Dynamic optical soliton solutions of M-fractional modify unstable nonlinear Schro<spacing diaeresis>dinger equation via two analytic methods

In optical fibers, the nonlinear Schro<spacing diaeresis>dinger equation (NLSE) serves as the primary model for studying the propagation of ultra-short waves. Specific instabilities in manipulated wave-trains are governed by a fundamental equation belonging to the class of nonlinear integrable systems, referred to as the modified unstable NLSE (mUNLSE). Both stable and unstable media experience time evolution perturbations described by the mUNLSE. Within this framework, the main objective of this work is to examine the M-fraction mUNLSE analytically by using two novel methods: the extended Jacobian elliptic function expansion and the unified approaches. These methods provide new insights into wave propagation and the behavior of optical solitons in various fields such as nonlinear optics, optical communications, quantum mechanics, plasma physics, and other engineering disciplines. Thus, this work will have a prominent significance in the development of sustainable cities and will be beneficial for the concerned communities. The techniques enable the derivation of innovative soliton solutions expressed through elliptic, trigonometric, hyperbolic, and rational functions. For specific parametric values, the extended Jacobian elliptic function expansion technique reveals solutions such as double-periodic waves, periodic waves, periodic lump waves, and cross-periodic waves. Additionally, the unified approach uncovers periodic breather solitons, periodic waves, kinky periodic waves, and periodic waves with lump waves. The effects of truncated M-fraction parameters are illustrated graphically through 3D and 2D plots. The findings have the potential to enhance the understanding of the physical characteristics of waves propagating in dispersive media.