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), a coherent multiscale structure emerges that persists for the lifetime of the simulation. This state features (i) a broad, spatially localized, and slowly evolving envelope and (ii) a train of shocks, approximately on the scale of the initial carrier wave. We loosely call this structure an envelope carrier-shock train. The structure of the solution violates the often 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 but the principle resonance induced by the initial carrier frequency. We derive, via a nonlinear geometrical optics expansion, a system of nonlocal integrodifferential equations governing the coupled evolution of backward and forward propagating waves. These equations incorporate 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 the fine scale carrier shocks of the nonlinear periodic Maxwell equations.