Elliptic function and solitary wave solutions of the higher-order nonlinear Schrodinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability

The higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion, cubicquintic terms, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. In this paper, the elliptic function, bright and dark solitons and solitary wave solutions of higher-order NLSE are constructed by employing a modified extended direct algebraic method, which has important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the bright and dark solitons for this equation. The modulation instability is utilized to discuss the stability of these solutions, which shows that all solutions are exact and stable. Many other higher-order nonlinear evolution equations arising in applied sciences can also be solved by this powerful, effective and reliable method.