On the dynamics of the generalized unstable nonlinear Schrödinger equation in dispersive media

The nonlinear Schrödinger equation (NLSE) is one of the most important physical model in optical fiber theory for explaining changes in optical soliton growth. Due to the large range of opportunities for ultrafast signal routing systems and brief pulses of light in communications, optical soliton transmission in nonlinear fibers is currently a topic of significant interest for researchers. In this article, we study the dynamics of the generalised unstable NLSE which specified the time evolution of perturbations in barely steady or unstable media. A set of efficient analytical approaches are implemented to develop various interesting travelling wave structures such as bright, dark, singular, optical, bell shaped and periodic solitons solutions. Additionally, the stability of the obtained results is validated. Finally, the modulation instability of the governing model has also been investigated to present change effects that occurs as a result of the irregular refractive indices in an optical fiber, leading to distinct behaviour patterns. The methods as reported are applicable in clarity, effectiveness, and simplicity, establishing their applicability to various sets of dynamic and static nonlinear equations concerning evolutionary phenomena in computational physics, as well as other practical domains and numerous areas of research.