New Analytical Soliton Molecule Solutions of the Conformable Time-Fractional Sharma-Tasso-Olver-Burgers Equation Using Modified Kudryashov Method

In this study, novel analytical soliton molecule solutions for the conformable time-fractional Sharma-Tasso-Olver-Burgers (STOB) equation have been obtained using the modified Kudryashov method for the first time. The STOB equation is an important nonlinear differential equation that combines the well-known Burgers and Sharma-Tasso-Olver (STO) equations, showing rich soliton molecular behavior. Our solutions reveal previously unknown soliton molecule structures with unique nonlinear dynamics that go beyond classical Burgers and STO equations. Through systematic application of the modified Kudryashov method, we obtain multiple families of exact solutions with different characteristics, including single-kink wave profiles and solitary wave solutions with different characteristics. The conformable fractional derivative approach used in our study allows accurate modeling of anomalous dynamic behaviors and fractional scaling in physical systems, while showing how changes in the fractional derivative order affect dynamic evolution. We discover different dynamical behaviors ranging from subdiffusive to classical dynamics across a broad spectrum, depending on the fractional derivative order value. The obtained solutions have potential applications in various physical systems, from mode-locked fiber lasers to anomalous diffusion processes, and from slow light propagation in optical fibers to Bose-Einstein condensates. The evolution of these solutions is examined through comprehensive graphical analysis. 3D surfaces, contour maps, and 2D cross-sectional plots are utilized to illustrate the physical impact of the fractional order α. These visualizations confirm that the wave profiles maintain their stability while exhibiting distinct fractional dynamics as they propagate. Furthermore, the obtained results make significant contributions to the understanding of soliton molecules in fractional-order systems, opening new research avenues in fields such as quantum field theory, plasma physics, and nonlinear optics.