The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields.

We investigate nonlinear magnetic excitations in the ( 2 + 1 ) -dimensional Heisenberg ferromagnetic spin equation using the n -fold Darboux transformation. Exact analytical solutions, including dark, bright, and breather-type solitons, are constructed in determinant form. By employing a degenerate spectral limit, we derive higher-order smooth positon and breather–positon solutions. The determinant representation enables systematic analysis of their decomposition, interaction behavior, and phase shifts. Furthermore, trajectory analysis on the Bloch sphere provides physical insight into the precessional dynamics of localized magnetization vectors. These results reveal rich nonlinear interaction patterns arising from higher-dimensional effects in ferromagnetic media.