Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

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 mode equations, xNLCME. Numerical studies support the existence of longlived spatially localized coherent structures, featuring a slowly varying envelope and 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 Schrödinger equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS , which interpolates between infinitely many decoupled NLS equations ( = 0) and xNLS ( = 1). Using bifurcation methods we show existence of solutions for a range of ∈ (− 0 , 0 ) and, by a numerical continuation method, establish the continuation of certain branches all the way to = 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.

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

“…These results are shown in Tables 1-3. From these tables, we clearly see that the error values are close to the theoretical values for = 0.1, while for a larger value of = 0.5, they tend toward the T A B L E 3 Maximum absolute error scaling parameters for the weakly-nonlinear solution including ( ) terms for the initial condition in (57). The domain lengths and parameters are = = = 1, = 40, and = 1∕ theoretical values with the increase of d. We also confirm that as increases, the values of 1 and 3 tend to the next largest integer value.…”

“…Rigorous estimates for the error terms were obtained, similarly to Ref. , and convergence rates predicted by this derivation were illustrated by partial numerical experimentation (restricted by the complicated formulas for the higher‐order corrections given in terms of Fourier coefficients).…”

“…Maximum absolute error scaling parameters for the weakly-nonlinear solution including ( √ ) terms for the initial condition in(57). The domain lengths and parameters are = = = 2, = 40, and = 1∕…”

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

“…with the same definition for f ± s as in (33). The closed form expressions for the initial conditions in terms of the leading order solutions f ± can be found in [18,19].…”

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

“…Here we explore a more general reduction to the NLS equation, which is also valid for perturbations of the solitary wave. Justification of these reductions to the NLS equation (in a more complex setting of infinitely many coupled NLS equations) can be found in the recent work [26].…”