<p>Nonlinear dispersive wave equations play a fundamental role in the mathematical description of pulse propagation in complex biological fluids, including arterial blood flow. In this work, a higher-order nonlinear Schrödinger equation is examined as a governing model for nonlinear wave evolution in elastic vessels. Despite the extensive literature on nonlinear Schrödinger-type systems, comprehensive analytical characterizations of exact traveling wave solutions for this higher-order hemodynamic framework remain relatively scarce. To address this issue, two complementary analytical schemes, namely the F-expansion technique and the complete discriminant system method, are employed to derive new closed-form wave structures. Several families of exact solutions are obtained, including soliton-type profiles, periodic trigonometric waves, and Jacobi elliptic function solutions, subject to suitable parameter constraints. The discriminant-based approach further provides a systematic classification of admissible solution patterns. In addition, graphical representations via three-dimensional surfaces, contour plots, and two-dimensional profiles are presented to demonstrate the influence of key parameters on wave amplitude, localization, and propagation features. The present analysis is limited to idealized parameter regimes of the higher-order nonlinear Schrödinger model; nevertheless, the derived results offer valuable theoretical insight into nonlinear arterial pulse dynamics and may be extended to other dispersive physical media. Overall, this study enriches the existing literature by expanding the spectrum of exact solutions for higher-order nonlinear evolution equations.</p>