Convergence to Scattering States in the Nonlinear Schr\"odinger Equation

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$ and \linebreak $u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-it\Delta}u(t)-u_ \pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{it\Delta}u_\pm\to0$ in $X$ as $t\to\pm\infty.$ We also study the converse. Finally, we estimate $|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:|$ under some less restrictive assumptions

Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation

We study the local well-posedness of the initial-value problem for the nonlinear "good" Boussinesq equation with data in Sobolev spaces \textit{$H^s$} for negative indices of $s$.Comment: Referee comments incorporate

On the Cauchy Problem for the Zakharov System

Abstract We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data ( u (0), n (0), ∂ t n (0))∈ H k ⊕ H lscr; ⊕ H l−1 , in arbitrary space dimension ν . We define a natural notion of criticality according to which the critical values of ( k , l) are ( ν /2−3/2, ν /2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of ( k , l). The results cover the whole subcritical range for ν ⩾4. For ν ⩽3, they cover only part of it and the lowest admissible values are ( k , l)=(1/2, 0) for ν =2, 3 and ( k , l)=(0, −1/2) for ν =1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of ( k , l)

Edge Current due to Majorana Fermions in Superfluid 3^3He A- and B-Phases

We propose a method utilizing edge current to observe Majorana fermions in the surface Andreev bound state for the superfluid $^3$He A- and B-phases. The proposal is based on self-consistent analytic solutions of quasi-classical Green's function with an edge. The local density of states and edge mass current in the A-phase or edge spin current in the B-phase can be obtained from these solutions. The edge current carried by the Majorana fermions is partially cancelled by quasiparticles (QPs) in the continuum state outside the superfluid gap. QPs contributing to the edge current in the continuum state are distributed in energy even away from the superfluid gap. The effect of Majorana fermions emerges in the depletion of the edge current by temperature within a low-temperature range. The observations that the reduction in the mass current is changed by $T^2$-power in the A-phase and the reduction in the spin current is changed by $T^3$-power in the B-phase establish the existence of Majorana fermions. We also point out another possibility for observing Majorana fermions by controlling surface roughness.Comment: 13 pages, 4 figures, published versio

The Variability of Sagittarius A* at Centimeter Wavelengths

We present the results of a 3.3-year project to monitor the flux density of Sagittarius A* at 2.0, 1.3, and 0.7 cm with the VLA. The fully calibrated light curves for Sgr A* at all three wavelengths are presented. Typical errors in the flux density are 6.1%, 6.2%, and 9.2% at 2.0, 1.3, and 0.7 cm, respectively. There is preliminary evidence for a bimodal distribution of flux densities, which may indicate the existence of two distinct states of accretion onto the supermassive black hole. At 1.3 and 0.7 cm, there is a tail in the distribution towards high flux densities. Significant variability is detected at all three wavelengths, with the largest amplitude variations occurring at 0.7 cm. The rms deviation of the flux density of Sgr A* is 0.13, 0.16, and 0.21 Jy at 2.0, 1.3, and 0.7 cm, respectively. During much of this monitoring campaign, Sgr A* appeared to be relatively quiescent compared to results from previous campaigns. At no point during the monitoring campaign did the flux density of Sgr A* more than double its mean value. The mean spectral index of Sgr A* is alpha=0.20+/-0.01, with a standard deviation of 0.14. The spectral index appears to depend linearly on the observed flux density at 0.7 cm with a steeper index observed during outbursts. This correlation is consistent with the expectation for outbursts that are self-absorbed at wavelengths of 0.7 cm or longer and inconsistent with the effects of simple models for interstellar scintillation. Much of the variability of Sgr A*, including possible time lags between flux density changes at the different wavelengths, appears to occur on time scales less than the time resolution of our observations (8 days). Future observations should focus on the evolution of the flux density on these time scales.Comment: 16 pages, 10 figures, accepted for publication in A

Successive transition from superconducting to antiferromagnetic phase in (Ca_6(Al, Ti)_4O_y)Fe_2As_2 studied via ^{75}As and ^{27}Al NMR

An unusual successive phase transition from superconducting (SC) to antiferromagnetic (AF) phases was discovered via ^{75}As and ^{27}Al nuclear magnetic resonance (NMR) in (Fe_2As_2)(Ca_6(Al, Ti)_4O_y) with four (Al, Ti)O layers intercalated between FeAs planes. Although the spatially-uniform AF ordering is clearly visible from ^{27}Al spectra, the ordered moments are very small and the low-frequency fluctuation is much suppressed, contrary to existing pnictides with localized magnetic elements. Furthermore, the temperature (T) dependence of the fluctuation at both nuclei is very similar throughout the entire temperature range. These facts suggest that some hybridization between Ti and Fe orbitals induces a uniform electronic state within FeAs and (Al, Ti)O layers accompanied by the SC and AF transitions. The iron-based pnictide with Ti-doped blocking layers is the first high-T_c compound having metallic blocking layers

Global Existence and Uniqueness of Solutions to the Maxwell-Schr{\"o}dinger Equations

The time local and global well-posedness for the Maxwell-Schr{\"o}dinger equations is considered in Sobolev spaces in three spatial dimensions. The Strichartz estimates of Koch and Tzvetkov type are used for obtaining the solutions in the Sobolev spaces of low regularities. One of the main results is that the solutions exist time globally for large data.Comment: 30 pages. In the revised version, the following modification was made. (1) A line for dedication was added in the first page. (2) Some lines were added at the bottom in page 4 and the top in page 5 in the first section to make the description accurate. (3) Some typographical errors were corrected throughout the pape

Acoustic radiation controls friction: Evidence from a spring-block experiment

Brittle failures of materials and earthquakes generate acoustic/seismic waves which lead to radiation damping feedbacks that should be introduced in the dynamical equations of crack motion. We present direct experimental evidence of the importance of this feedback on the acoustic noise spectrum of well-controlled spring-block sliding experiments performed on a variety of smooth surfaces. The full noise spectrum is quantitatively explained by a simple noisy harmonic oscillator equation with a radiation damping force proportional to the derivative of the acceleration, added to a standard viscous term.Comment: 4 pages including 3 figures. Replaced with version accepted in PR