Multi-form solitary wave solutions of the KdV-Burgers-Kuramoto equation

This work is dedicated to the construction of solitary wave solutions of the KdV-Burgers-Kuramoto equation. The peculiarity of the solutions obtained for this purpose is that they result from the combination of solitary waves of the bright and dark type thus generating multi-form solutions which are also called hybrid solitary waves. The Bogning-Djeumen Tchaho-Kofané method is used to obtain the results. The reliability and feasibility of these results are tested using numerical simulations.


Introduction
The universe in its complex formation is constituted by physical systems whose deployments generate nonlinear and exciting phenomena which arouse in a growing way the curiosity and the desire of comprehension by human beings. Thus, the human being in general and the physicist in particular model the dynamics of nonlinear phenomena by mathematical equations of all outputs, among which are the nonlinear differential equations. They vary most often according to the physical system studied [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. If one thing is to get these equations, in order to analyze and understand the dynamics of these physical systems, another thing is to solve them and get solutions that are closer to reality. But the general observation is that nonlinear partial differential equations (NPDEs) are for the most difficult to integrate. This justifies the proliferation of techniques used by researchers to provide solutions to these different equations and the multiplicity of solutions sometimes resulting from the resolution of a single NPDE. It is in this perspective that physicists and mathematicians in recent years are multiplying the approaches and methods to study analytically and numerically these equations. This last decade and very recently new methods and approaches have been developed [16][17][18][19][20][21][22][23][24][25][26][27][28]. This approach will not stop because of many phenomena whose dynamics still escape our understanding. In this work, we subscribe to this approach to build multiple solitary wave solutions of the KdV-Burgers-Kuramoto equation (KBK) given by where u xxxx is the stability and energy dissipation term, u xxx represents the dispersion term, u xx accounts for the instability and energy production term, uu x the dominant nonlinear term and u x t , ( ) a function characterizing some physical processes in unstable systems [4], with b a parameter which measures the relative importance of dispersion, , a g are real constants [1], x the coordinate, t is the time. It can be used to describe unstable draft waves in plasma [4], long waves in a viscous fluid flowing down along an inclined plane and turbulent cascade model in a barotropic atmosphere. For β=0, equation (1) is referred too as the Kuramoto-Sivashinsky equation (KSE), which is a canonical nonlinear evolution equation arising in a variety of physical contexts [15].
The solutions of KBK equation possess their actual physical applications. The peculiarity of the solutions sought in this work comes from the fact that they result from the combination of solitary wave packages [29,30].
To achieve our goal, the work that we propose in this article is organized as follows: in section 2, we return to the Bogning-Djeumen Tchaho-Kofané method that will be used to determine the solutions of the KBK equation. Section 3 proposes the different analytical solutions coupled to numerical simulations in order to verify the reliability and practical feasibility of these analytically results. Section 4 is devoted to discussions on the results obtained. Finally we end the work with a conclusion.

The Bogning-Djeumen Tchaho-Kofané method (BDKm)
It is a method developed on the theory of construction of nonlinear partial differential equation solutions [31][32][33][34][35][36][37][38][39][40][41][42]. This method is interested particularly the NPDEs of the form [31][32][33][34][35][36][37][38][39][40][41][42] H u u u u u u u u u u , , , , , where u x t , ( ) is an unknown function to be determined, H is some function of u and its derivatives with respect to x and H includes the highest order derivatives and the nonlinear terms Under the travelling wave transformation u x t , x t x n =where n is the wave speed, equation (2) is transformed to the following ordinary differential equation (ODE)   H , , , , , y y y y y where , y y ¢  represent respectively the first and second derivatives of the envelope ψ with respect to ξ. We construct the solutions of equation ( where α is a real constant and b ij are the unknown constants to be determined. Thus, inserting equation (4) in to equation (3) permits to have the main equation under the form å a n ax a n ax ax a n ax a n ax ax a n where i j k l , , , are positive natural integers and n m , the real numbers [29,30]. Through equation (5), we obtain, the coefficients b , , ij a n and some constraints that may result, by identify at zero the functions F b G b , , , , , , ij ij a n a n ( ij a n a n a n ( ) ( ) ( ) . Thus, the ansatz given in equation (4) can be supported by equation (2) where , , y y y ¢  ¢¢¢ and y⁗ denote first, second, third and fourth order partial derivatives of ψ with respect to ξ, respectively. We focus on the case 0 gba ¹ in equation (1). Based on the BDKm largely explained in [31][32][33][34][35][36][37][38][39][40][41][42], equation (6) may have the following ansatz(which can be supported by equation (1) The transformations of equations (8) and (9) clearly show that equation (8) is constituted of a bright wave package and equation (9) is constituted of a package solitary waves of dark type. Thus, the insertion of equation (7) into equation (6) leads to the coefficient range equation a, b and c whose identification of the coefficients of terms in sech j qx j 2; 4; 6; ...; 14 = ( ) and sinh sech i k qx qx (i=1 and k 3; 5; 7; ...; 15 = ) at zero leads to series of following equations the term in sech 14 qx, . the term in sech 12 qx, bc ab . the term in sech 10 qx, . the term in sech 6 qx, c  5  260  76  3496  796  16  6 29 0, 14 . the term in sech 4 qx, . the term in sech 2 qx , . the term in sinh sech 15 qx qx, . the term in sinh sech 13  q g q a q q a q g q . the term in sinh sech 5  q g q a q q g q a -- . the term in sinh sech 3 qx qx, from which expansion coefficients a b , and c can be determined under certain conditions satisfied by parameters α, β, θ and γ. One notices that, from equation (10) 3.1.1. First family; case: b ¹ 0, and c ¹ 0 From equations (11), (12) and (13) where a and b are given by equations (25) and (27), respectively. Here, it is important to emphasize that equation (28)  y q x q x q g q g q qx qx where a is given by equation (33) which takes into account equation (36). One can notices that in the case of this third family, it is a question of multiform analytical solutions induced by the combination of two sub-families of hybrid solutions of dark and dark types respectively which, according to the values of parameters a, c , α, θ and γ in turn generates the multi-form solutions of the bright-dark type or another hybrid solitary wave solution (see figure 4). q a q g - From equation (42), it is easy that α, β and θ satisfy the following condition  (40) and (41), we obtain the condition satisfy by ν, β, θ and γ as follows So the fourth family of solutions of equation (1)   It should be noted that all families of analytical solutions obtained in this section are different from those found in [11,17,18,[24][25][26][27][28]34]

Numerical simulations
Now, we test the feasibility, reliability and even robustness of some of the solutions obtained in this work through intense numerical simulations that will allow their observations during laboratory propagation tests. Thus, they may be useful in detecting so many new phenomena that are simultaneously involved in nonlinearity, dissipation, dispersion and instability. To achieve this, we used the MATLAB toolbox pdepe [43] which  solves initial-boundary value problems for parabolic-elliptic PDEs in 1-D with zero flux boundary conditions . We also used large enough spatial grids in other to limit reflections at the boundaries which could bring spurious effects. These boundary conditions well suit the profiles of the solutions investigated in this work as there are not identical at the two boundaries, instead of the well known periodic boundary conditions which imply that when a wave passes through one end of the computational spatial grid it reappears on the opposite end with the same properties.
It is worth mentioning that, all the curves obtained in this section were from the same approach. That is to say, one fixes some parameters of the system modeled by equation (1) and those of the wave given by the ansatz (7) and one deduces the others. As an example, in figure 1 Top row (Left), when fixed the parameters b ; c ; θ; γ; β to   (25), (26), (27) respectively. These profiles are different from those obtained in [17,18,[24][25][26][27][28]34].

Discussions
The numerical study carried out allowed to obtain profiles of the multi-form solitary wave solutions or hybrid solitary wave solutions resulting from the combination of solitary wave packages [29,30] of the dark solitary wave family and the bright solitary wave family. The concept package is used here because on the one hand the second term of equation (7) (sech tanh 2 4 qx qx) can be decomposed into a sum of analytic sequences specific to bright solitary waves and on the other hand the third term (sech tanh 2 5 qx qx) can also be broken down into a sum of analytic sequences specific to solitary waves of the dark type. We also use the notion of solitary wave family because in a common way, the sech analytic sequence is known as that which indicates the fundamental solitary wave of bright type and the analytical sequence tanh is known as that which indicates the fundamental solitary wave of dark type. In contrast to the previous work and the different representations of solitary wave profiles obtained in the past [17,18,[24][25][26][27][28]34], the analytical sequence sech tanh 2 4 qx qx is that of a family solitary waves of bright type or of the large family n sech 0 n qx  ( )and therefore the representative is sechqx and the analytic representation sech tanh 2 5 qx qx that of a dark type solitary wave or the great family n sinh sech 0 n qx qx  ( )and therefore the representative is tanh qx . Figures 1 and 2 show the hybrid solitary wave profiles resulting from the combination of the solitary wave packets of the dark family and the bright family, the analytical sequence of which is given by equation (28). One notices that, the solutions obtained in the bottom row of figure 1 are symmetric with respect to the plane x=0. In contrast, figure 3 shows the hybrid character of one of these families represented by the first term of equation (7) (sech tanh 4 qx qx). Further on, figure 4 presents hybrid solitary wave profiles derived from the combination of solitary wave packets of the dark and bright family whose analytic sequence is given by equation (37). One also notices that, by setting γ=−1, then derived from equations (32) and (34) α=−0.016; c=0.001 2 , other parameters as in figure 4 Top row (Right), one obtains the same profile as in figure 3, (Right), which we did not consider necessary to plot again. When focusses on figure 4, it is easy to observe that γ increases 17; 1.75; 8; 30 g Î -( { } ) with the structure formation of a multiform solitary wave solution from Bottom row (Right) to Top row (Left). Thus, this observation allows to choose in a simple manner the solution profile (already obtained in the above figures) by acting on the γ value. So, it would be useful and would also save time in the laboratory during the study of propagation test. One hopes that hybrid solitary wave may in general found its application field in telecommunication technologies and will further revolutionize the energy and information transport.

Conclusion
In this work, we have investigated approximate 'New' multi-form solitary wave solutions of the KdV-Burgers-Kuramoto equation. By using the straightforward Bogning-Djeumen Tchaho-Kofané method based on the construction of solitary wave solutions of certain types of Nonlinear Partial Differential equations(which are an appropriate models to describe many phenomena which are simultaneously involved in non-linearity, dissipation, dispersion and instability(self-excitation), especially at the description of turbulence process), we have proposed multi-form solitary wave solutions that consist of a combination of hybrid solitary waves called bright-dark solitary wave profiles. One reliable feature of the solutions proposed here is the possibility to alter the amplitude of each individual hybrid solitary wave or the possibility to alter the value of γ allowing to generate 'new' multi-form structures. The analytical predictions are confirmed by the numerical integrations of equation (1). By further numerical simulations, we have also shown that different robust multi-form structures might be obtained when one inserts as initial conditions the combined hybrid solitary waves proposed in equation (7) with relative values of amplitudes a; b and c . Some solutions expressed as travelling waves (solitary waves, periodic waves and so on) of the KBK equation have been found in [11,17,18,[24][25][26][27][28]34] which are different from those obtained in the present work. According to the obtained profiles in figure 4, one could numerically track as a function of γ the bifurcations of the solutions. So that our result may be useful in detecting many new phenomena that are simultaneously involved in non-linearity, dissipation, dispersion and instability. However, many issues suggest that there are the dynamic properties of the analytical solutions which remain to be unveiled in details such as the bifurcation theory for dynamical systems may be used to precise exact explicit parametric representations of the dynamical solutions [28]. In the same vein, further development and analysis should be used to unveiled many awaiting issues.