Modulation Instability, Analytical, and Numerical Studies for Integrable Time Fractional Nonlinear Model through Two Explicit Methods

The present work deals with the investigation of the time-fractional Klein–Gordon (K-G) model, which has great importance in theoretical physics with applications in various fields, including quantum mechanics and field theory. The main motivation of this work is to analyze modulation instability and soliton solution of the time-fractional K-G model. Comparative studies are investigated by β-fraction derivative and M-fractional derivative. For this purpose, we used unified and advanced exp−ϕξ-expansion approaches that are highly important tools to solve the fractional model and are used to create nonlinear wave pattern (both solitary and periodic wave) solutions for the time-fractional K-G model. For the special values of constraints, the periodic waves, lumps with cross-periodic waves, periodic rogue waves, singular soliton, bright bell shape, dark bell shape, kink and antikink shape, and periodic wave behaviors are some of the outcomes attained from the obtained analytic solutions. The acquired results will be useful in comprehending the time-fractional K-G model’s dynamical framework concerning associated physical events. By giving specific values to the fractional parameters, graphs are created to compare the fractional effects for the β-fraction derivative and M-fractional derivative. Additionally, the modulation instability spectrum is expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third-order dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to theoretical physics. This work provided the effect of the recently updated two fraction forms, and in the future, we will integrate the space–time M-fractional form of the governing model by using the extended form of the Kudryashov method. Maple 18 is utilized as the simulation tool.