In this paper, the higher nonlinear Schrödinger equation is solved by the Hirota method. As an example, the exact grey one- and two-soliton solutions in explicit forms are generated analytically under the continuous-wave background. The results reveal that the velocity of the grey soliton is clearly affected by the higher order effects, yet the grey soliton propagates without any change in their shape and intensity. The higher order term and the phase velocity play the important role for the maximum valley of grey soliton, i.e., the intensity of grey soliton. For the black soliton, the velocity of soliton is determined only by the higher order effects. The analysis of the asymptotic behavior of a two grey soliton solution shows the collision is elastic.
Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects