Closed‐Form Exact Solitary‐Wave Solutions for the Time‐Fractional Rosenau‐KdV‐RLW Equation

ABSTRACT In this study, we investigate exact solitary‐wave solutions of the time‐conformable Rosenau Korteweg de Vries (R‐KdV) regularized long‐wave equation, extended through a power‐law nonlinearity. A specific form of the modified R‐KdV equation is addressed, and closed‐form solitary‐wave solutions are derived using the well‐established generalized Riccati equation mapping (GREM) technique. Initially, the conformable time‐fractional R‐KdV equation is transformed into a nonlinear ordinary differential equation (ODE) through a nonlinear conformable time transformation. The GREM method is then applied to obtain exact analytical solutions of the resulting model. This approach yields various classes of solutions, including both solitons and solitary waves. The derived solutions include singular waves, bright and dark solitons, as well as other localized wave structures. To support the physical interpretation of the results, we present two‐dimensional (2D), three‐dimensional (3D), and contour plots for different parameter settings. These graphical representations illustrate the qualitative behavior of the waveforms and highlight the impact of model parameters on the soliton dynamics.