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.
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Xu et al. (2003) studied this question.
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