Soliton solutions of the Sasa–Satsuma equation in the monomode optical fibers including the beta-derivatives
Introduction
During the last decades, many efforts have been done to conduct a series of research works on soliton solutions of nonlinear partial differential (NLPD) equations and their generalized forms [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]]. Solitons are a class of exact solutions that play a key role in many branches of scientific disciplines. There are effective techniques to look for soliton solutions; some of the reliable methods that have attracted a lot of attention are new versions of Kudryashov and exponential methods [[22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]]. Recently, Hosseini et al. [24,25,[28], [29], [30], [31], [32], [33], [34]] demonstrated repeatedly the efficiency and capability of these methods to extract soliton solutions of NLPD equations and their generalized forms.
The principal purpose of this study is to consider the following nonlinear model termed as the Sasa–Satsuma equation including the beta-derivatives [[35], [36], [37], [38]]and retrieve its soliton solutions by using new versions of Kudryashov and exponential methods. In the model (1) that describes the propagation of short light pulses, and are the coefficients of the group velocity dispersion and the Kerr effect whereas and are the coefficients of the third-order dispersion and the self-steepening effect; signifies the coefficient of the stimulated Raman scattering. The Sasa–Satsuma equation has received newly a lot of interest owing to its application in the monomode optical fibers. For instance, optical solitons of the SS equation were obtained by Yıldırım [35] using the modified simple equation method. Yıldırım [36] also found optical soliton pulses of the SS equation by applying the trial equation method. Khater et al. considered the SS equation in [37] and obtained its dispersive optical soliton solutions by means of a new auxiliary equation method. Baleanu et al. [38] applied the generalized projective Riccati method to derive soliton solutions of the SS equation with the beta time derivative. Very recently, exact solitary wave solutions to the 2D-SS equation were obtained by Mvogo et al. in [39] using the Jacobi elliptic function method.
Considering NLPD equations with the beta-derivatives and investigating the dynamics of their soliton solutions have been the major concern of many studies. The beta derivative which is a generalization of the classical derivative is defined as [[40], [41], [42], [43], [44], [45], [46], [47], [48]]where includes the following interesting properties
It should be noted that the first property gives the relation of the beta and classical derivatives whereas the second property provides us with an easy way for converting a NLPD equation with the beta-derivative to a NLOD equation of integer-order.
The remainder of this article is as follows: In Section 2, new versions of Kudryashov and exponential methods are explained in detail. In Section 3, a traveling wave transformation is adopted to reduce the SS equation with the beta-derivatives to a nonlinear ODE of integer-order. In Section 4, soliton solutions of the SS equation including the beta-derivatives are derived using new versions of Kudryashov and exponential methods. A brief review of the results is presented in the last section.
Section snippets
New versions of Kudryashov and exponential methods
The main steps of new versions of Kudryashov and exponential methods are summarized formally in this section.
Reducing the Sasa–Satsuma equation with the beta-derivatives
In order to reduce the SS equation with the beta-derivatives to a nonlinear ODE of integer-order, we consider a traveling wave transformation as bellowwhere represents the speed of the wave whereas , , and indicate one-to-one frequency, wave number, and phase constant.
By replacing Eq. (5) in the SS equation with the beta-derivatives and distinguishing the real and imaginary components, one can get
Optical solitons of the Sasa–Satsuma equation with the beta-derivatives
New versions of Kudryashov and exponential methods are formally adopted in this section in order to get soliton solutions of the SS equation in the presence of the beta-derivatives.
Conclusion
A detailed study was conducted in the present work to investigate the dynamics of soliton solutions of a nonlinear model occurred in the monomode optical fibers. About this, soliton solutions of the governing model namely the Sasa–Satsuma equation describing the propagation of short light pulses were firstly extracted in the presence of the beta-derivatives; then, the dynamics of soliton solutions in the monomode optical fibers was analyzed for different values of the parameter . The soliton
Declaration of Competing Interest
No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.
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