D.E. Pelinovsky, G. Simpson, and M.I. Weinstein

Polychromatic Solitons in a Periodic and Nonlinear Maxwell System

SIAM J. Applied Dynamical Systems 11 (2012), 478-506

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so called extended nonlinear coupled equations, xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, consisting of a slowly varying envelope of a train of carrier shocks. Thus, it is natural to study the localized coherent structures of xNLCME.
In this paper we explore, by analytical, asymptotic and numerical methods, the existence and properties of spatially localized structures of the xNLCME system, which arises for a refractive index profile consisting of periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a band-edge. In this case, stationary xNLCME is well-approximated by an infinite system of coupled, stationary, nonlinear Schrodinger equations, the extended nonlinear Schrodinger system, xNLS. We embed xNLS in a one-parameter family of equations, which interpolates between infinitely many decoupled NLS equations (eps = 0) and xNLS (eps = 1). Using bifurcation methods we show existence of solutions for small values of eps and, by a numerical continuation method, establish the continuation of certain branches all the way to eps = 1. Finally, we perform time-dependent simulations of truncated xNLCME and find the small-amplitude near-band-edge gap solitons to be robust to both numerical errors and the NLS approximation.

Coupled-mode equations, wave resonances, gap solitons, existence of homoclinic orbits, numerical and variational approximations