21 research outputs found
Waves in Honeycomb Structures
We review recent work of the authors on the non-relativistic Schr\"odinger
equation with a honeycomb lattice potential, . In particular, we summarize
results on (i) the existence of Dirac points, conical singularities in
dispersion surfaces of and (ii) the two-dimensional Dirac
equations, as a large, but finite time, effective description of
, for data , which is spectrally localized at a Dirac
point. We conclude with a formal derivation and discussion of the effective
large time evolution for the nonlinear Schr\"odinger - Gross Pitaevskii
equation for small amplitude initial conditions, . The effective
dynamics are governed by a nonlinear Dirac system.Comment: 11 pages, 2 figures, 39 \`emes Journ\'ees EDP - Biarretz. arXiv admin
note: text overlap with arXiv:1212.607
Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime
We apply the two-scale formulation approach to propose uniformly accurate
(UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic
limit regime. The nonlinear Dirac equation involves two small scales
and with in the nonrelativistic
limit regime. The small parameter causes high oscillations in time which brings
severe numerical burden for classical numerical methods. We transform our
original problem as a two-scale formulation and present a general strategy to
tackle a class of highly oscillatory problems involving the two small scales
and . Suitable initial data for the two-scale
formulation is derived to bound the time derivatives of the augmented solution.
Numerical schemes with uniform (with respect to )
spectral accuracy in space and uniform first order or second order accuracy in
time are proposed. Numerical experiments are done to confirm the UA property.Comment: 22 pages, 6 figures. To appear on Communications in Mathematical
Science
Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations
In a recent article [10], the authors proved that the non-relativistic
Schr\"odinger operator with a generic honeycomb lattice potential has conical
(Dirac) points in its dispersion surfaces. These conical points occur for
quasi-momenta, which are located at the vertices of the Brillouin zone, a
regular hexagon. In this paper, we study the time-evolution of wave-packets,
which are spectrally concentrated near such conical points. We prove that the
large, but finite, time dynamics is governed by the two-dimensional Dirac
equations.Comment: 34 pages, 2 figure
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
We propose and rigourously analyze a multiscale time integrator Fourier
pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless
parameter which is inversely proportional to the speed of
light. In the nonrelativistic limit regime, i.e. , the
solution exhibits highly oscillatory propagating waves with wavelength
and in time and space, respectively. Due to the rapid
temporal oscillation, it is quite challenging in designing and analyzing
numerical methods with uniform error bounds in . We
present the MTI-FP method based on properly adopting a multiscale decomposition
of the solution of the Dirac equation and applying the exponential wave
integrator with appropriate numerical quadratures. By a careful study of the
error propagation and using the energy method, we establish two independent
error estimates via two different mathematical approaches as
and ,
where is the mesh size, is the time step and depends on the
regularity of the solution. These two error bounds immediately imply that the
MTI-FP method converges uniformly and optimally in space with exponential
convergence rate if the solution is smooth, and uniformly in time with linear
convergence rate at for all and optimally with
quadratic convergence rate at in the regimes when either
or . Numerical results are
reported to demonstrate that our error estimates are optimal and sharp.
Finally, the MTI-FP method is applied to study numerically the convergence
rates of the solution of the Dirac equation to those of its limiting models
when .Comment: 25 pages, 1 figur
Spatial algebraic solitons at the Dirac point in optically induced nonlinear photonic lattices
The discovery of a new type of soliton occurring in periodic systems is reported. This type of nonlinear excitation exists at a Dirac point of a photonic band structure, and features an oscillating tail that damps algebraically. Solitons in periodic systems are localized states traditionally supported by photonic bandgaps. Here, it is found that besides photonic bandgaps, a Dirac point in the band structure of triangular optical lattices can also sustain solitons. Apart from their theoretical impact within the soliton theory, they have many potential uses because such solitons are possible in both Kerr material and photorefractive crystals that possess self-focusing and self-defocusing nonlinearities. The findings enrich the soliton family and provide information for studies of nonlinear waves in many branches of physics
Double Conical degeneracy on the band structure of periodic Schr\"odinger operators
Dirac cones are conical singularities that occur near the degenerate points
in band structures. Such singularities result in enormous unusual phenomena of
the corresponding physical systems. This work investigates double Dirac cones
that occur in the vicinity of a fourfold degenerate point in the band
structures of certain operators. It is known that such degeneracy originates in
the symmetries of the Hamiltonian. We use two dimensional periodic
Schr\"odinger operators with novel designed symmetries as our prototype. First,
we characterize admissible potentials, termed as super honeycomb lattice
potentials. They are honeycomb lattices potentials with a key additional
translation symmetry. It is rigorously justified that Schr\"odinger operators
with such potentials almost guarantee the existence of double Dirac cones on
the bands at the {\Gamma} point, the origin of the Brillouin zone. We further
show that the additional translation symmetry is an indispensable ingredient by
a perturbation analysis. Indeed, the double cones disappear if the additional
translation symmetry is broken. Many numerical simulations are provided, which
agree well with our analysis