This study investigates the extended \((3+1)\) -dimensional Kairat-II and Kairat-X equations, which simulate the propagation of ultrashort pulses in optical fibers, planar waveguides, photonic crystals, plasma, and fluid media. The Kairat equations are well-known because they involve numerous equivalency relations and second-order spatiotemporal dispersion and group velocity dispersion, which demonstrate curve differential geometry. We utilized the modified simplest equation method to find soliton solutions in trigonometric and hyperbolic forms, which depict dark soliton, singular soliton, periodic wave solitons, mixed solitons in the shape of bright-dark solutions, singular-periodic soliton, and dark-singular solitons solutions. It is used in traffic flow modeling and environmental systems, aiding in the development of transportation networks and the control of pollutants and population dynamics. This method provides a systematic and efficient approach for analyzing nonlinear wave dynamics. To illustrate the physical behavior of these solutions, we present two-dimensional, three-dimensional, and contour plots, highlighting the influence of key parameters. The two-dimensional plots are used to illustrate the comparisons between different values of t. In particular, this study investigates the extended Kairat equations and derives classes of exact soliton solutions using the modified simplest equation method, which have not been reported previously in this general form. Furthermore, stability analysis and detailed graphical visualizations are presented to support the analytical results and to enhance the physical interpretation of the obtained solutions.
