With the increasing input power in optical fibers, the dispersion problem is becoming a severe restriction on wavelength division multiplexing (WDM). With the aid of solitons, in which the shape and speed can remain constant during propagation, it is expected that the transmission of nonlinear ultrashort pulses in optical fibers can effectively control the dispersion. The propagation of a nonlinear ultrashort laser pulse in an optical fiber, which fits the high-order nonlinear Schr？dinger equation (NLSE), has been solved using the expansion method. Group velocity dispersion, self-phase modulation, the fourth-order dispersion, and the fifth-order nonlinearity of the high-order NLSE were taken into consideration. A series of solutions has been obtained such as the solitary wave solutions of kink, inverse kink, the tangent trigonometric function, and the cotangent trigonometric function. The results have shown that the expansion method is an effective way to obtain the exact solutions for the high-order NLSE, and it provides a theoretical basis for the transmission of ultrashort pulses in nonlinear optical fibers. 1. Introduction It is understood that a soliton is excited by a nonlinear field, and its energy is relatively concentrated in a small area. The elastic scattering phenomenon occurs during the interaction between two solitons. The energy is nondispersed, so that the shape and speed can remain unchanged during its propagation. The history of the studies of solitons can be traced back to 1834, when James Scott Russell, a British scientist, accidentally observed that a bulge of water in Edinburgh-Glasgow Canal was propagating undistorted over several kilometers [1]. In 1973, Hasegawa and Tappert [2] first proposed the idea of applying “optical solitons” to photonic communication. After rigorous theoretical and mathematical deduction, he predicted that both bright and dark soliton pulses were present in an optical fiber. He also proved that any nondestructive optical pulse could travel as stable as a soliton during its transmission in an optical fiber. As an intrinsically nonlinear phenomenon, a soliton is the product of the optical fiber dispersion and the nonlinear interaction. It obeys the NLSE and is controlled by the optical fiber dispersion and the nonlinear effects of self-phase modulation. In 1980, bright solitons were observed for the first time in an optical fiber by Mollenaure [3] and his colleagues during their experiments at the Bell Laboratory of the United States. In 1987, Emplit observed dark solitons in a fiber by using an amplitude
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