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We consider the one-dimensional propagation of electromagnetic waves in aweakly nonlinear and low-contrast spatially inhomogeneous medium with no energydissipation. We focus on the case of a periodic medium, in which dispersionenters only through the (Floquet-Bloch) spectral band dispersion associatedwith the periodic structure; chromatic dispersion (time-nonlocality of thepolarization) is neglected. Numerical simulations show that for initialconditions of wave-packet type (a plane wave of fixed carrier frequencymultiplied by a slow varying, spatially localized function) very long-livedspatially localized coherent soliton-like structures emerge, whose character isthat of a slowly varying envelope of a train of shocks. We call this structurean envelope carrier-shock train.  The structure of the solution violates the oft-assumed nearly monochromaticwave packet structure, whose envelope is governed by the nonlinear coupled modeequations (NLCME). The inconsistency and inaccuracy of NLCME lies in theneglect of all (infinitely many) resonances except for the principle resonanceinduced by the initial carrier frequency. We derive, via a nonlineargeometrical optics expansion, a system of nonlocal integro-differentialequations governing the coupled evolution of backward and forward propagatingwaves. These equations incorporate effects of all resonances. In a periodicmedium, these equations may be expressed as a system of infinitely many coupledmode equations, which we call the extended nonlinear coupled mode system(xNLCME). Truncating xNLCME to include only the principle resonances leads tothe classical NLCME.  Numerical simulations of xNLCME demonstrate that it captures both large scalefeatures, related to third harmonic generation, and fine scale carrier shocksfeatures of the nonlinear periodic Maxwell equations.

Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations