INVESTIGATING THE FRACTIONAL INTEGRABLE CALOGERO–BOGOYAVLENSKII–SCHIFF EQUATION: EXPLORING THE SOLITARY WAVE SOLUTIONS

Solitary wave solutions captivate the attention of mathematicians and other scientists because they provide a foundational elucidation of nonlinear phenomena that have numerous pragmatic applications. Due to their excellent persistence and behavior, they provide a solid base for the development of innovative nonlinear models in many areas of physical, biological, and medical modeling. This study presents novel soliton solutions for the generalized [Formula: see text]-dimensional beta fractional Calogero–Bogoyavlenskii–Schiff equation using advanced methodologies: the modified generalized exponential rational function method, the modified generalized Riccati equation mapping method, and the Riccati modified extended simple equation method. The equation models the interaction between Riemann wave propagation along the [Formula: see text]-axis and long-wave dynamics along the [Formula: see text]-axis. Exact traveling wave solutions are pivotal for advancing analytical and numerical studies in wave phenomena. By employing a fractional derivative-based wave transformation, the model is converted into a nonlinear ordinary differential equation. A diverse array of soliton solutions is derived, including mixed, dark, bright–dark, bright, complex, and combined solitons. The utilized methods demonstrate exceptional computational efficacy and precision, enabling systematic exploration of exact solutions with high accuracy in such nonlinear systems. These findings enhance the understanding of wave interactions and provide valuable insights for applications in mathematical physics and engineering. Moreover, we plot 2D and 3D graphs with specific parameters to show the solution’s dynamics at numerous parameters. In the physical world, various types of partial differential equations may be solved using the suggested novel techniques. The obtained soliton solutions may contribute to a better understanding of wave behaviors and therefore support further research in many areas.