This work examined new analytical soliton solutions of the conformable fractional nonlinear (1+1)-dimensional Akbota problem utilizing the modified extended direct algebraic technique and a new Kudryashov method. This study investigated the application of the fractional Akbota equation in optical fiber telecommunications, where soliton solutions are crucial for maintaining signal integrity over long distances. Akbota is an integrable equation of the Heisenberg ferromagnetic variety, and it holds considerable importance for surface geometry and curve analysis in optics and magnetism. The derived soliton solutions might be characterized as dark, bright, periodic, or in other forms. The results collected are validated and shown in three-dimensional and two-dimensional graphs. The utilization of fractional derivatives has yielded results that are more contemporary than those presently found in the literature. The findings indicate that the fractional variant of the Akbota equation enhances modeling precision for nonlinear phenomena in optical fibers, rendering it an essential instrument for improving fiber optic networks. Consequently, the derived answers are beneficial for subsequent investigations of this model. The utilized methodologies yield a variety of solutions. In conclusion, the applied techniques are straightforward, effective, and dependable for solving other different models in mathematical physics. The novelty of this work lies in the application of the conformable fractional approach to the Akbota equation system, along with the implementation of new analytical methods that reveal a broader spectrum of soliton solutions, including bright, dark, periodic-singular, and breather structures, many of which have not been previously reported for this model.

Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.