Search

We consider the stability of bound-state solutions of a family of regularized nonlinear Schr\"odinger equations which were introduced by Dumas, Lannes and Szeftel as models for the propagation of laser beams. Among these bound-state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schr\"odinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states., Comment: 39 pages

We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state., Comment: 38 pages. Appendices D and E are adapted from arXiv:2011.04570. To appear in Communications in Partial Differential Equations

For Schroedinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside the ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness., Comment: Minor changes to the presentation and the proof of Theorem 2.1

We consider the mean-field dynamics of Bose-Einstein condensates in rotating harmonic traps and establish several stability and instability properties for the corresponding solution. We particularly emphasize the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case., Comment: 13 pages; corrected a systematic typo

We study a class of focusing nonlinear Schroedinger-type equations derived recently by Dumas, Lannes and Szeftel within the mathematical description of high intensity laser beams [7]. These equations incorporate the possibility of a (partial) off-axis variation of the group velocity of such laser beams through a second order partial differential operator acting in some, but not necessarily all, spatial directions. We study the well-posedness theory for such models and obtain a regularizing effect, even in the case of only partial off-axis dependence. This provides an answer to an open problem posed in [7]., Comment: 19 pages. Revised version

We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point., Comment: 19 pages. Updated references and minor revisions

We report a novel radio autocorrelation (AC) search for extraterrestrial intelligence (SETI). For selected frequencies across the terrestrial microwave window (1-10 GHz) observations were conducted at the Allen Telescope Array to identify artificial non-sinusoidal periodic signals with radio bandwidths greater than 4 Hz, which are capable of carrying substantial messages with symbol-rates from 4-1000000 Hz. Out of 243 observations, about half (101) were directed toward sources with known continuum flux > ~1 Jy over the sampled bandwidth (quasars, pulsars, supernova remnants, and masers), based on the hypothesis that they might harbor heretofore undiscovered natural or artificial, repetitive, phase or frequency modulation. The rest of the targets were mostly toward exoplanet stars with no previously discovered continuum flux. No signals attributable to extraterrestrial technology were found in this study. We conclude that the maximum probability that future observations like the ones described here will reveal repetitively modulated emissions is less than 1% for continuum sources and exoplanets, alike. The paper concludes by describing a new approach to expanding this survey to many more targets and much greater sensitivity using archived data from interferometers all over the world., Comment: 33 pages, 4 figures, 3 tables