<p>This research paper is used to examine the generalized third-order nonlinear Schrödinger equation that is important for the description of complex nonlinear wave propagation in optical fibers, plasma physics, and in fluid mechanics. By using the generalized auxiliary equation method and improved modified Sardar-sub equation method various analytical soliton solutions are obtained. The Periodic, bell, anti-bell, dark, W-type, kink, anti-kink, and M-shape solitons solutions of the problem are obtained by using proposed methods. The behavior of several soliton solutions is depicted graphically. The findings are important for applications in engineering and mathematical physics. A key component of planar dynamical system theory is the bifurcation of the dynamical system of the governing equation caused by the Galilean transformation. By emphasizing how susceptible the system is to its initial conditions and how uncertain its long-term evolution is, the study also investigates chaotic behavior. Lastly, sensitivity analysis is performed to examine the effects of altering system parameters on the stability and results of the solutions. By taking these things into account, this work aids in our comprehension of the proposed model’s dynamical characteristics and offers helpful insights into its use in nonlinear media and other fields of research. In contrast to previous research, this work is novel because it applies enhanced analytical methods to the generalized third-order nonlinear Schrödinger equation, resulting in a wider range of solution classes.</p>