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

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Polychromatic Solitons in a Periodic and Nonlinear Maxwell System

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

**Abstract:**

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.

**Keywords**:

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