Higher Order Solitary Wave Solutions of the Standard KdV Equations

Considered under their standard form, the fifth-order KdV equations are a sort of reading table on which new prototypes of higher order solitary waves residing there, have been uncovered and revealed to broad daylight. The mathematical tool that made it possible to explore and analyze this equation is the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning' functions. The analytical form of the solutions chosen in this manuscript is particular in the sense that it contains within its bosom, a package of solitary waves made up of three solitons, especially, the bright type soliton, the hybrid soliton and the dark type soliton which we estimate capable in their interactions of generating new hybrid or multi-form solitons. Existence conditions of the obtained solitons have been determined. It emerges that, these existence conditions of the chosen ansatz could open the way to other new varieties of fifth-order KdV equations including to which it will be one of the solutions. Some of the obtained solitons are exact solutions. Intense numerical simulations highlighted numerical stability and confirmed the hybrid character of the obtained solutions. These results will help to model new nonlinear wave phenomena, in plasma media and in fluid dynamics, especially, on the shallow water surface.


Introduction
Many natural phenomena regularly occur in the universe. These phenomena can in certain cases be destructive for the environment in which they take place. For example, the cases of the propagation of the nonlinear excitations in fluid dynamics which sometimes, are manifested by tsunamis, hurricanes, tidal bore, and so on, and which very often are devastating when surge is across on the continent without warning. These different phenomena as presented in this case (among so many others) constitute real and permanent threats to our existence on the earth planet. It is following this observation that, a great number of researchers in particular, physicists and mathematicians, deploy themselves in everyday life in order to design and make available mathematical models [1] [2] [3] [4] [5] in an attempt to analyze, understand and explain these phenomena that can be observed in various fields such as nonlinear optics, mechanics, biology, and so on. Most of these models are nonlinear partial differential equations (NPDEs). From all these equations, we are interested, in this manuscript, in the one whose history throughout the last two centuries is intimately linked from the outset to the remarkable scientific discovery made in August 1834 [6] by John Scott Russell and who shows up under the form Equation (1) is the standard form of the well-known fifth-order KdV (fKdV) equations [7]- [13] where , , α β γ are real and nonzero arbitrary parameters, olution in order to understand and explain phenomena which occur in systems whose dynamics are governed by these NLPDEs. Despite all this progress, a lot of work remains to be done because many other phenomena at the origin of the new predictable or unpredictable behaviors of these different systems still have to be detected, understood and explained in order to guarantee all of humanity a future safely.
So, in this manuscript, One tracks down, using the Bogning-Djeumen Tchaho-Kofané method (BDKm) extended to the new implicit Bogning' (iB) functions, new prototypes of solitary waves of the standard KdV equation while revealing the hybrid character of these waves. Section 2 will be devoted to the brief description of the BDKm including new implicit Bogning' functions (iB-functions) while, Section 3 will focus on finding the analytical solutions coupled with an intense numerical simulation. Section 4 will provide the gist of discussions. A conclusion followed by a perspective will complete this work in Section 5.

The BDKm Theory
This section takes care of highlighting all the mechanisms necessary for the implementation of the BDKm including iB-functions which will allow in section 3, to unearth new prototypes of the solitary wave solutions of Equation (1).

iB-Functions
These iB-functions [23] [27] [28] [29] have been highlighted thanks to the multiple research works [22] [23] [28]- [44] produced for more than a decade. It is during the repeated constructions of solitary wave solutions of certain types of equations in wave mechanics presenting dispersion terms coupled with non-linear terms (which can be of different orders) via the BDKm, that the fascinating properties of these functions were detected by Bogning Jean Roger.
These iB-functions are noted [23] [27] [28] [29] ( ) ( ) The left member is the implicit form and the right member is the explicit form of the iB-functions, where ( ) where m and n are keeping the same characteristics as in Equation (2), α is a constant associated to the independent variable x. We associate here some of the It is important to note here that this function in its trigonometric form is written as although we should not use it for this work, but for the knowledge of the reader.
For a better understanding of the properties of these functions, it is needful to refer to [21] [25] [26] [27] where they are widely explained.

Implementation of the BDKm
The BDKm which has been proposed by three Cameroonian researchers finds its implementation field in nonlinear physics, wave mechanics, mathematics physics, and others. It is better suited for solving certain types of NPDEs of the form [22] [23] [28]- [44] ( ) ( ) 2 2 , , , , is an unknown function to be determined, X is some function of Φ and its derivatives with respect to x and X includes the highest order derivatives and the nonlinear terms. Generally, the solution sought is of the form where ν is the speed of the wave and Equation (13) where η is a real constant and ij µ are the unknown constants to be determined. So, the combination of Equations (14) and (15) gives the main equation where , , , i j k l are positive natural integers and , n m the real numbers [23] [27] [28] [29]. It can be noted here that Equation (16)

Results
This part of the work is grouped into two subsections. The first subsection deals with the construction of the analytical hybrid solitary wave solutions of Equation (1). The second sub-section, for its part, is working on an intense numerical simulation in order to reassure itself of the stability of the obtained solutions with a view to a probable future application and for a possible confirmation of the hybrid characters (planned when choosing of the ansatz given by Equation (18) below) of these obtained solutions.

Analytical Higher Order Solitary Wave Solutions
By considering the change of variable ( ) ( ) , and ν the wave speed, Equation (1) becomes We are looking for the solutions under the form where a,b and c are real constants to be determined later, η , the inverse of the width of each component of Equation (18). It should be noted here that Equation (18) has a peculiarity in the sense that it is a collection of solitons taken individually in the two main families of existing solitary waves, namely the bright family and the dark family to make it a package [23] [27] [28] [29]. To be more precise, the first term of Equation (18) is a bright represented by the sech function, the second term is a hybrid soliton obtained by performing the product of a bright by a dark represented by sech tanh × and the third term is a dark represented by tanh 2 . One just has to represent each of the terms to realize it. In this context, we believe that this mixture can be at the origin of interactions (or competitions) between these different components and generate hybrid structures, this, in relation to the values taken by the coefficients , , a b c and the parameters , η ν of the wave, as well as those taken by the parameters , , α β γ of the considered system. Following all that has just been said, we continue with our investigations by inserting Equation (18) into Equation (17) 10 30 10 0, c ab η β η γ ηα 14 0. c c a ηα η γ η νη  (33) and (35) on the one hand, then, between Equations (34) and (35) The insertion of Equation (36) into one of Equations (33), (34) and (35) gives It emerges from Equation (37) that, the coefficient c is a linear function of parameter γ or that c is a hyperbolic function of parameter ( ) 0 α α ≠ or that c is a parabolic function of the inverse of the width at half height η of the solitons, that one is searches to constructing according to the fact that the choice of the variable be worn on one of the parameters , γ α or η . These observations sufficiently show what will be the importance of the impact of the variations of the coefficient c on the formation of the wave structures that we will obtain. By continuing, the insertion of Equation (37) into Equation (20) and Equation (26) gives, successively and 4 4 6 .
Equation (40) reveals that the speed ν has almost the same variations as in the case of Equation (37) with respect to the parameters , γ β and η . Taking into account Equations (37) and (38) in the Equation (22) gives, respectively Open Journal of Applied Sciences It is important to emphasize here that, the two conditions which validate Equations (39) and (42) [5] including her will be solution.  It should be said here that the solutions given by Equations (63) and (66) are exact solutions of Equation (1) of which one of the classes had been proposed in [12] through the tanh method. We can retain here that, for certain solutions constructed in this subsection by using the discriminant ∆ , only the cases 0 ∆ ≥ are interesting because complex solutions are not part of the aims of this work.

Numerical Simulations
This section is dedicated to the numerical simulations aiming to make observable, reliable and applicable the obtained solutions. They (numerical simulations) also aim to confirm the theoretical predictions on the hybrid characters of the new prototypes of the obtained solitary waves within the framework of this work. To implement all of this, and knowing that the boundary conditions of the profiles of the solutions constructed in this manuscript are not identical at the two borders. We have used the MATLAB toolbox pdepe [45] which solves initial-boundary value problems for parabolic-elliptic PDEs in 1-D, with zero flux boundary conditions. We have also used spatially extended grids in others to minimize boundary reflections that could induce spurious effects. It is also necessary to point out here that, these boundary conditions are appropriate to the profiles of the solutions studied in this work, instead of the periodic boundary conditions which require that when a wave passes from one end (of the computational spatial grid) to the other which is opposite to it, it should keep the same properties.
We cannot continue this study without providing the reader with elements useful for a good understanding of the different spatiotemporal evolution curves obtained. For example, to obtain Figure 1

Discussions
From all that has been formulated in the previous lines, two poles clearly emerge and on which our discussions will focus. The first pole will be on the analytical results, while, the second pole will revolve around results revealed by numerical simulations. Thus:  to develop varieties of the fifth-order KdV equations by simply modifying [7] the real values of the parameters , α β and γ . Note here that all these varieties obtained must admit the ansatz given by Equation (18)   4 β γ α α = − = and whose Equation (18) is an approximate solution.
-next, the solution given by Equation (52) whose solution given by Equation (18) is an approximate solution.
-eventually, a similar reasoning can also be carried out with the existence condition ( ) - Figure 2 shows the dominant interactions of each of the terms making up Equation (18). We can observe this by noting that Figure 2    could be of capital importance in the understanding and the explanation of certain phenomena which occur in systems whose dynamics are described by Equation (1).
-Finally, Figure 5 completes the advanced forms presented in Figure 4. This is how Figure 5 (1) in [13]. Again, Abdul-Majid Wazwaz had used in [12] the sine-cosine and the tanh methods to successively derive exact periodic and soliton solutions of Equation (1). Bell shape multi-solitons, dark solitons and dark periodic cusp solutions of Equation (1) were found in [10] [11] while, exponential function solutions were revealed in [9]. It becomes easy to realize that, all these proposed solutions of Equation (1) including solutions found in [19] [25], are almost entirely different from those obtained in this manuscript.

Conclusion
In the final analysis, we can say with enthusiasm that, the application of the  [16], during the transport of data through non-linear optical fibers and many other phenomena in quantum mechanics, in order to better secure our existence. One also hopes that, the properties of symmetry which certain obtained structures have, could be of capital importance in the understanding and the explanation of certain phenomena which occur in systems whose dynamics are described by Equation (1). It should also be pointed out that, the possibilities of formulating new varieties of fKdV-type equations offered by the results of this manuscript will help in the future, to devise new shallow fluid media with improved properties. However, the universe as a whole is being perpetually dynamic. We must further explore these models in order to obtain every day new