{c}_6}{{1-2n}^{2}1+n{c}_2^2}t+\theta .\end{array}$$

  We can retrieve dark and bright soliton waves from solution (41) according to the values of arbitrary constants represented in this solution. Examples of these bright and dark soliton solutions are illustrated in Figures 7 and 8.

[figure(s) omitted; refer to PDF]

4. Conclusion

In this paper, the Lie point symmetry method has enabled us to obtain some new optical soliton solutions for the GKE in equation (1) with an arbitrary power nonlinearity. First, the GKE in equation (1) is transformed into the nonlinear ODE in equation (3) using the traveling wave transformation in equation (2). Then, the Lie point symmetry method is used to reduce the order of Equation (3) to obtain the first order ODE in equation (12). The first-order ODE in equation (12) is solved in some cases to retrieve some explicit forms of dark, bright, and kink soliton solutions for the GKE in equation (1). We mention here that some bright, dark, and singular soliton solutions are obtained in reference [18] at the strong condition $c_2=c_4=c_5=c_7=0$, whereas, in this paper, the new bright soliton solution in equation (21), kink soliton solution in equation (26), and dark soliton solution in equation (31) are obtained at the weak condition $c_1=c_2=0$. Also, the new bright and dark soliton solution in equation (41) is obtained at the weak condition $c_7=c_8=0.$

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