Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schr & ouml;dinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.