Direct perturbation analysis on the localized waves of the modified nonlinear Schrödinger equation under nonvanishing boundary condition

In this paper, the modified nonlinear Schrödinger equation is investigated via the direct perturbation method, which can describe the femtosecond optical pulse propagation in a monomodal optical fiber. Considering the quintic nonlinear perturbation, we obtain the approximate solution with the first-order correction, which can be expressed by the solution and symmetry of the derivative nonlinear Schrödinger equation. Under the nonvanishing boundary conditions, the approximate dark and anti-dark soliton solutions are derived and the existence conditions are also given. The effects of the perturbation on the propagations and interactions of the solitons on the nonzero background are discussed by comparing the physical quantities of solitons with the unperturbed case. It is found that the quintic nonlinear perturbation can lead to the change of the velocity as well as the pulse compression, but has no influence on the dynamics of the elastic interactions. Finally, numerical simulations are performed to support the theoretical results.