<p>In this paper, the high dispersive nonlinear Schrödinger equation with parabolic law nonlinearity, which is significant in the study of nonlinear optics, is discussed using two different approaches, namely the unified method and the improved <i>F</i>-expansion method. As an outcome, scores of complex analytical exact optical solutions are offered. Specifically, three types of solutions in polynomial form are constructed via the unified method. A succession of hyperbolic trigonometric, trigonometric and rational solutions are built with the assistance of the improved <i>F</i>-expansion method. Further, we draw two-dimensional, three-dimensional and contour images for some selected solutions so as to make a more comprehensive and systematic exploration for wave propagation.</p>