Wave patterns and dynamical properties of optical propagation by a higher order nonlinear Schrödinger equation

•The existences of soliton and periodic solution are proved.•The topological stability of each chirped solution is analyzed.•All the traveling wave solutions are constructed to verify the above conclusions, some of them are initially given. It is of great significance to explore the physical properties of optical propagation, thus the scholars are keen on the physical essence represented by classical mathematical physics equations. In this paper, a higher order nonlinear Schrödinger equation describing the behavior of polarization mode in optical fibers is firstly analyzed qualitatively, and the existence of periodic and soliton solutions is proved by using bifurcation method. Secondly, all chirped wave patterns with special form for the equation are obtained by using the complete discrimination system for polynomial method and direct integral method, the chirped rational functions and some elliptic functions wave patterns are initially found. As a result, the parameter stability of these patterns is given for the first time, which shows the variant of patterns as the parameters change, and the graphs of several typical patterns are drawn.