Dark solitary waves in a generalized version of the nonlinear Schrödinger equation

We investigate dark-wave solutions of the nonlinear parabolic equation with a nonlinearity of rather general type and nonzero boundary conditions at infinity. Traveling-wave solutions of some polynomial models are presented in the evident forms. The bistability of dark pulses as the possibility for two different waves to exist under the same boundary conditions is discussed. The consideration is supported by the numerical treatment of the polynomial nonlinearity allowing the bistability regime. It is found that some different boundary conditions can be coordinated with the nonlinear equation in the case of an ssN-shaped nonlinearity. In the small-amplitude limit the nonlinear Schr\"odinger equation is reduced to the Korteveg--de Vries equation. The stability of small-amplitude solutions is determined by both the kind of the nonlinearity and the intensity level at infinity. The occurrence of antidark pulses is pointed out.