65 research outputs found
A Bloch wave numerical scheme for scattering problems in periodic wave-guides
We present a new numerical scheme to solve the Helmholtz equation in a
wave-guide. We consider a medium that is bounded in the -direction,
unbounded in the -direction and -periodic for large ,
allowing different media on the left and on the right. We suggest a new
numerical method that is based on a truncation of the domain and the use of
Bloch wave ansatz functions in radiation boxes. We prove the existence and a
stability estimate for the infinite dimensional version of the proposed
problem. The scheme is tested on several interfaces of homogeneous and periodic
media and it is used to investigate the effect of negative refraction at the
interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure
Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation
We rigorously analyze the bifurcation of stationary so called nonlinear Bloch
waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a
periodic potential, in arbitrary space dimensions. These are solutions which
can be expressed as finite sums of quasi-periodic functions, and which in a
formal asymptotic expansion are obtained from solutions of the so called
algebraic coupled mode equations. Here we justify this expansion by proving the
existence of NLBs and estimating the error of the formal asymptotics. The
analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In
addition, we illustrate some relations of NLBs to other classes of solutions of
the GP equation, in particular to so called out--of--gap solitons and truncated
NLBs, and present some numerical experiments concerning the stability of these
solutions.Comment: 32 pages, 12 figures, changes: discussion of assumptions reorganized,
a new section on stability of the studied solutions, 15 new references adde
Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation
We consider the existence of localized modes corresponding to eigenvalues of
the periodic Schr\"{o}dinger operator with an interface.
The interface is modeled by a jump either in the value or the derivative of
and, in general, does not correspond to a localized perturbation of the
perfectly periodic operator. The periodic potentials on each side of the
interface can, moreover, be different. As we show, eigenvalues can only occur
in spectral gaps. We pose the eigenvalue problem as a gluing problem for
the fundamental solutions (Bloch functions) of the second order ODEs on each
side of the interface. The problem is thus reduced to finding matchings of the
ratio functions , where are
those Bloch functions that decay on the respective half-lines. These ratio
functions are analyzed with the help of the Pr\"{u}fer transformation. The
limit values of at band edges depend on the ordering of Dirichlet and
Neumann eigenvalues at gap edges. We show that the ordering can be determined
in the first two gaps via variational analysis for potentials satisfying
certain monotonicity conditions. Numerical computations of interface
eigenvalues are presented to corroborate the analysis.Comment: 1. finiteness of the number of additive interface eigenvalues proved
in a remark below Corollary 3.6.; 2. small modifications and typo correction
Eigenvalue Bifurcation in Doubly Nonlinear Problems with an Application to Surface Plasmon Polaritons
We consider a class of generally non-self-adjoint eigenvalue problems which
are nonlinear in the solution as well as in the eigenvalue parameter ("doubly"
nonlinear). We prove a bifurcation result from simple isolated eigenvalues of
the linear problem using a Lyapunov-Schmidt reduction and provide an expansion
of both the nonlinear eigenvalue and the solution. We further prove that if the
linear eigenvalue is real and the nonlinear problem -symmetric, then the bifurcating nonlinear eigenvalue remains real. These
general results are then applied in the context of surface plasmon polaritons
(SPPs), i.e. localized solutions for the nonlinear Maxwell's equations in the
presence of one or more interfaces between dielectric and metal layers. We
obtain the existence of transverse electric SPPs in certain -symmetric configurations.Comment: Minor corrections in accordance to the referees' suggestion
- …