29,191 research outputs found
Parabolic Metamaterials and Dirac Bridges
A new class of multi-scale structures, referred to as `parabolic
metamaterials' is introduced and studied in this paper. For an elastic
two-dimensional triangular lattice, we identify dynamic regimes, which
corresponds to so-called `Dirac Bridges' on the dispersion surfaces. Such
regimes lead to a highly localised and focussed unidirectional beam when the
lattice is excited. We also show that the flexural rigidities of elastic
ligaments are essential in establishing the `parabolic metamaterial' regimes.Comment: 14 pages, 4 figure
Purcell Enhancement of Parametric Luminescence: Bright and Broadband Nonlinear Light Emission in Metamaterials
Single-photon and correlated two-photon sources are important elements for
optical information systems. Nonlinear downconversion light sources are robust
and stable emitters of single photons and entangled photon pairs. However, the
rate of downconverted light emission, dictated by the properties of
low-symmetry nonlinear crystals, is typically very small, leading to
significant constrains in device design and integration. In this paper, we show
that the principles for spontaneous emission control (i.e. Purcell effect) of
isolated emitters in nanoscale structures, such as metamaterials, can be
generalized to describe the enhancement of nonlinear light generation processes
such as parametric down conversion. We develop a novel theoretical framework
for quantum nonlinear emission in a general anisotropic, dispersive and lossy
media. We further find that spontaneous parametric downconversion in media with
hyperbolic dispersion is broadband and phase-mismatch-free. We predict a
1000-fold enhancement of the downconverted emission rate with up to 105 photon
pairs per second in experimentally realistic nanostructures. Our theoretical
formalism and approach to Purcell enhancement of nonlinear optical processes,
provides a framework for description of quantum nonlinear optical phenomena in
complex nanophotonic structures.Comment: 29 pages, 10 figure
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
The fully nonlinear and weakly dispersive Green-Naghdi model for shallow
water waves of large amplitude is studied. The original model is first recast
under a new formulation more suitable for numerical resolution. An hybrid
finite volume and finite difference splitting approach is then proposed. The
hyperbolic part of the equations is handled with a high-order finite volume
scheme allowing for breaking waves and dry areas. The dispersive part is
treated with a classical finite difference approach. Extensive numerical
validations are then performed in one horizontal dimension, relying both on
analytical solutions and experimental data. The results show that our approach
gives a good account of all the processes of wave transformation in coastal
areas: shoaling, wave breaking and run-up
A unified hyperbolic formulation for viscous fluids and elastoplastic solids
We discuss a unified flow theory which in a single system of hyperbolic
partial differential equations (PDEs) can describe the two main branches of
continuum mechanics, fluid dynamics, and solid dynamics. The fundamental
difference from the classical continuum models, such as the Navier-Stokes for
example, is that the finite length scale of the continuum particles is not
ignored but kept in the model in order to semi-explicitly describe the essence
of any flows, that is the process of continuum particles rearrangements. To
allow the continuum particle rearrangements, we admit the deformability of
particle which is described by the distortion field. The ability of media to
flow is characterized by the strain dissipation time which is a characteristic
time necessary for a continuum particle to rearrange with one of its
neighboring particles. It is shown that the continuum particle length scale is
intimately connected with the dissipation time. The governing equations are
represented by a system of first order hyperbolic PDEs with source terms
modeling the dissipation due to particle rearrangements. Numerical examples
justifying the reliability of the proposed approach are demonstrated.Comment: 6 figure
Polarized high-frequency wave propagation beyond the nonlinear Schrödinger approximation
This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes KleinâGordon equations and the MaxwellâLorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear SchroÌdinger equation. The given construction of polarized solutions is explicit, uses in addition a linear SchroÌdinger equation for each further order of approximation, and is accessible to direct numerical approximation
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