Abstract This paper presents the first application of the Modified Extended Direct Algebraic (MEDA) method to the (2+1)-dimensional Wazwaz-Kaur-Boussinesq equation, a model governing wave dynamics in shallow waters. The approach successfully uncovers previously unreported classes of exact solutions, including combo dark–singular solitons and Jacobi elliptic function solutions. The spectrum of obtained solutions–which also encompasses bright, dark, and singular solitons, as well as hyperbolic, periodic, exponential, and rational functions–reveals rich and complex soliton dynamics. A comprehensive stability analysis confirms the robustness of these solutions under perturbation. These results significantly advance the understanding of wave propagation in nonlinear systems, providing valuable insights for applications in fluid dynamics, nonlinear optics, and plasma physics, while demonstrating the efficacy of the MEDA method for tackling complex nonlinear evolution equations.
Elsonbaty et al. (Wed,) studied this question.