In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.

It is well known that viral infections have a high impact on public health in multiple ways, including disease burden, outbreaks and pandemic, economic consequences, emergency response, strain on healthcare systems, psychological and social effects, and the importance of vaccination. Mathematical models of viral infections help policymakers and researchers to understand how diseases can spread, predict the potential impact of interventions, and make informed decisions to control and manage outbreaks. In this work, we formulate a mathematical model for the transmission dynamics of COVID-19 in the framework of a fractional derivative. For the analysis of the recommended model, the fundamental concepts and results are presented. For the validity of the model, we have proven that the solutions of the recommended model are positive and bounded. The qualitative and quantitative analyses of the proposed dynamics have been carried out in this research work. To ensure the existence and uniqueness of the proposed COVID-19 dynamics, we employ fixed-point theorems such as Schaefer and Banach. In addition to this, we establish stability results for the system of COVID-19 infection through mathematical skills. To assess the influence of input parameters on the proposed dynamics of the infection, we analyzed the solution pathways using the Laplace Adomian decomposition approach. Moreover, we performed different simulations to conceptualize the role of input parameters on the dynamics of the infection. These simulations provide visualizations of key factors and aid public health officials in implementing effective measures to control the spread of the virus.