90 research outputs found
Surface gap solitons at a nonlinearity interface
We demonstrate existence of waves localized at the interface of two nonlinear
periodic media with different coefficients of the cubic nonlinearity via the
one-dimensional Gross--Pitaevsky equation. We call these waves the surface gap
solitons (SGS). In the case of smooth symmetric periodic potentials, we study
analytically bifurcations of SGS's from standard gap solitons and determine
numerically the maximal jump of the nonlinearity coefficient allowing for the
SGS existence. We show that the maximal jump vanishes near the thresholds of
bifurcations of gap solitons. In the case of continuous potentials with a jump
in the first derivative at the interface, we develop a homotopy method of
continuation of SGS families from the solution obtained via gluing of parts of
the standard gap solitons and study existence of SGS's in the photonic band
gaps. We explain the termination of the SGS families in the interior points of
the band gaps from the bifurcation of linear bound states in the continuous
non-smooth potentials.Comment: 23 pages, 6 figures, 3 tables corrections in v.2: sign error in the
energy functional on p.3; discussion of the symmetries of Bloch functions on
p. 5-6 corrected; derivative symbol missing in (3.5) and in the formula for
\mu below (3.6
Dispersive homogenized models and coefficient formulas for waves in general periodic media
We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix that is periodic with
characteristic length scale ; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
well by a non-dispersive wave equation on fixed time intervals
. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions
describe well on time intervals . More
precisely, we provide a norm and uniform error estimates of the form for . They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
-independent equation of third order that describes dispersion
along rays and we present numerical examples.Comment: 28 pages, 7 figure
pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual
pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path
for elliptic systems of PDEs over bounded 2D domains, based on the Matlab
pdetoolbox. The new features include a more efficient use of FEM, easier
switching between different single parameter continuations, genuine
multi-parameter continuation (e.g., fold continuation), more efficient
implementation of nonlinear boundary conditions, cylinder and torus geometries
(i.e., periodic boundary conditions), and a general interface for adding
auxiliary equations like mass conservation or phase equations for continuation
of traveling waves. The package (library, demos, manuals) can be downloaded at
www.staff.uni-oldenburg.de/hannes.uecker/pde2pat
Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media
Traveling modulating pulse solutions consist of a small amplitude pulse-like
envelope moving with a constant speed and modulating a harmonic carrier wave.
Such solutions can be approximated by solitons of an effective nonlinear
Schrodinger equation arising as the envelope equation. We are interested in a
rigorous existence proof of such solutions for a nonlinear wave equation with
spatially periodic coefficients. Such solutions are quasi-periodic in a
reference frame co-moving with the envelope. We use spatial dynamics, invariant
manifolds, and near-identity transformations to construct such solutions on
large domains in time and space. Although the spectrum of the linearized
equations in the spatial dynamics formulation contains infinitely many
eigenvalues on the imaginary axis or in the worst case the complete imaginary
axis, a small denominator problem is avoided when the solutions are localized
on a finite spatial domain with small tails in far fields.Comment: 30 pages; 5 figures
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