| Surface Plasmon Polaritons (SPPs) are the coupling of an electromagnetic wave with collective charge oscillations along the surface between a dielectric and a metal. SPPs propagate along the surface while remaining confined to it. As their wavelength decreases, SPPs transition from propagating without dispersion, to propagating with dispersion, to finally becoming standing waves. In this dissertation we focus on the evolution of small amplitude SPPs in the short-wave limit when coupled with nonlinear media.;Nondimensionalizing Maxwell's equations and using the method of multiple scales, we derive an equation modeling the amplitude's evolution. Referred to as the bidirectional equation, it includes a cubic nonlinearity that couples left and right moving waves. Deriving inequalities for the nonlinear terms, we prove short time existence for initial data in Hs( T) for s≥1/2 using a fixed point method. Taking advantage of the Hamiltonian structure of the equations, we derive conserved quantities. Assuming the initial data to have purely positive Fourier components, the bidirectional equation reduces to the unidirectional equation. We prove the global existence of weak solutions for the unidirectional equation with initial data in L2(T).;Using pseudo-spectral and finite difference methods we generate numerical solutions. These solutions display strong spatial focusing, for unidirectional data solutions appear to remain smooth after focusing, but whether singularities are formed for bidirectional initial data is unclear. |
