Quantitative and qualitative analyses of the mKdV equation and modeling nonlinear waves in plasma

Abstract

In this paper, nonlinear electrostatic structures on the ion time scale in plasma consisting of two populations of electrons (cold and hot), positrons, and warm adiabatic ions are investigated. The multiple scale method is used to derive the modified Korteweg–de Vries equation (mKdVE). The Jacobi elliptic function expansion method (JEFEM) is employed to find some exact analytical solutions such as periodic, solitonic, and shock solutions. It is shown that the variation in the plasma parameters of interest, for our model, allows the existence of solitary and periodic structures and no shocks. It is also shown that the most important plasma parameters for the plasma model under consideration are positron concentration, α, and the percentage of cold and hot electrons, represented by the parameters μ and ν, respectively. Additionally, the qualitative behavior of the mKdVE is studied using dynamical system theory. The topological structure of the solution is discussed in the phase plane. In this work, the phase plane analysis, which is restricted to the discrete values of the parameter, is extended to the continuous range of the parameter using a bifurcation diagram. Bifurcation diagrams are drawn to forecast the behavior of the solution for specifically chosen essential plasma parameters. The analytical solution and the qualitative behavior of the solution presented in this paper are shown to be compatible with each other. The results presented here are general and can be gainfully employed to study a variety of nonlinear waves in space, laboratory plasmas, and astrophysical plasmas.