Gray and black optical solitons with quintic nonlinearity

A higher-order nonlinear Schrödinger equation with the third-order dispersion, the cubic-quintic nonlinearities, derivative non-Kerr nonlinear terms, Kerr dispersion, and stimulated inelastic scattering is considered. The model can be used to describe the subpicosecond or femtosecond optical pulse propagation in highly nonlinear optical fibers. We find the exact gray solitary wave solution (i.e., a dark localized structure with nonzero minimum in intensity) on a continuous-wave background for the equation by adopting an appropriate ansatz solution. The propagation characteristics of such solitary pulses are determined in the presence of all higher-order effects. We also find black solitary wave solutions for the higher-order evolution equation under specific conditions for its coefficients. The parametric conditions on fiber parameters for the existence of both types are given.