Directed electromagnetic wave propagation in 1D metamaterial: Projecting operators method

• The problem of boundary regime propagation is solved by a systematic dynamic projecting method.
• By this method a hybrid amplitude is introduced and used for derivation of nonlinear equation of opposite directed waves.
• The equations are specified for Drude metamaterial dispersion and Kerr nonlinearity.
• It is shown that one of unidirection waves in the metamaterial is specified as Shafer–Wayn integrable equation.
• A stationary wave solution is approximately expressed in terms of elliptic functions.

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.