Global Behavior Of Finite Energy Solutions To The dd-Dimensional Focusing Nonlinear Schr\"odinger Equation

We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t u+\Delta u+ |u|^{p-1}u=0,$ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with \ME[u_0]<1 (\ME[u_0] stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient \g_u of a solution $u$ to NLS is initially less than 1, i.e., \g_u(0)<1, then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr\"odinger evolution as $t\to\pm\infty$); if the renormalized gradient \g_u(0)>1, then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle.Comment: 57 pages, 4 figures and updated reference

Surface Waves and Forced Oscillations in QHE Planar Samples

Dispersion relations and polarizations for surface waves in infinite planar samples in the QHE regime are explicitly determined in the small wavevector limit in which the dielectric tensor can be considered as local. The wavelength and frequency regions of applicability of the results extends to the infrared region for typical experimental conditions. Then, standard samples with millimetric sizes seem to be able to support such excitations. Forced oscillations are also determined which should be generated in the 2DEG by external electromagnetic sources. They show an almost frequency independent wavevelength which decreases with the magnetic field. A qualitative model based in these solutions is also presented to describe a recently found new class of resonances appearing near the edge of a 2DEG in the QHE regime.Comment: latex file, 18 pages, 3 figures, spelling correcte

Dynamics of conduction blocks in a model of paced cardiac tissue

We study numerically the dynamics of conduction blocks using a detailed electrophysiological model. We find that this dynamics depends critically on the size of the paced region. Small pacing regions lead to stationary conduction blocks while larger pacing regions can lead to conduction blocks that travel periodically towards the pacing region. We show that this size-dependence dynamics can lead to a novel arrhythmogenic mechanism. Furthermore, we show that the essential phenomena can be captured in a much simpler coupled-map model.Comment: 8 pages 6 figure