This work investigates the Triki–Biswas equation (TBE), a notable generalization of the nonlinear Schrödinger equation that models nonlinear wave propagation in optical fibers, shallow water, and plasma systems. The TBE plays a crucial role in describing the transmission of ultrashort pulses in optical networks and the dynamics of localized excitations in dispersive media. To explore its solitary wave structures, we apply the generalized \(\phi ^{6}-\) model expansion method, an advanced analytical approach that enables the derivation of diverse solution families. Through systematic reduction, the TBE is transformed into nonlinear ordinary differential equations, from which explicit solutions are constructed under appropriate constraint conditions, ensuring physical relevance. The obtained results include periodic, bright, dark, kink-type, anti-peaked, and smooth solitary wave solutions, many of which are novel contributions. Their dynamics are further illustrated through 2D, 3D, and contour visualizations, providing clear insights into pulse transmission behavior. These findings significantly enrich solitary wave theory, deepen the understanding of nonlinear wave dynamics, and open new pathways for applications in optical communication, fluid dynamics, and plasma physics.
