Comprehensive Analytical Investigation of Optical and Plasma Soliton Solutions for Nonlinear Evolution Equations via the Extended Khatir Method with Stability and Dynamical Behavior Analysis

Abstract

The general equations of nonlinear evolution play the basic role in modeling the wave propagation phenomena in optical fibers and plasma media, where the delicate fusion of dispersion and nonlinearity means that it produces a stable organized localized structure called soliton. In this study research a rather comprehensive analytical work of optical and plasma soliton solutions is carried out using Extended Khatir Method. The nonlinear Schrodinger equation and modified Korteweg- de Vries equation are made as typical representatives of the models that represent the dynamics of the pulses in nonlinear optical fibers and ion acoustic waves in plasma settings, respectively. With the partial differential control equations, in place of enriched expansion and powered with positive and negative values of an auxiliary control, through traveling wave transformation the partial differential governing equations are dramatically simplified to nonlinear ordinary differential equations which are then resolved by enriched expansion using either positive or negative values and powers of the auxiliary control. The resultant proposed structure results in a wide range of precise solutions, including light and dark, periodic and breather type solitons. Deduced are explicit parametric conditions under which physically meaningful localized structures are present, that the dispersion and non-linear coefficients play a significant role in the control of the amplitude and width of the wave. Physical viability, in linear stability analysis, in the analysis of eigenvalue spectrum, systematically, energies are applied. The findings validate the fact that the bright and dark solitons are recoverable in the proper dispersion - nonlinearity regimes as opposed to periodic and breather solutions which exhibit conditional stability based on parameters of the system. Moreover, phase-plane and bifurcation studies provide a geometrical description of the solutions observed and it can be demonstrated that localized waves are related to homoclinic and heteroclinic orbits of the model dynamical system. The findings form the Extended Khatir Methodology as a coherent and dynamically compatible analytical scheme whose solutions may provide structurally rich and physically robust soliton solutions of nonlinear dispersive systems. The duality between the construction of the exact solution and stability analysis and the dynamics analysis contribute to the enhancement of the theoretical rigor and the practical interest in the subject area of the nonlinear optics and the plasma physics.