The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation

A set of subsidiary ordinary differential equations (sub-ODEs for short) are firstly introduced. And then the exact solutions (including the kink type solitary wave, the bell type solitary wave, the algebraic solitary wave and the sinusoidal traveling wave) of the higher order dispersive nonlinear Schrödinger equation with both fourth-order dispersion effects and a quintic nonlinearity describing the propagation of optical pulse in a medium are derived with the aid of the sub-ODEs, the Riccati equation and the homogeneous balance principle.