Optical soliton solutions: the evolution with changing fractional-order derivative in Biswas–Arshed and Schrödinger Kerr law equations

The space–time fractional Biswas–Arshed and Schrödinger Kerr law equations featuring beta derivative hold substantial application in nonlinear optics, optical solitons, ultrafast optical signal, nonlinear photonics, quantum optics, biophotonics, photonic crystals photonics, etc. In this study, a wide variety of geometric shape solitons have been established that include hyperbolic, exponential, trigonometric, and rational functions, as well as their assimilation to the considered equations, through the two-variable (\(R^{\prime } /R, 1/R\))-expansion approach. The implication of the fractional parameter \(\mu \) on the wave shape has also been examined by depicting two-dimensional and three-dimensional plots for particular parameter values. The solitons include irregular periodic, pulse like, V-shaped, bell-shaped, positive periodic, asymptotic, general solitons, and some others. It is significant to note that the changes in the wave pattern result from the adjustments to substantive and auxiliary parameters. The outcomes demonstrate the efficiency, acceptability, and dependability of the (\(R^{\prime } /R, 1/R\))-expansion approach for obtaining solutions to the fractional-order evolution equations in the domains of engineering, technology, and sciences. It is evident from the graph that changing the value of μ results in a change in the shape of the soliton. The study explores how these equations change as fractional-order derivatives vary. Soliton solutions, which are stable, localized waveforms, are crucial in optical communication systems. Understanding their behavior under changing fractional-order derivatives is essential for advancing optical signal processing and communication technologies.