The principal objective of this work focuses on achieving a class of advanced and impressive N-dark–dark solitons estimations to the coupled nonlinear Schr¨odinger system (CNLSS) that represents the propagation of optical pluses in the model-locked fiber lasers. The achieved results will perfectly describe different nonlinear coherent structures for the behavior of the resultant solitons to this model as well as ottosecond pulses in optical fibers that play fundamental rule for all recent telecommunications. The N-dark-dark solitons arise when nonlinearities are all removed, or when there is mixture between both focusing and defocusing nonlinearities. If various focusing and defocusing solitons components are collide with each other, the energies will emerge. The N–dark-dark solitons for this model will be established by using five significant schemas that were checked before to extracting the bright, dark and other types of solitons for many nonlinear problems that arise in many branches of science. The proposed methods that are bided for this goal are the extended direct algebraic method (EDAM), the (G′ G ) -expansion method, the extended simple equation method (ESEM), the Paul-Painleve approach method (PPAM) and the Riccati-Bernoulli Sub-ODE method (RBSODM). The suggested methods have been checked before for many other NLPDE and continuously achieve distinct results. The achieved results will add future studies for all recent applications of the internet communications.