The fractional soliton solutions of the dynamical system of equations for ion sound and Langmuir waves: a comparative analysis

In light of the ponderomotive force, this article focuses on establishing the exact wave structures of the ion sound system. It is the result of non-linear force and affects a charged particle oscillating in an inhomogeneous electromagnetic field. By using the Riemann-Liouville operator, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> -operator, and Atangana-Baleanu fractional analysis, the examined equation-which consists of the normalized electric field of the Langmuir oscillation and normalized density perturbation-is thoroughly examined. The solutions can be obtained with the help of a relatively new integration tool, the new extended direct algebraic method. We extract various wave structures in bright, dark, combo, ark, bright, and singular soliton solutions, among other forms, from soliton solutions. This method is simple and quick, and it can be used with more non-linear models or systems. Using the Mathematica software package, a graphically comparative analysis of some solutions is also presented here, taking appropriate parametric values into consideration.