Broad Band Solitons in a Periodic and Nonlinear Maxwell System
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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 it extended nonlinear
coupled mode equations, xNLCME. Numerical studies support the existence of
long-lived spatially localized coherent structures, featuring a slowly varying
envelope and a train of carrier shocks.
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 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 Schr\"odinger equations, the extended
nonlinear Schr\"odinger system, xNLS. We embed xNLS in a one-parameter family
of equations, xNLS$^\epsilon$, which interpolates between infinitely many
decoupled NLS equations ($\epsilon=0$) and xNLS ($\epsilon=1$). Using
bifurcation methods we show existence of solutions for a range of
$\epsilon\in(-\epsilon_0,\epsilon_0)$ and, by a numerical continuation method,
establish the continuation of certain branches all the way to $\epsilon=1$.
Finally, we perform time-dependent simulations of truncated xNLCME and find the
small-amplitude solitons to be robust to both numerical errors and the NLS
approximation.
