61 research outputs found
Wave dynamics on networks: method and application to the sine-Gordon equation
We consider a scalar Hamiltonian nonlinear wave equation formulated on
networks; this is a non standard problem because these domains are not locally
homeomorphic to any subset of the Euclidean space. More precisely, we assume
each edge to be a 1D uniform line with end points identified with graph
vertices. The interface conditions at these vertices are introduced and
justified using conservation laws and an homothetic argument. We present a
detailed methodology based on a symplectic finite difference scheme together
with a special treatment at the junctions to solve the problem and apply it to
the sine-Gordon equation. Numerical results on a simple graph containing four
loops show the performance of the scheme for kinks and breathers initial
conditions.Comment: 31 pages, 9 figures, 2 tables, 41 references. Other author's papers
can be downloaded at http://www.denys-dutykh.com
Berry phase in superconducting multiterminal quantum dots
We report on the study of the non-trivial Berry phase in superconducting
multiterminal quantum dots biased at commensurate voltages. Starting with the
time-periodic Bogoliubov-de Gennes equations, we obtain a tight binding model
in the Floquet space, and we solve these equations in the semiclassical limit.
We observe that the parameter space defined by the contact transparencies and
quartet phase splits into two components with a non-trivial Berry phase. We use
the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if
the quantum dot level sits at zero energy, then the Berry phase takes the
values or . We demonstrate that this non-trivial
Berry phase can be observed by tunneling spectroscopy in the Floquet spectra.
Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting
multiterminal quantum dots are shifted by half-a-period if . Our
numerical calculations based on Keldysh Green's functions show that this Berry
phase spectral shift can be observed from the quantum dot tunneling density of
states.Comment: 15 pages, 7 figures. Supplemental Material as ancillary file (3
pages, 5 figures), manuscript in final for
Stopping a reaction-diffusion front
We revisit the problem of pinning a reaction-diffusion front by a defect, in
particular by a reaction-free region. Using collective variables for the front
and numerical simulations, we compare the behaviors of a bistable and
monostable front. A bistable front can be pinned as confirmed by a pinning
criterion, the analysis of the time independant problem and simulations.
Conversely, a monostable front can never be pinned, it gives rise to a
secondary pulse past the defect and we calculate the time this pulse takes to
appear. These radically different behaviors of bistable and monostable fronts
raise issues for modelers in particular areas of biology, as for example, the
study of tumor growth in the presence of different tissues
Vaccination strategy on a geographic network
We considered a mathematical model describing the propagation of an epidemic
on a geographical network. The initial growth rate of the disease is the
maximal eigenvalue of the epidemic matrix formed by the susceptibles and the
graph Laplacian representing the mobility. We use matrix perturbation theory to
analyze the epidemic matrix and define a vaccination strategy, assuming the
vaccination reduces the susceptibles. When the mobility is small compared to
the local disease dynamics, it is best to vaccinate the vertex of least degree
and not vaccinate neighboring vertices. Then the epidemic grows on the vertex
corresponding to the largest eigenvalue. When the mobility is comparable to the
local disease dynamics, the most efficient strategy is to vaccinate the whole
network because the disease grows uniformly. However, if only a few vertices
can be vaccinated then which ones do we choose? We answer this question, and
show that it is most efficient to vaccinate along the eigenvector corresponding
to the largest eigenvalue of the Laplacian. We illustrate these general results
on a 7 vertex graph, a grid, and a realistic example of the french rail
network
Engineering the Floquet spectrum of superconducting multiterminal quantum dots
Here we present a theoretical investigation of the Floquet spectrum in
multiterminal quantum dot Josephson junctions biased with commensurate
voltages. We first draw an analogy between the electronic band theory and
superconductivity which enlightens the time-periodic dynamics of the Andreev
bound states. We then show that the equivalent of the Wannier-Stark ladders
observed in semiconducting superlattices via photocurrent measurements, appears
as specific peaks in the finite frequency current fluctuations of
superconducting multiterminal quantum dots. In order to probe the
Floquet-Wannier-Stark ladder spectra, we have developed an analytical model
relying on the sharpness of the resonances. The charge-charge correlation
function is obtained as a factorized form of the Floquet wave-function on the
dot and the superconducting reservoir populations. We confirm these findings by
Keldysh Green's function calculations, in particular regarding the voltage and
frequency dependence of the resonance peaks in the current-current
correlations. Our results open up a road-map to quantum correlations and
coherence in the Floquet dynamics of superconducting devices.Comment: 13 pages, 7 figures, Supplemental Material as ancillary file (7
pages), revised manuscript, Physical Review Editors' suggestio
Stability analysis of static solutions in a Josephson junction
We present all the possible solutions of a Josephson junction with bias
current and magnetic field with both inline and overlap geometry, and examine
their stability. We follow the bifurcation of new solutions as we increase the
junction length. The analytical results, in terms of elliptic functions in the
case of inline geometry, are in agreement with the numerical calculations and
explain the strong hysteretic phenomena typically seen in the calculation of
the maximum tunneling current. This suggests a different experimental approach
based on the use, instead of the external magnetic field the modulus of the
elliptic function or the related quantity the total magnetic flux to avoid
hysteretic behavior and unfold the overlapping curves.Comment: 36 pages with 17 figure
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