Statistical approach of modulational instability in the class of nonlocal NLS equation involving nonlinear Kerr-like responses with non-locality: Exact and approximated solutions

The purpose of this work is the investigation of the behavior of modulated signal light in the class of optical waveguide whose the dynamics is governed by the nonlocal nonlinear Schrödinger (NNLS) equation involving nonlinear Kerr-like responses with nonlocality. Firstly, from the NNLS equation, the modulational instability (MI) is investigated for both the coherent and partially-coherent cases. The coherent MI is investigated by using the deterministic approach, i.e. the well-known Benjamin–Feir method, while for the partially coherent case (The propagating signal lights are not always monochromatic in real applications), the statistical approach is used starting from the Wigner Moyal transform, also known as Klimontovich’s statistical average method, and all these studies are checked by numerical investigations. The modulated bright soliton as well as periodic signal as solutions of the waveguide equation are next found. Then we find the implicit exact solutions, that the profiles are just obtained numerically which are not new in the literature. In addition, we use the perturbation method to find the approximated explicit bright and periodic solutions, where profiles are compared to the implicit ones.