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Using the Jacobi Elliptic Functions, an analytical solution is developed for the nonlinear amplitude equation of Surface Plasmon Polaritons (SPPs) in a graphene-dielectric waveguide. It is shown that the field localization of SPPs coupled with TM polarized terahertz light can be enhanced if the nonlinearity is increased. On the side, a numerical solution based on Split Step Beam Propagation Method (SSBPM) suggests that spatial Modulation Instabilty (MI) can be dominant. Accordingly, larger nonlinearity leads to the generation of discrete plasmon solitons rather than the diffracted profile resulted for the modest nonlinearity. Adding then the temporal variations to the nonlinear amplitude equation and solving numerically by predictor-corrector method, it is revealed that temporal MI appears as ultrashort pulse trains with multi-periodic behavior. Evoking the similarity with a laser cavity, the waveguide can be assumed as a spasing system-assuming the large nonlinearity regime-in which if the coupling depth is raised, the character of MI will be changed from the convective to absolute and the amplitude of SPPs will grow fast. The procedure can be even chaotic and unpredictable. This spasing system is suitable for applications of the electro-optical circuitry, optical amplification, optical communication and biomedical sensing through which the low power consumption and non-destructivity are of important traits.