Modulation instability and solitons on a cw background in an optical fiber with higher-order effects

We construct the Lax pair for a higher-order nonlinear Schrödinger equation that includes terms accounting for the third-order dispersion, the self-steepening effect, and the delayed nonlinear response effect. Two exact analytic solutions that describe (i) modulation instability and (ii) soliton propagation on a continuous wave background are obtained by using the Darboux transformation. In addition, we analyze the amplification-absorption and quintic nonlinearity effects on the second solution in the adiabatic approximation.