General equation for directed Electromagnetic Pulse Propagation in 1D metamaterial: Projecting Operators Method

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.Comment: 14 pages, Workshop "Waves in inhomogeneous media and integrable systems" dedicated to Professor Sergey Leble on the occasion of his 70th birthday Gda\'nsk University of Technolog