Nonlinear dispersive wave propagation pattern in optical fiber system

Nonlinear fractional-order partial differential equations are an important tool in science and engineering for explaining a wide range of nonlinear processes. We consider the nonlinear space-time fractional modified Korteweg de Vries and sine-Gordon equations in this article and extract diverse sorts of traveling wave as well as soliton solutions using a typical approach, namely the generalized $\left(G^{\prime }/G\right)$-expansion approach. These equations are used to model a wide range of nonlinear phenomena, including plasma physics, high-intensity laser-generated plasma, ultra-small electronic devices, optical fibers, control theory, turbulence, acoustics, and others. The fractional derivative is defined in the sense of the beta-derivative established by Atangana and Baleanu. Some standard shapes of waveforms, including kink type, singular-kink type, bell-shaped, periodic-type, single soliton, multiple soliton type, and several other types of solitons, have been established. To validate the physical compatibility of the results, the 3D, contour, and vector plots have been delineated using consistent values of the parameters. The approach used in this study to extract inclusive and standard solutions is approachable, efficient, and speedier in computing.