Dynamical behaviors and soliton solutions of a generalized higher-order nonlinear Schrödinger equation in optical fibers

Under study in this paper is a generalized higher-order nonlinear Schrödinger (GHNLS) equation with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering effects , which describes the propagation of ultrashort pulses in optical fibers. Via the phase plane analysis, both the homoclinic and heteroclinic orbits are found in the two-dimensional plane autonomous system reduced from the GHNLS equation, which proves the existence of bright and dark soliton solutions from the viewpoint of nonlinear dynamics. Furthermore, through the method of binary Bell polynomials and auxiliary function, the explicit bright and dark soliton solutions under certain conditions are obtained. Particular analysis is made to study the effects of the higher-order on the double-hump bright and double-hole dark solitons. The results show that the self-phase modulation and SS parameters determine the interval between two humps for the double-hump bright soliton, while the one for the double-hole dark soliton is related with the TOD and SS effects. Moreover, numerical simulations show that the double-hump bright soliton and the double-hole dark soliton are more stable when the amplitude or depth is comparably small.