Role of parametric gain operator for the higher-order nonlinear Schrodinger equation

Based on the separate discussions on dispersion and nonlinear effects, the parametric gain operator is established to exhibit the pump field's impact in the process of the fast and wide spectral expansion. The condition of inducing sideband is that sin[∑n=2∞(−1)n+1n!βnωndz]=0 has a solution with the intensity fluctuations or cos[∑n=2∞(−1)n+1n!βnωndz]=0has a solution with a initial phase. The initial intensity fluctuation results in a coherent white source while the initial phase modulation cuts down this band's width but brings the nearby blue-shift or red-shift solitons which depend on the dispersion coefficients. Furthermore, this parametric gain operator can be used to describe the phenomenon of CW pump inducing solitons.