The seventh-order Sawada-Kotera-Ito equation is a fundamental nonlinear partial differential equation that arises in modeling complex wave phenomena in fluid dynamics and other physical systems. In this study, two analytical techniques, namely, the \(\phi ^6\) -model expansion method and the extended simplest equation method are employed to derive exact analytical solutions to the seventh-order Sawada-Kotera-Ito equation. As a result, we construct explicit solutions that describe solitary waves, kink and anti-kink waves, breather-type waves, and other wave structures. The efficiency and versatility of these methods are demonstrated through the systematic derivation of solutions, which are further validated through graphical representations. The obtained solutions have diverse applications in various areas of applied sciences, and the graphical structures assist researchers in understanding the physical phenomena underlying this dynamical model. Modulational instability (MI) analysis is performed to investigate the stability of the model and its solutions. The computational results confirm that these methods are simple, straightforward, and efficient. These findings provide new insights into the dynamics of higher-order nonlinear wave equations and broaden their potential applications in mathematical physics and engineering. Moreover, these methods can be applied to other nonlinear wave equations arising in mathematical physics and related fields of applied sciences.
