Stability analyses have been widely used to better understand the mechanism of traffic jam formation. In this paper,we consider the impact of cooperative systems (a.k.a. connected vehicles) on traffic dynamics and, more precisely, on flow stability. Cooperative systems are emerging technologies enabling communication between vehicles and/or with the infrastructure. In adistributed communication framework, equipped vehicles are able to send and receive information to/from other equipped vehicles. Here, the effects of cooperative traffic are modeled through a general bilateralmultianticipative car-following law that improves cooperative drivers' perception of their surrounding traffic conditions within a given communication range. Linear stability analyses are performed for a broad class of car-following models. They point out different stability conditions in both multianticipative and nonmultianticipative situations. To better understand what happens in unstable conditions, information on the shock wave structure is studied in the weakly nonlinear regime by the mean of the reductive perturbation method. The shock wave equation is obtained for generic car-following models by deriving the Korteweg de Vries equations. We then derive traffic-state dependent conditions for the sign of the solitary wave (soliton) amplitude. This analytical result is verified through simulations. Simulation results confirm the validity of the speed estimate. The variation of the soliton amplitude as a function of the communication range is provided. The performed linear and weakly nonlinear analyses help justify the potential benefits of vehicle-integrated communication systems and provide new insights supporting the future implementation of cooperative systems.