  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.
