Sasa-Satsuma’s Dynamical Equation and Optical Solitary Wave Solutions

This work proposes the construction of a prototype of pulse-kink hybrid solitary waves with a strong Kink dosage of the Sasa-Satsuma equation which describes the dynamics of the wave propagating in an optical fiber where the stimulated Raman scattering effect is bethinking during modeling. The ulti-mate goal of this work is to propose a plateful of solutions likely to serve as signals during studies on computer or laboratory propagation studies. The resolution of such an equation is not always the easiest thing, and we used the Bogning-Djeumen Tchaho-Kofané method extended to the implicit functions of Bogning to obtain the results. The flexibility of the iB-functions made it possible to deduce the trigonometric solutions, from the obtained solitary wave solutions with a hyperbolic analytical sequence of the studied Sa-sa-Satsuma equation. A better appreciation of the nature of the solutions obtained is made through the profiles of some solutions obtained during the different analyses.


Introduction
Nowadays, nonlinear optics is a field that increasingly instigates the curiosity of many researchers around the world. This curiosity can be justified by the fact that many telecommunication industries use optical fibers more as a medium for transmitting or transporting large quantities of data over long distances, especially transoceanic, transcontinental and many other distances. This understanding can also be extended to the fact of the wide application of which optical solitons constitute the basic element of data transmission technology. It is undoubtedly for this reason that a good number of works encountered in the literature track down and reveal the optical solitons which coexist with mathematical models such as the nonlinear Schrödinger equation [1] [2] [3] [4], the Fokas-Lenells, Drinfeld-Sokolov-Wilson equations [5], and the generalized Sasa-Satsuma equation [6], to name just a few. Although we are particularly interested in optical solutions, many other relevant previous works have focused on models studied in hydrodynamics [7] [8] and many others. It turns out that these works for the most part only offer exact solutions of the single soliton type and periodic solutions [9]. Thereby, it emerges that very few of the previous works, or almost none, offer solutions of the exact, approximate or forced multiple optical solitary waves type [10] [11] [12], and this comprises hyperbolic functions. This is a problem if we stick to the fact that a new solution of a nonlinear partial differential equation is a new behavior likely to be developed by physical systems whose dynamics are described by the considered mathematical model. As corollaries, we will miss the behaviors which make it possible to analyze and explain new phenomena that occur in physical systems described by the studied Sasa-Satsuma equation, in particular certain propagations and interactions regimes of robust waves of multi-soliton types in nonlinear optical fibers. It is in response to these shortcomings that this work fits and chooses the dynamical model of Sasa-Satsuma.
The aim of this work is to construct new prototypes of optical solitary waves of the Sasa-Satsuma dynamical equation in order to further enrich the literature with new varieties of more robust solitary wave solutions. And at the same time, allow new investigations in laboratories during the propagation tests which will lead to the understanding of new phenomena that occur in the studied model. Owing to all of this, our manuscript is organized as follows: Section 2 gives a brief description of the chosen model while; Section 3 explains the outline of the Bogning-Djeumen Tchaho-Kofané method (BDKm) [13]- [18] used; Section 4 in its content delivers the results obtained; the discussions are carried out in Section 5; a conclusion coupled with perspectives is recorded in Section 6.

Chosen Model
The Sasa-Satsuma equation which is the chosen model is written as being [19]- [23]  0.
The first term describes the temporal evolution of the optical soliton molecules the optical soliton profile, the factor of the imaginary i through the coefficients , , β θ α sequentially provides the self-steepening, stimulated Raman scattering in additionally third-order dispersion. Two mathematical techniques namely, improved F-expansion and improved auxiliary methods was used in [19] to construct several types of solitons such as dark soliton, bright soliton, periodic soliton, elliptic function and solitary waves solutions of Equation (1), while in [22], it was used to introduce and discussed the Sasa-Satsuma model in berefringent fibers without of four-wave mixing terms (FWM). Equation (1) is used to describe the propagation of femtosecond pulses in optical fibers as well as the propagation and interaction of the ultrashort pulses in the sub-picosecond or femtosecond regime. Let us take a look what it is about the used method.

Used Method
The  [36] and used within the framework of this work applies to some partial differential equation types in which coexist the nonlinear terms and the dispersive terms (and others) under the form: is an unknown function to be determined, X is some function of Φ and its derivatives with respect to , , , x y z t and X includes the highest order derivatives and the nonlinear terms. Most often, we use the change of variables In the case where Φ is a function of , , x y z and t, ξ becomes x y z t ξ ν = + + − , where ν is the wave speed. In this context, Equation (2) gives rise to the ordinary differential equation (ODE) below: where , ′ ′′ Ω Ω represent respectively the first and second derivatives of the envelope Ω with respect to ξ . Then, the solution we are looking for can be  , m and n are powers of both terms of Equation (5). For more details, see [34] [35] [36]. So, the combination of Equations (3) and (4) gives rise to the main equation of ranges  (6) where , , , i j k l are positive natural integers and , n m the real numbers [30] [31] [34]. It can be noted here that Equation (16)  Here, the priority makes reference to the serie that permits to obtain good results or merely that tends more to the sought exact solution. Very often, the series of equations obtained by identify at zero the coefficient of ( )

Results
Now, we address the resolution of Equation (1) by applying the BDKm with certain transformations specific to traveling waves.

Analytical Optical Solitary Wave Solutions
The traveling wave solutions that we seek to construct can be considered in the form below x t ξ ν = − ; ω is the angular frequence; k the wave number and ν the wave speed. Thus, the insertion of Equation (7) into Equation (1) yields the following equation J ηξ are hyperbolic iB-functions; , , a b c are constant real coefficients, η the inverse of the width at half height of each of the solitons contained in Equation (9), and i an imaginary such that Equation (9) is a complex multi-soliton whose basic form is represented by the first two terms of coefficient a and b, respectively. This basic shape is disturbed by the addition of a hybrid soliton of amplitude c. We believe that it is this disturbance term that is at the origin of the emergence of new hybrid structures in the propagation media. In this context, inserting of Equation (9)  to best approach the solutions given by Equation (9) in the case of this Family I.
The cape being fixed, one obtains from the identification at zero of the ranges of equations offered by the coefficients l P and j R : -From the real part the term in ( ) the term in ( ) -From the imaginary part the term in ( ) From the structuring of the above equations, it appears from Equations (11) and (17) (12), (17), (19) and (21)  , which means in this case that, the contribution of this equation to this order of power of the constitutive hyperbolic functions of the ansatz given by Equation (9) is not significant, and therefore negligible. Continuing our analysis, Equations (14) and (16) Given Equation (23), the successive combinations of Equations (22), (24) and (25) provides the two respective constraints  and (17) whose are constraints imposed by the studied system.) and the constraints given by Equations (27) and (28) respectively. It should be emphasized here that the third term of Equation (29) is a hybrid soliton of the bright-dark or dark-bright type. By carefully observing this Equation (29), we realize that it is a complex multi-soliton prototype whose third term is at the origin of the appearance in the propagation medium of multiform solitary wave structures, this in relation with the values taken by the parameters of the wave. This can be justified in the sense that the magnitude of the amplitude Ψ stages the squares of the first and the second term, which, in the absence of the third term, generates either a bright soliton or a classic dark soliton. It is necessary to note in this case that, the dynamics of the amplitude given by the Equation (9) is only described by the traveling wave equation provided by the imaginary part of the Equation (8) and which is under the form: Equation (30) compared with Equation (8) suggests that the dynamics of the amplitude given by Equation (29) A thorough observation of the two ranges of equations above shows that, apart from a factor, Equations (32) and (36) are identical. The same is true for Equations (33) and (37). Under this observation including the structuring of the rest of the above equations, and taking into account the fact that we are looking for the non-trivial solutions of Equation (1) (34) and (35) gives, respectively ( ) ( ) On the other side, the combination of Equations (38) and (39) offers the constraint where 0 β ≠ . So, we obtain Subfamily I of Family II of the solutions of Equation (1) and which is represented by the exact solution below with constraints obtained for the cause. Equation (44) is also a kind of complex multi-soliton, which due to the squares existing in the modulus of the amplitude Ψ , makes appear in the propagation medium either a bright-soliton, or a dark-soliton, and this according to the values taken by the wave parameters. This gives freedom of choice of the structure that one would like to use during laboratory propagation tests.
 . The solution given Equation (52) shows that the Sasa-Satsuma dynamical equation admits a bright-soliton as an exact pure real solution.

Trigonometric Solutions
The transition from hyperbolic forms to trigonometric forms of iB-functions is done by means of the relation [34] [35] [36] wave solutions given by Equations (29) and (44), respectively. The graphical tool which made it possible to achieve this result is MAPLE. Thus, we have, respectively. Note here that, the choice of values is linked to constraints given by the obtained analytical expressions. At this level, several values of variables can be defined. This is for example, the case of Figure 1

Discussions
In this section, it is important to note that the obtained analytical or graphical results corroborate with the theoretical predictions about the multi-soliton characters which consist of the proposed ansatz, and this with a more or less good accuracy. An illustration of this corroboration may be observed through: -the different profiles of Figure 1 where Figure 1(a) reveals a bright soliton while Figure 1(b) presents a dark soliton. This is a consequence of the fact that the plotted module involves the sum of two squares, one of which is a bright soliton and the other a dark solid. Thus, depending on the values assigned to each of the coefficients , a b of the wave given by Equation (44), one or the other structure is obtained.
-the different profiles of Figure 2 where Figure 2(a) displays a bright-dark soliton structure while Figure 2(b) shows a dark-bright soliton structure.
These two structures are all hybrids and have equal bright and dark tendencies respectively. This is the direct result of the fact that the analytic form given by Equation (29)   the third term of Equation (29)), are equals tendencies of bright-dark or dark-bright soliton. On the other hand, Figure 2(c) and Figure 2(d) display two bright-dark soliton structures with a strong bright soliton tendency. This is also due to the different values taken by the wave parameters.
In summary, we note that the values assigned to each of the parameters of the wave in particular, then extended to the parameters of the system studied in general, are fundamental in the formation of the resulting structure. In addition, Figure 1 and Figure 2 confirm one of the a priori ideas that loum during the conception of the ansatz given by Equation (9) and which reported that the disturbance term of amplitude c is at the origin of the emergence of the new hybrid structures displayed by Figure 2(c) and Figure 2(d). It should also be noted here that the results obtained are new and different from those proposed in [19] [20] [21] [22] [23], at least in the mathematical form, and through the displayed profiles. This being the case, one conclusion is in order.

Conclusion
In our previous work [20], we constructed the solitary wave multi-solutions of the modified Sasa-Satsuma equation describing the dynamics of sea waves. The method used for this purpose was the BDKm and it had shown its full effectiveness. The transformations associated with the method in its initial form were very cumbersome and required a lot of care in their management when we came to the idea of constructing the solitary wave solutions of the Sasa-Satsuma equation. But, this time, with the one that describes the wave dynamics in optical systems and in particular the optical fiber having the particularity of taking into account the Self-Steeping effect, the third-order dispersion and especially the stimulated Raman scattering effect, we opted to use the BDK method extended to iB-functions. The idea of the method to be adopted has been decided upon. We