Rogue waves for a generalized nonlinear Schrödinger equation with distributed coefficients in a monomode optical fiber

Investigated in this paper is the generalized nonlinear Schrödinger equation with distributed coefficients, which describes the amplification or absorption of pulses propagating in a monomode optical fiber with distributed group-velocity dispersion and self-focusing Kerr nonlinearity. By virtue of the Kadomtsev-Petviashvili hierarchy reduction, we obtain the rogue waves based on rogue-wave solutions in terms of the Gramian under certain constraint. We study the effects of group-velocity dispersion, nonlinearity and amplification/absorption coefficients on the rogue waves with the help of figures. Amplitudes of the rogue waves are independent with the group-velocity dispersion and nonlinearity coefficients. The first-order rogue wave with an eye-shaped distribution density and the second-order rogue waves with the highest-peak amplitude and with the triple-peak structure are presented. Both the intermingled or separated composite rogue waves are derived. Periodic rogue waves are obtained and period of the periodic rogue wave increases with the period of the group-velocity dispersion. Furthermore, nonlinear tunneling of the rogue waves is observed: rogue waves get amplified when they reach to the dispersion barriers and recover their original shapes after passing through the barriers, while amplitudes of the rogue waves decrease inside the dispersion wells. Amplification/absorption coefficient influence the background and amplitude of the rogue wave, and three types of the backgrounds are discussed due to different amplification/absorption coefficients.