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 [1–5]. 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. (paper)