Hybrid Dispersive Optical Solitons in Nonlinear Cubic-Quintic-Septic Schrödinger Equation

Certain hybrid prototypes of dispersive optical solitons that we are looking for can correspond to new or future behaviors, observable or not, developed or will be developed by optical media that present the cubic-quintic-septic law coupled, with strong dispersions. The equation considered for this purpose is that of non-linear Schrödinger. The solutions are obtained using the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning’ functions. Some of the obtained solutions show that their existence is due only to the Kerr law nonlinearity presence. Graphical representations plotted have confirmed the hybrid and multi-form character of the obtained dispersive optical solitons. We believe that a good understanding of the hybrid dispersive optical solitons highlighted in the context of this work allows to grasp the physical description of systems whose dynamics are governed by nonlinear Schrödinger equation as studied in this work, allowing thereby a relevant improvement of complex problems encountered in particular in nonliear optaics and in optical fibers.


Introduction
Human beings, in search of well-being, face daily the multiple obstacles (or difficulties) imposed on them by the universe in its complexity. Therefore, a momentum of enormous progress in the advancement of essential knowledge and understanding of the natural phenomena born of these obstacles becomes essential. It is undoubtedly in this context that, for decades, several fields of research have emerged with the aim of providing adequate responses in order to preserve lives and improve living conditions. At the heart of these fields of science, in particular in physics, we can cite among others fluid mechanics, solid state physics, nonlinear optics, plasma physics, data transmission and so on. Since the world around us is intrinsically nonlinear, nonlinear partial differential equations (NLPDEs) appear to be the best adapted and are widely used to describe complex phenomena [1] [2] [3] [4] [5] in these different fields. Among these NLPDEs, those describing the dynamics of dispersive optical solitary wave attract our attention, in particular, the Schrödinger-Hirota equation and the Fokas-Lenells equation. Although these two models have been studied in depth [6]- [11] in the past, several behaviors developed by physical systems whose dynamics are described by these latter (models) still remain to be highlighted in the concern of possible improvement of the living conditions of citizens of the world, for example, the long range worldwide traffic of data (voice, images, etc.) and at very high speed through optical fibers. As part of the most exciting advances in nonlinear science and theoretical physics, one has attracted considerable the attention from many researchers around the globe, including the development of methods [12]- [28] to search for exact solutions of the nonlinear partial differential equations. But very few of these methods offer approximate solutions, or even forced solutions, because these exact solutions are not always easy to construct given that, such requires a deep understanding of mathematics. In this setting, one quickly realizes by visiting the literature that, in the field of nonlinear optics, the theory of optical solitons occupies a spot of choice and, in recent decades, a large amount of results have been highlighted. This mobilization of researchers to produce more results on this subject continues until now. However, in this dynamic, an attention must be more worn paid to the area of dispersive optical solitons because of their significance in the transmission of transoceanic data by optical fibers. These dispersive optical solitons inevitably appearing in systems whose dynamics are governed by nonlinear Schrödinger-Hirota and Fokas-Lenells type equations. An explicit analytical study of these solitons becomes a necessity in the concern of a possible detection of new behaviors which could be developed by propagation media of these dispersive optical solitons. It is on this track that this manuscript sign up, in order to enrich the literature with not only dispersive optical solitons, but also with novel prototypes of hybrid dispersive optical solitons.

Brief Presentation of the Mathematical Model
In the context of this work, our choice is carried on the nonlinear Schrödinger equation (NLSE) presenting the cubic-quintic-septic (CQS) law coupled with terms of dispersion of several orders, and which appears in the form [16] [17] [18] [19] [20] ( ) It is needful to note here that Equation (1)  This importance granted by theoretical physics and mathematical to this equation sure finds its explanation for the fact that currently, all optical communications used for transcontinental and transoceanic data transfer, are done through long distance optical fibers. In a context of highly dispersive solitons, by the time the group velocity dispersion (GVD) is low, an appearance of the terms IMD, 3OD, 4OD, 5OD and 6OD becomes essential to establish the balance between the non-linearity and dispersion, and thus ensuring the stability of the soliton. It follows for this purpose an improvement in performance during the propagation of solitons over long distances [10] [29]. In the following paragraph, we present the implementation of the BDKm which we will use in section 4 to construct hybrid dispersive optical soliton solutions of Equation (1).

The BDKm Theory Extended to the iB-Functions
Here, a brief presentation of the iB-functions will be followed by the implemen-

iB-Functions
These iB-functions [25] [30] [31] The member on the left is the implicit form and the member on the right is the explicit form of the function, where ( ) where α is a constant associated to the independant variable x. We associate here two of the fundamental properties of this function which will be useful in the rest of this manuscript in the respective forms It is important to note here that this function in its trigonometric form is written as For a better understanding of the properties of these functions, it is needful to refer to [25] [30] [31] where they are widely explained.

Implementation of the BDKm
The BDKm finds its implementation field in nonlinear physics, wave mechanics, is an unknown function to be determined, T is some function of φ and its derivatives with respect to x and t; and T includes the highest order derivatives and the nonlinear terms. Generally, the solution sought is of the form Equation (8) is an ordinary differential equation(ODE), where ′ Θ , ′′ Θ represent respectively the first and second derivatives of the envelope Θ with respect to ξ . According to Equation (3), the solution we are trying to construct can be expressed as where η is a real constant and ij λ are the unknown constants to be determined. So, the combination of Equations (9) and (8) gives the main equation where , , , i j k l are positive natural integers and , n m the real numbers [29] [34] [35]. It can be noted here that Equation (10) is the one from which all the possible analyzes result. The identification of coefficients λ η ν λ η ν λ η ν λ η ν at zero makes it possible to obtain the ranges of equations whose resolutions could allow to obtain the expressions of the unknown coefficients ij λ . Solving these series of equations can lead to exact, approximated or forced solutions [34] [35] [36] [37] depending on the models and the form of the considered ansatz. In the case of approximate or forced solutions, the priority in the order of resolution is given to those from the highest clues of

Results
We are exploited the BDKm to approach and unearth the new dispersive optical solitons that we qualify as hybrids due to the design and constitution of the ansatz (see Equation (13)) that we will use in this part of the work.

Analysis of the Range Equations
At first glance, it is easy to see that Equations (15) and (22), respectively, lead to  (15), (16) and (17) (20) and (21) due to their complexities require to By summarizing all this mainly to 3 It is very important to note here that this analysis gives rise to two families of non-trivial solutions to which we will focus all of our attention throughout the following paragraph.

Analytical Hybrid Dispersive Optical Solitons
In this subsection, we group the obtained solutions into two large families that In this case, the substitution of Equation (39) in Equation (37) Equation (33) and Equation (40) give, respectively, where d is given by Equation (50). One should note here that, Equations (48) and (52) (21), (18) and (15), and therefore their structure has suggested the choice of c d = condition.
2) Second subfamily of the first family of solutions; case: We note here that we face the same difficulties as in the case of Equation (44).
Which suppose that we will use the same approach for its resolution.
-First associated conditions 2: where a is given by Equation (55) where d is given by Equation (59). One should also note here that, Equations (57) and (61) are the second prototypes of hybrid dispersive optical solitons which we believe the last two terms resulting from equation (13) (21), (18) and (15), and therefore their structure has suggested this time the choice of c d = − condition.

Graphical Verification of the Hybrid Trait of the Obtained Solutions
This section getting charge to unveil the real nature of certain structures which are concealed behind the obtained analytical solutions and given by Equations

Discussions
From the previous study, it should be emphasized here that Equation (13) is a complex hybrid prototype of dispersive optical solitons which seems to harbor within it particular interactions between terms taken two by two of the same odd power 1;3 m = and contained in the different sub-packages (bright and dark mentioned above), and directly related to the cubic-quintic-septic law. These observations are reinforced by the appearance in the range Equations (15) to (29) of the expressions ( )   3  1  2  2  2  2  2  2  2  2  2  2  3  3  1  3  3  2  3  3  3  3  3  1  3  3   ;  ; ; ; ; , where we have imagined an emergence at an order j of the terms ( )     Following the previous scenery, we estimate that such results will have a very great advantage during the propagation tests in laboratories where the researchers will be able to choose the form of the signal which they will want to inject into the system which they study quite simply by operating an appropriate choice of the values of these coefficients , , , , a b c d α and ν of the wave. In addition to all that has just been said, we believe that some of these new prototypes of hybrid dispersive optical solitons may find their applications in nonlinear fiber optics when the refractive index of light is proportional to the intensity; others in neuroscience where optical solitons have been spotted [44] [45] [46] and in the fluid media, more precisely in the context of waves in deep waters, etc. And thereafter, these solutions may also make it possible to detect new behaviors in the propagation media in which the Cubic-Quintic-Septic law appears, and thus contributes to the progress of technologies of the information. At the end of the discussions, it should be noted that most of the obtained results are different from those proposed in [5] [47]. This difference can be observed at two levels: -first from the analytical point of view by its mathematical form given by Equation (13). It is a package [25] [30] [31] that contains within its bosom four terms of which the first and third terms are representative of the bright type soliton, the second is a kink while the fourth term is a double-kink [34].   Whereas, in [7] and [18], the authors proposed exact bright, dark solitons and the singular optical solitons while in [17], authors proposed the highly dispersive singular optical solitons.

Conclusion
Ultimately, it should be pointed out that, thanks to the BDKm theory, we were able to locate and unearth new hybrid prototypes of dispersive optical solitons as much on their mathematical forms as on their profiles. This method has revealed a certain affinity coupled (between certain terms of the ansatz (13) those of parameters ν and α of the initial wave given by Equation (13). Thus, as the form of the signal which one wants to obtain varies according to the values of the real constants , , , , a b c d ν and α , that supposes that one can control the energies of the nonlinear systems whose the dynamics are governed by the Schrödinger Equation (1) by simply playing on the values of these real constants.
These new hybrid prototypes of dispersive optical solitons translate the new behaviors that can be developed by systems whose dynamics are described by Equation (1). Since the transmission of data through optical fibers is the responsibility of optical solitons [ [29], we believe that the new hybrid prototypes of dispersive optical solitons that we have proposed in the framework of this work will be able to respond to the requirement in recent years, of information technology which is, the improvement of optical fiber transmission systems associated with the extraction of optical solitons. We believe that a good understanding of the hybrid dispersive optical solitons highlighted in the context of this work may also allow to grasp the physical description of systems whose dynamics are governed by Equation (1)