Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

The major goal of the present paper is to construct optical solitons of the Ginzburg–Landau equation including the parabolic nonlinearity. Such an ultimate goal is formally achieved with the aid of symbolic computation, a complex transformation, and Kudryashov and exponential methods. Several numerical simulations are given to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. To the best of the authors’ knowledge, the results reported in the current study, classified as bright and kink solitons, have a significant role in completing studies on the Ginzburg–Landau equation including the parabolic nonlinearity.


Introduction
Nonlinear partial differential (NLPD) equations have long been regarded as useful tools for describing a wide range of phenomena in a variety of scientific disciplines. As models for exploring real-world phenomena, NLPD equations play a fundamental role in the development of the contemporary world. Over the last several decades, one of the most significant challenges has been the development of new methods to construct exact solutions for NLPD equations. In recent years, several new, more powerful, and effective approaches have been established to retrieve exact solutions of NLPD equations, the sine-Gordon expansion method (Yan 1996;Yıldırım et al. 2021aYıldırım et al. , 2021bAkbar et al. 2021), the ( G � ∕G)-expansion method (Wang et al. 2008;Bekir 2008;Siddique et al. 2021;Bekir et al. 2021), the Sardar sub-equation method (Rezazadeh et al. 2020a(Rezazadeh et al. , 2020bAkinyemi 2021;Akinyemi et al. 2021a), the Kudryashov method (Kudryashov 2020a(Kudryashov , 2020b(Kudryashov , 2020cHosseini et al. 2021a), and the exponential method (He and Wu 2006;Ali and Hassan 2010;Hosseini et al. 2020a), are examples to mention.
As it is evident, the nonlinear Schrödinger equation is often used to simulate soliton dynamics in nonlinear optics. Many additional models, in contrast to the nonlinear Schrödinger equation, can be used as an alternative to such a classical model, for example, the Schrödinger-Hirota equation, the Chen-Lee-Liu equation, and many more. In the present study, the authors aim to conduct a study on the following Ginzburg-Landau equation including the parabolic nonlinearity (Biswas 2018;Arshed et al. 2019;Elboree 2020) and acquire its optical solitons using Kudryashov and exponential methods. In Eq. (1),u(x,t) indicates the wave profile, and x and t denote spatial and temporal coordinates, respectively. Besides, 1 is the GVD while 4 is the coefficient of nonlinear terms, 5 is the coefficient of detuning, and 2 and 3 relate to the parabolic nonlinearity. Optical solitons of the GL equation including the parabolic nonlinearity were derived by Biswas in (2018) with the help of the semi-inverse method. Arshed et al. (2019) employed the exponential method to obtain a series of optical soliton of the GL equation including the parabolic nonlinearity. Elboree (2020) used the exp(− ( )) method to report optical solitons of the GL equation including the parabolic nonlinearity. More works regarding the Ginzburg-Landau equation and its solitons can be found in Mirzazadeh et al. (2016);Rezazadeh 2018;Sulaiman et al. 2018;Osman et al. 2019;Hosseini et al. 2020b;Hosseini et al. 2021b;Ouahid et al. 2021).
Kudryashov and exponential methods have been designed as newly well-established methods to derive solitons of NLPD equations. In recent years, these methods have achieved much attention, especially from mathematicians and physicists. Akinyemi et al. (2021b) used the Kudryashov method to derive solitons of a Schrödinger equation involving spatio-temporal dispersions. Nisar et al. (2021) extracted solitons of a population equation with the beta-time derivative using the exponential method.
The structure of the present paper is as follows: In Sect. 2, a full description of Kudryashov and exponential methods are provided. In Sect. 3, the GL equation including the parabolic nonlinearity is reduced in a 1D regime using a complex transformation. In Sect. 4, Kudryashov and exponential methods are used to retrieve optical solitons of the GL equation including the parabolic nonlinearity. Furthermore, Sect. 4 presents several numerical simulations to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. The article is concluded in Sect. 5.

Kudryashov and exponential methods
The current section gives a full description of Kudryashov and exponential methods. The Kudryashov method recommends a series as follows as the solution of In series (2), a i , i = 0, 1, ..., N are derived later, N is found through the balance principle, and K(∈) is satisfying From Eqs. (2) and (3), we reach a consistent nonlinear system whose solution leads to solitons of Eq. (3).
Compared to the Kudryashov method, the exponential method seeks the following nontrivial solution as the solution of Eq. (3). In Eq. (4), the coefficients are acquired later and N ∈ ℕ.
As before, from Eqs. (4) and (3), we arrive at a consistent nonlinear system whose solution yields solitons of Eq. (3).

The model in its 1D regime
To reduce the governing model in a 1D regime, it is assumed that the model solution has the form where U(∈) indicates the shape of the pulse and and V 4 (∈) in Eq. (7), it is found that

The model and its solitons
In the current section, Kudryashov and exponential methods are applied to acquire optical solitons of the GL equation including the parabolic nonlinearity. Furthermore, the present section gives several numerical simulations to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons.

Employing the Kudryashov method
Based on Eqs.

Conclusion
In the present paper, the authors acquired optical solitons to the Ginzburg-Landau equation including the parabolic nonlinearity by employing the Kudryashov and exponential methods. As a result, a series of optical solitons, classified as bright and kink solitons, to the governing model was formally listed. Some numerical simulations were considered to examine the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. The current study's findings proved the superior performance of Kudryashov and exponential methods in dealing with the Ginzburg-Landau equation including the parabolic nonlinearity. It is worth mentioning that the authors' task for future works is adopting other well-designed methods Inc 2015, 2017;Inc et al. 2016;Tchier et al. 2016Tchier et al. , 2017 to seek new optical solitons of the Ginzburg-Landau equation including the parabolic nonlinearity.
Funding The authors have not disclosed any funding.

Conflict of interest
The authors declare no conflict of interest.