Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations

We consider the one-dimensional propagation of electromagnetic waves in a weakly nonlinear and low-contrast spatially inhomogeneous medium with no energy dissipation. We focus on the case of a periodic medium, in which dispersion enters only through the (Floquet-Bloch) spectral band dispersion associated with the periodic structure; chromatic dispersion (time-nonlocality of the polarization) is neglected. Numerical simulations show that for initial conditions of wave-packet type (a plane wave of fixed carrier frequency multiplied by a slow varying, spatially localized function) very long-lived spatially localized coherent soliton-like structures emerge, whose character is that of a slowly varying envelope of a train of shocks. We call this structure an envelope carrier-shock train. The structure of the solution violates the oft-assumed nearly monochromatic wave packet structure, whose envelope is governed by the nonlinear coupled mode equations (NLCME). The inconsistency and inaccuracy of NLCME lies in the neglect of all (infinitely many) resonances except for the principle resonance induced by the initial carrier frequency. We derive, via a nonlinear geometrical optics expansion, a system of nonlocal integro-differential equations governing the coupled evolution of backward and forward propagating waves. These equations incorporate effects of all resonances. In a periodic medium, these equations may be expressed as a system of infinitely many coupled mode equations, which we call the extended nonlinear coupled mode system (xNLCME). Truncating xNLCME to include only the principle resonances leads to the classical NLCME. Numerical simulations of xNLCME demonstrate that it captures both large scale features, related to third harmonic generation, and fine scale carrier shocks features of the nonlinear periodic Maxwell equations.