Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrödinger equation and its stability

The higher-order nonlinear Schrödinger equation with fourth-order dispersion, cubic-quintic terms, nonlinearity, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. The extended form of simple equation method is proposed to construct exact soliton and solitary wave solutions of higher-order nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity. These new exact solutions are expressed in the forms of trigonometric, hyperbolic, exponential and rational functions. These solutions are also presented graphically. Furthermore, many other higher-order nonlinear evolution equations arising in mathematical physics and other areas of applied sciences can also be solved by this powerful, reliable and capable method.