We consider a boundary problem for 1D electrodynamics modeling of a pulse propagation in a metamaterial medium. We build and apply projecting operators to a Maxwell system in time domain that allows to split the linear propagation problem to directed waves for a material relations with general dispersion. Matrix elements of the projectors act as convolution integral operators. For a weak nonlinearity we generalize the linear results still for arbitrary dispersion and derive the system of interacting right/left waves with combined (hybrid) amplitudes. The result is specified for the popular metamaterial model with Drude formula for both permittivity and permeability coefficients. We also discuss and investigate stationary solutions of the system related to some boundary regimes.

In this paper, a theoretical study of wave propagation in 1D metamaterial is presented. A system of evolution equations for electromagnetic waves with both polarizations account is derived by means of projection operators method for general nonlinearity and dispersion. It describes interaction of opposite directed waves with a given polarization. The particular case of Kerr nonlinearity and Drude dispersion is considered. In such approximation, it results in the corresponding system of nonlinear equations that generalizes the Schäfer–Wayne one. Traveling wave solution for the system of equation of interaction of orthogonal-polarized waves is also obtained. Dependence of wavelength on amplitude is written and plotted.