Modulational instabilities and dark solitons in a generalized nonlinear Schrödinger equation

The generalized nonlinear Schrödinger equation with the nonlinear term of a general form is considered. In particular cases this equation covers a saturable nonlinearity as well as higher-order nonlinearities. Modulational instability is investigated, and the possibility of dark soliton propagation is analyzed. It is shown that in the small-amplitude limit the generalized dark solitons are stable, and they are described by the Korteweg-de Vries equation. The condition to create anti-dark solitons is also formulated.