Epidemic model on a network: Analysis and applications to COVID-19.

We analyze an epidemic model on a network consisting of susceptible-infected-recovered equations at the nodes coupled by diffusion using a graph Laplacian. We introduce an epidemic criterion and examine different isolation strategies: we prove that it is most effective to isolate a node of highest degree. The model is also useful to evaluate deconfinement scenarios and prevent a so-called second wave. The model has few parameters enabling fitting to the data and the essential ingredient of importation of infected; these features are particularly important for the current COVID-19 epidemic.

Radial sine-Gordon kinks as sources of fast breathers.

We consider radial sine-Gordon kinks in two, three, and higher dimensions. A full two-dimensional simulation showing that azimuthal perturbations remain small allows us to reduce the problem to the one-dimensional radial sine-Gordon equation. We solve this equation on an interval [r(0),r(1)] and absorb all outgoing radiation. As the kink shrinks toward r(0), before the collision, its motion is well described by a simple law derived from the conservation of energy. In two dimensions for r(0)≤2, the collision disintegrates the kink into a fast breather, while for r(0)≥4 we obtain a kink-breather metastable state where breathers are shed at each kink "return." In three and higher dimensions d, an additional kink-oscillon state appears for small r(0). On the application side, the kink disintegration opens the way for new types of terahertz microwave generators.

Fourier-mode dynamics for the nonlinear Schrödinger equation in one-dimensional bounded domains.

We analyze the 1D focusing nonlinear Schrödinger equation in a finite interval with homogeneous Dirichlet or Neumann boundary conditions. There are two main dynamics, the collapse which is very fast and a slow cascade of Fourier modes. For the cubic nonlinearity the calculations show no long-term energy exchange between Fourier modes as opposed to higher nonlinearities. This slow dynamics is explained by fairly simple amplitude equations for the resonant Fourier modes. Their solutions are well behaved so filtering high frequencies prevents collapse. Finally, these equations elucidate the unique role of the zero mode for the Neumann boundary conditions.

Stimulated Raman scattering with strong damping: a simple theory of the spike phenomenon.

The classical stimulated Raman scattering system describing the resonant interaction between two electromagnetic waves and a fast relaxing medium wave is studied by introducing a systematic perturbation approach in powers of the relaxation time. We separate amplitude and phase effects for these complex fields. The analysis of the former shows the existence of a stagnation distance after which monotonic energy transfer begins from one electromagnetic wave to the other, and this quantity is calculated. Concerning phase effects we give the conditions for the formation of a Raman spike from an initial fast and large phase jump in one of the waves. The spike evolution and width estimated from the reduced model agree with the results from numerical simulations of the original system.