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 $\varphi_B=0$ or $\varphi_B=\pi$. 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 $\varphi_B=\pi$. 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 $I_{max}(H)$ curves.Comment: 36 pages with 17 figure