The present work aims to investigate solitary wave solutions for two recently developed extended equations in the context of (2+1)-dimensional and (3+1)-dimensional structures. The equations under consideration are of the Korteweg–de Vries (KdV) type, which are well-recognized as significant aspects of fluid dynamics. These equations have broad applications in physics, mathematics, and other scientific disciplines, particularly in the study of waves, soliton theory, plasma physics, biology and chemistry, and nonlinear phenomena. Its soliton solutions and integrability properties make it a fundamental model in various areas of research. This serves as the main motivation for our research work. To analyze these equations, we employ an advanced direct algebraic equation method capable of generating several sorts of solutions, including solitary and shock wave solutions, as well as their combination. In addition to these wave phenomena, singular solitons and solutions expressed in Jacobi and Weierstrass doubly periodic types have also been observed. The utilization of this outstanding technique and the subsequent acquisition of novel solutions demonstrate the originality of our study. This also allows further exploration of nonlinear models that accurately depict significant physical processes in our everyday existence.
