Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrödinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.