Analytical investigation and in-depth analysis of the new concatenated derivative nonlinear Schr $\ddot{o}$dinger equation in plasma physics

Recently, Zayed et al. ^[4] unified the three well-known derivative nonlinear Schr $\ddot{o}$dinger equations (DNLSEs) into a single concatenated DNLSE that preserves the respective properties of the individual models while introducing additional complexities and flexibilities relevant to plasma waves. This new model was thoroughly examined in the present manuscript, with a focus on unraveling some of its salient properties. The study was therefore directed toward constructing various solitonic and periodic solutions using two promising analytical methods, analyzing the resulting linearized dispersion relation, examining the possibility of modulation instability, and, lastly, analyzing the bifurcation dynamics in the posed coupled nonlinear dynamical system. In addition, based on the aforementioned analyses and appropriate numerical simulations, this study reported that the complex-valued wave profile of the new model is strongly influenced by temporal variations and by parameters arising from the adopted analytical methods. Additionally, the linearized dispersion relation has been noted to mainly disturb by the linearization parameter and the coefficient of the group-velocity dispersion. Moreover, the discovered non-singular dynamical system revealed convergent periodic and heart-like limit cycles, while firmly remaining responsive to the initial conditions change. Indeed, across the established results from the constructed soliton solutions, dispersion relation analysis, modulation instability, and bifurcation analysis, the take-home remained the attainment of optimal dispersion dynamics for plasma waves, like the Alf $\acute{v}$en and Langmuir waves through the unified concatenation controlling parameters, including cold-plasma environments. Finally, this study recommends further undertakings from different perspectives. In particular, one may extend the model by incorporating higher-order and perturbation terms, or extending the new equation to a higher-dimensional form, capable of modeling several nonlinear complex physical scenarios.