Construction of Breather soliton solutions of a modeled equation in a discrete nonlinear electrical line and the survey of modulationnal Instability

The most used signals nowadays for the propagation of information in the different transmission lines are solitons because of the simple fact that they are waves of steady state that maintain their forms, their velocity and resist best on dissipative factors [–]. Contrary to the other signals, a soliton has a mathematical analytic expression obtained from nonlinear partial differential equations of integrated physical systems and permits the easy access to information relative to the type of signal, to its velocity, to its wave vector and even the characteristics of the transmission line. In this article, we are using a nonlinear line made up of a sequence of identical discrete LC electrical networks to model a discrete nonlinear differential equation which govern the dynamics of Breather solitons in the line. We then construct some solitary wave solutions of type Dark, Bright, and the combined Bright and Dark solitons of that equation by using the direct and effective mathematical method of Bogning-Djeumen Tchaho-Kofane. A numerical simulation has permitted to draw and observe the different profiles of obtained real solitons and the different profiles of their intensity. We use the analytical expressions of each of those obtained Breather solitons and the technique of perturbing steady state solution to study their modulational instability. This has permitted to obtain information on the factors that perturbate these solitons in the course of their propagation in the electrical line notably the domain of stability or the domain of instability.


Introduction
The propagation of a soliton is due to interaction between nonlinearity and dispersion which necessitate that the transmission line must be a nonlinear and a dispersive medium [6][7][8][9][10][11][12]. This is why we consider in this study a discrete LC electrical line in other to find out how one can change the physical properties of capacitors notably their charge so that these capacitors become nonlinear components and permit the propagation of Breather solitons. The line obtained after the process is a discrete nonlinear capacitive electrical line; reason being that the only nonlinear components in the discrete line is capacitors [13]. The application of Kirchhoff laws to the circuit of the LC discrete nonlinear capacitive electrical line has permitted one to obtain the discrete nonlinear differential equation which govern the dynamics of solitary waves of type Breather in the line. Let us note that it is easy to make a theoretical description of solitons but mathematically, the research of the analytical expression from the modeled equation has not been an easy task [14,15] in spite of the existence of some mathematical techniques such as: Hirota's bilinear method, Painleve expansions, the inverse scattering transform, homogeneous balance method, F-expansion method, Jacobi elliptic function method, tanh-function method [4,7,[16][17][18][19][20][21][22][23][24]. However, the usage of the direct and effective Bogning-Djeumen Tchaho-Kofane methods [25][26][27][28][29][30] based on the identification of basic hyperbolic functions coefficients permits the construction of solitary wave solitons of type Breather of modeled discrete nonlinear differential equation. Solitary wave solutions has permitted one to draw their profiles and to deduce their natures. To have supplementary information concerning the stability domain or the instability domain of obtained solitons, it is necessary to study one of the factors as a modulational instability that perturbate these solitons during their propagation. The work we are presenting in this paper is divided as follows: in section two one will introduce the general presentation of Bogning-Djeumen Tchaho-Kofane methods, in section three of the work, we will present the discrete nonlinear capacitive electrical line for the study then, we model the discrete nonlinear differential equation which govern the dynamics of solitary waves of type Breather in the line. In section four, we use the Bogning-Djeumen Tchaho-Kofane method to construct solitary wave solutions of type Bright, Dark and combined Bright and Dark of the modeled equation then, we draw real profiles of those solitons and the profile of their intensity. In section five, we use the method of perturbation steady state solution to study modulational instability of obtained Bright and Dark solitons. In section six, we present the conclusion.

General presentation of Bogning-Djeumen Tchaho-Kofane method
Bogning, Djeumen and Kofane have developed an analytical method for obtaining solution of shape sech n in certain class of nonlinear partial differential equations. This method is focused on the construction of solitary wave solution and has been adopted to facilitate the resolution of certain type of nonlinear partial differential equations where the nonlinear terms and dispersive terms coexist. This method of construction of the solitary wave solutions intends to look for the solutions of certain categories of nonlinear partial differential equations on the form | | u the unknown to be determine and u 2 | | the magnitude of u. One looks for solution of equation (1) under the shape of a linear combination of the hyperbolic functions as follows Where α is a constant which depends on the system parameters which model the nonlinear partial differential equation and a ij , the constants to be determined. Thus the combination of equations (1) and (2) This equation presents five ranges of equations of coefficients a ij which are : the range of the coefficients F (a ij ) of power , ( ) the equations that best seeks the solutions are those raised to the must elevated powers. In the ranges of coefficients of cos h k (αx) and sin h (αx) cos h l (αx) priority is given to the equations of coefficients of low powers. The last range of equations of coefficients w (a ij ) is not very important because it is considered like a confused domain for the correct solutions obtainable. One can classify these equations of coefficients in order of decreasing priority F (a ij ), G (a ij ), H (a ij ), T (a ij ) and w (a ij ). Here the importance or priority makes reference to the range that permits to obtain good results or merely that which tends more to the exact value. While identifying the coefficients of equation (1) to zero, one gets the first range from most elevated. But care must be taken because it is not the most elevated power which gives the best solution directly; it depends on the shape of solution considered from the onset, the symmetry of the equation to solve as well as from its nonlinearity degree. In these conditions, one moves directly to the equations of lower powers until the good equation to solve is obtained. In the case where the first two set of equations (4) and (5) don't give a satisfactory solution, one moves to the set of equations of the following range In the set of equations (6) and (7), priority is given to the equations of low powers of cos h k (αx) and sin h (αx) cos h l (αx). In general, the first two set of equations (4) and (5) permit to find the solution of the problem. In the case where that they don't give satisfactory solution, it would be cautious to change the shape of the solution or merely the form of solution we want to construct or simply the method. The ranges w (a ij ) is considered now as the one that brings no reliable information. It is important to mention that this method appears complicated in the case where the properties of transformations of hyperbolic functions are not mastered. A mastery of these transformations reduces the difficulties considerably as regard to the calculations.

Modeling of discrete nonlinear partial differential equation
Recently, T T Guy and J R Bogning have studied a nonlinear capacitive electrical line in the continuum domain [35]. They have defined that line as an identical series of RC electrical networks where capacitors are nonlinear components, well defined in such a way that the line accepts to propagate certain types of solitary waves. In this work we are studying a discrete nonlinear capacitive electrical line made as an identical series of LC electrical networks.
Let's consider a discrete nonlinear capacitive electrical line that we present in figure 1. This line is a series of identical LC electrical networks where each is numbered by the positive integer n. L stands for the inductance, q (u n ) stands for the charge of capacitors connected to the network order n and varies in nonlinear manner in terms of voltage u n across that capacitor. i n stand for the current that flows through LC network order n. Applying Kirchhoff laws to the circuit figure 1 we obtain the following equations We define the nonlinear charge of the capacitors under the analytical shape as follows Where C 0 stand for the capacitance of capacitors in linear state, A 1 and A 2 are real numbers which stand for the nonlinear coefficients. A combination of equations (11) and (10) permits to obtain discrete nonlinear differential equation - In linear state A 1 =A 2 =0 and equation (12) becomes One can find the solution of equation (13) under the shape Where k p is the wave vector of sinusoidal wave with the weak amplitude B and w p its frequency. Equations (13) and (14) has permitted to obtain the relation between the frequency w p and the wave vector k p under the form The sinusoidal wave with weak amplitude propagates with group velocity v . By considering now the nonlinear state, sinusoidal wave with weak amplitude is perturbed and henceforth constitutes the carrier that propagates in discrete domain with the same frequency w p and the same wave vector k p but its amplitude B is modulated and becomes an envelope that propagates slowly than the carrier in time and space and can be studied in continuum domain. As such, considering the solution of the discrete nonlinear differential equation (12) under the form Where X=ε (n−v g t), T=ε 2 t are slow variables and ε is a real parameter less than one. The substitution of u n (t) given by (16) in (12) by retaining all the term order ε 2 and ε 3 proportional to exp (i(k p n−w p t)) permits one to obtain the equation that describes the dynamics of the envelope in the discrete nonlinear capacitive electrical line which is given by [31][32][33] i

Construction of solitary wave solution of type Breather of modeled equation and presentation of their profiles
In this section, we construct the different analytical expressions of Breather soliton which are exact solutions of differential equation (12) notably Bright soliton, Dark soliton and a combined Dark and Bright solitons. For this reason, we are finding out for each case envelope solution before deducing the analytical expression of Breather soliton susceptible to propagate in the line.

Construction of solitary wave solution of type Bright
Let us find out the envelope solution B (X, T) under the analytical form Where a, α e , v e , α p and v p are real numbers to be determined in terms of system parameters. The substitution of B (X, T) in (17) has permitted to obtain the equation Each basics hyperbolic function coefficient of (19) must be equal to zero. This permits to obtain the set of four equations as follows The solving of (20) permits to obtain the following results  Figure 2 shows effectively a real profile of a solitary wave of type Bright whose carrier is in rapid motion in an envelope of type Pulse that propagates very slowly.

Construction of solitary wave solution of type Dark
We find out the analytical expression of the envelope under the form Where a, α e , v e , α p and v p are real numbers to be determined in relation to discrete electrical line parameters. A combination of equations (23) and (17) permits one to obtain the following relation  (24) is verified if and only if each of its basic hyperbolic function coefficient is equal to zero. This permits to obtain a set of four equations given by The expression of B (X, T) given by (26) permits to deduce from equation (16) the analytical solution of equation (12) in the following manner The analytical expression of u n (t) given by (27) is that of Dark soliton which is an exact solution of the discrete nonlinear differential equation (12). Considering the values of the following parameters ε=0, 1,  )| This permits to obtain respectively in figure 3 the representation of real profile and intensity profile of that Dark soliton.
The diagram of figure 3 shows a real profile of Dark soliton with a carrier which propagates faster in its envelope of type Kink that trends it moderately.

Construction of solitary wave solution of combined Bright and Dark soliton
Let's find out an analytical expression of envelope B (X, T) under the form Where a, b, α e , v e , α p and v p are real numbers to be determined in relation to the parameters of the line. By substituting B (X, T) of (28) in (17) one obtains the following equations The expression of B (X, T) given by (31) permits to deduce from equation (16) the analytical solution of equation (12) in the following manner The expression of u n (t) given by (32) is a combined Dark and Bright soliton solution of discrete nonlinear differential equation (12).  Figure 4 shows a solitary wave of combined Bright and Dark solitons. The real profile shows a sinusoidal wave without the deformation of its amplitude and can be explained by the simple fact that the envelope of Bright soliton which is a Pulse perturbates the envelope of Dark soliton which is a Kink or inversely. This trends the envelope in an equilibrium state which becomes a constant where the amplitude of each Pulse soliton and Kink soliton are almost equal. This is why the carrier has a sinusoidal motion without the deformation of its amplitude.

Modulational instability of obtained Bright and Dark solitons
In this section, we use the Bogning-Djeumen Tchaho-Kofane method to study the modulational instability of Dark and Bright solitons susceptible of propagating in the discrete nonlinear electrical line. The application of this method is simply due to the fact that it is very effective for the perturbation of steady state solution and gives addition information on the stability or on the instability of the signal. The modulational instability starts from the perturbation of solitary wave taken in its steady state with the help of a wave with weak amplitude. This permits one to obtain perturbation equations after neglecting higher-order perturbation terms and the terms proportional to the solitary wave amplitude taken in its steady state. Perturbation equations permits to have a dispersion relation and by consequence information on the instability of the signal, which are obtained in the domain where the frequency is complex and causes an abnormal increase of soliton amplitude. Modulational Since the perturbation δ (X, T) is supposed to be of a very weak amplitude, then we find the solution of perturbation equation (34) under the shape of sinusoidal wave as

Conclusion
The aim of this study was to construct Breather soliton solutions of a modeled equation in a discrete nonlinear electrical line and the survey of modulational instability, we have used a Discrete nonlinear capacitive electrical line and Kirchhoff laws to model a discrete nonlinear differential equation which describes the movement of solitary waves of type Breather having an envelope whose dynamics is governed by the higher-order nonlinear Schrodinger equation. The application of Bogning-Djeumen Tchaho-Kofane method has permitted the construction of analytical expression of some exact solutions of the modeled discrete nonlinear differential equation which are Breather soliton of type Bright, Dark, combined Bright and Dark. A numerical simulation has permitted to represent, to observe real profiles and intensity profiles of those found solitary waves. To have addition information on the dynamics of those Breather solitons, the study of one of the factors that perturbate those signals in the course of their propagation as a modulational instability has permitted one to discover that those solitary waves notably Dark solitons and Bright solitons are generally modulationaly stable. It is necessary to recall that the application of results of this study by industries for the manufacturing of electrical line that accepts to propagate those solitons will have purely economic advantages for the fact that these electrical lines are relatively cheaper and easy to manufacture than other transmission lines. In addition, the stable and nondissipative nature of those solitons will reduce the usage of amplification station in the new electrical line. It is important to mention that the obtained Breather soliton which is a combined Bright and Dark solitons is very interesting as it has a particularity of having an envelope in an equilibrium state permitting it to maintain it constant value which it had in linear domain. This combined Bright and Dark solitons has a particularity of protecting sinusoidal wave that propagate as such without deformation and without the loss of energy.