<p>The integrable Kuralay equations (IKEs) represent a new class of nonlinear partial differential equations (NLPDEs) that capture the delicate balance between dispersion and nonlinearity in complex physical systems such as nonlinear optical fibers, plasmas, and ferromagnetic spin chains. From a geometric standpoint, these equations correspond to the motion of space curves, where soliton solutions describe the curvature and torsion of evolving wave structures. In this work, we employ the Complete Discrimination System for Polynomial Method (CDSPM) to systematically construct and classify a rich family of optical soliton solutions, including Jacobian elliptic, double-hyperbolic (DH), double-periodic (DP), and trigonometric forms. By exploring the limiting behaviors of elliptic solutions, we also derive solitary wave (SW) structures that exhibit kink-type, periodic, and localized pulse geometries. Furthermore, the bifurcation and stability analyses reveal intricate quasi-periodic and transitional behaviors, highlighting the strong coupling between the system’s geometry and its nonlinear dynamics. The findings not only deepen the theoretical understanding of the Kuralay framework but also offer practical applications in optical communication, plasma energy transport, and nonlinear circuit design. Overall, the study presents a unified analytical and geometrical framework that bridges mathematical theory with real-world physical phenomena.</p>