In order to reduce the power and realize ultrafast response time, a plasmonic waveguide system based on two stub cavities side-coupled is designed, and a plasmon induced transparency effect is investigated. The plasmonic waveguide based on graphene-Ag composite material structures is tuned by the optical Kerr effect. An ultrafast response time of the order of 1 ps is reached. With dynamically tuning the propagation phase of the plasmonic waveguide, π-phase shift of the transmission spectrum in the plasmon induced transparency system is achieved under excitation of a pump light with an intensity as low as 5.83 MW/cm². The pump light intensity is reduced by adopting graphene-Ag composite material structures. The reason is that the optical Kerr effect is enhanced by the local electromagnetic field of surface plasmon polaritons, the slow light of the plasmon induced transparency effect and the plasmonic waveguide based on graphene-Ag composite material structures with giant effective Kerr nonlinear coefficient. The tunable bandwidth of about 40 nm is obtained. The group delay is controlled between 0.15 ps and 0.85 ps. Moreover, for the indirect coupling between two stub cavities or the phase coupling scheme, the phase shift multiplication effect of the plasmon induced transparency effect is found. The theoretical results are in good agreement with finite difference time domain simulations. Research results are of reference significance in design and fabrication of nanoscale integration plasmonic photonic devices with low power consumption and ultrafast nonlinear responses.