Numerical aspects of nonlinear Schrodinger equations in the presence of caustics

The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schrodinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proven rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed.Comment: 20 pages. To appear in Math. Mod. Meth. Appl. Sc

Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Lévy-driven SDEs. We construct multivariate infinitesimally over dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.Continuous time, Counting Markov process, Birth-death process, Environmental stochasticity, Infinitesimal over-dispersion, Simultaneous events

Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential

We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical regimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result

Geometric optics and instability for semi-classical Schrodinger equations

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrodinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the problem cannot be reduced to the study of an o.d.e.Comment: 22 pages. Corollary 1.7 adde

C ion-implanted TiO2 thin film for photocatalytic applications

Third-generation TiO2 photocatalysts were prepared by implantation of C+ ions into 110 nm thick TiO2 films. An accurate structural investigation was performed by Rutherford backscattering spectrometry, secondary ion mass spectrometry, X-ray diffraction, Raman-luminescence spectroscopy, and UV/VIS optical characterization. The C doping locally modified the TiO2 pure films, lowering the band-gap energy from 3.3 eV to a value of 1.8 eV, making the material sensitive to visible light. The synthesized materials are photocatalytically active in the degradation of organic compounds in water under both UV and visible light irradiation, without the help of any additional thermal treatment. These results increase the understanding of the C-doped titanium dioxide, helpful for future environmental applications. (C) 2015 AIP Publishing LLC