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

. 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 diferential equation (ODE). Using Lie point symmetry analysis, the nonlinear second-order ODE is reduced to a frst-order ODE. Tis frst-order ODE is solved in two cases to retrieve some new bright, dark, and kink soliton solutions of the GKE. Tese soliton solutions for the GKE are obtained here for the frst time.

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][32][33][34][35]. Details of this method can be found in reference [31]. Tis method will be used in this paper to fnd some new optical soliton solutions for equation (1).
Tis paper is organized as follows: In Section 2, equation (1) will be transformed into a second ODE using a traveling wave transformation. Ten, this second-order ODE will be reduced to a frst-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.

Lie Point Symmetry Method
For solving equation (1), we use the traveling wave transformation as follows: (2) where y(z) is the amplitude of the soliton, k and μ are constants, ω represents the frequency of soliton, k represents the wave number, and θ 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: and the imaginary part gives Here, we use Lie symmetry analysis [31][32][33][34][35] to reduce equation (3) to a frst-order ODE with well-known solutions. Te autonomous ODE in equation (3) admits the Lie symmetry generator [31].
Using equation (6) and (9), equation (11) becomes Equation (12) has many solutions, but we are interested in soliton solutions only which will appear in the next section.

Soliton Solutions
(i) In order to obtain the soliton solutions, assume that Substituting equation (13) into equation (12), we obtain Let Assume that Hence, equation (15) can be reduced to the following: Journal of Mathematics      Journal of Mathematics 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). Te bright soliton solution.
(31) 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: 8

Journal of Mathematics
Substituting equation (32) into equation (12), we obtain Assume that Hence, equation (34) can be reduced to the following: It is well known that equation (36) has many solutions mentioned in reference [7], we choose from them the following solution: where Journal of Mathematics (38) Solution (37) satisfes equation (36) when 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 Substituting equation (40) into equation (2), we obtain

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). Ten, the Lie point symmetry method is used to reduce the order of Equation (3) to obtain the frst order ODE in equation (12). Te frst-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.

Data Availability
No data were used in this article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.