Solitons on a continuous-wave background and collision between two dark pulses: some analytical results

First, we recall how two exact solutions of the nonlinear Schrödinger equation that describe (i) soliton propagation on a cw background and (ii) collision between two dark pulses can be obtained by using a common direct integration method. Second, we analyze the amplification–absorption effects on the first solution in the adiabatic approximation.