Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method

By using the extended hyperbolic auxiliary equation method, we present explicit exact solutions of the high-order nonlinear Schrödinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. These solutions include trigonometric function type and exact solitary wave solutions of hyperbolic function type. Among these solutions, some are found for the first time.