Advanced soliton structures and elliptic wave patterns in a sixth-order nonlinear Schrödinger equation using improved modified extended tanh function method

In this work, a sixth-order extension of the nonlinear Schrödinger equation (NLSE) within its integrable hierarchy is investigated to model higher-order nonlinear and dispersive effects relevant to optical fiber systems and nonlinear wave propagation. By employing the Improved Modified Extended Tanh Function Method, a comprehensive family of exact analytical solutions is derived, encompassing bright and dark solitons, singular soliton structures, and singular periodic solutions. In addition, solution families expressed in terms of Jacobi elliptic functions, Weierstrass doubly periodic elliptic functions, and exponential profiles are obtained. The novelty of this study lies in extending the analytical framework of the NLSE hierarchy to its sixth-order integrable form and in uncovering new soliton and elliptic wave structures. The obtained results reveal rich nonlinear dynamics associated with higher-order dispersion and nonlinearity and clarify the transition between periodic and localized behaviors. Two- and three-dimensional graphical simulations further illustrate the spatiotemporal evolution of the derived solutions. Overall, the findings deepen the understanding of advanced nonlinear wave mechanisms and offer potential implications for ultrafast optical and nonlinear waveguide systems.