Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

Emergence of a non-Fowler-Nordheim-type behavior for a general planar tunneling barrier

In this work we investigate a generalized tunneling barrier for planar emitters at zero-temperature. We present the evidence of the emergence of a non-Fowler-Nordheim-type general behavior for the field emission current density in the case that the Fermi energy (μ) is comparable with or smaller that the decay width ($d_F$). Therefore, for some non-metals or materials that have very small Fermi energy the standard Fowler-Nordheim-type theory may require a correction. In the opposite regime, i.e., for μ much larger that $d_F$, we confirm that the conventional theory is suitable for metals.     

Critical behavior of a heavy particle in a granular fluid

A second order phase transition is observed for the homogeneous cooling state of a heavy impurity particle in a granular fluid. The order parameter straight phi is the ratio of impurity mean square velocity to that of the fluid, with a conjugate field h proportional to the mass ratio. A parameter beta, measuring the fluid cooling rate relative to the impurity-fluid collision rate, is the analog of the inverse temperature. For beta<1 the fluid is "normal" with straight phi = 0 at h = 0, as in the case of elastic collisions. For beta>1 an "ordered" state with straight phi not equal0 occurs at h = 0, representing an extreme breakdown of equipartition. Critical slowing and qualitative changes in the velocity distribution function near the transition are noted.     

Exciton optics of a disordered semiconductor

Interband light absorption in a disordered semiconductor is considered with due allowance for exciton effects. Allowance is made for the scattering of the electron and hole in a weak static random field and also for collisions of the second kind experienced by the exciton on interaction with a random field. It is shown that the contribution of these latter to the half width of the absorption line of the exciton spectrum (even at temperatures of the order of the exciton ionization energy) is considerably smaller than the contribution arising from the scattering of the particles forming the exciton.     

Surface geometry and shadow of a Schwarzschild black hole with a halo

We studied the surface geometry and shadows of a Schwarzschild black hole with a halo containing quadrupolar and octopolar terms. It was found that the quadrupole term causes a Schwarzschild black hole to be prolate at the quadrupole strength \begin{document}$ {\cal{Q}}<0 $\end{document} and oblate at \begin{document}$ {\cal{Q}}>0 $\end{document}, and the octopole term causes to the shadow stretch upward at the octopolar strength ngth \begin{document}$ {\cal{O}}<0 $\end{document} and downward at \begin{document}$ {\cal{O}}>0 $\end{document}. The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal direction at \begin{document}$ {\cal{Q}}<0 $\end{document} and \begin{document}$ {\cal{Q}}>0 $\end{document}, respectively. Meanwhile, the black hole shadow shifts upward at \begin{document}$ {\cal{O}}<0 $\end{document} and downward at \begin{document}$ {\cal{O}}>0 $\end{document}. We present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms. From the observable width W, height H, oblateness K, and distortion parameter \begin{document}$ \delta_{c} $\end{document} of the black hole shadow, we can determine the quadrupole and octopolar strengths of a Schwarzschild black hole with a halo. Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo.     

Levitation of a rapidly oscillating magnetic dipole above a metallic sheet

We show that such a magnetic dipole suspended at a height h above a conducting sheet experiences a lift force proportional to $1/h^2$. This represents an order of magnitude improvement over the $1/h^4$ lift obtained in the quasistatic limit.     

Convergence of the cluster-variation method for a system on a triangular lattice

The paper studies the convergence of the cluster-variation method to the rigorous result as the cluster size increases. The calculation is done on the phase boundary at T = 0 between the A₂B-type ordered phase and the disordered phase on a two-dimensional triangular lattice with nearest-neighbor interaction. It is shown that the phase boundary at (T = 0) is obtained by maximizing the entropy under the constraint that only a limited number of atomic configurations are allowed. Formulations are developed for clusters of n = 3, 5, 7, 9,11, and 13 points. When thermodynamic quantities which are calculated using these clusters are plotted against 1/n, they approach the known rigorous (n = ∞) results more or less linearly but with a pseudo-period of δn = 6. An exception is the square of the long-range order, which bends down as 1/n tends to zero.     

Effect of channel-protein of a protein-like chain interaction on translocation through a finite channel

We study the translocation of a protein-like chain through a finite cylindrical channel using the pruned-enriched Rosenbluth method(PERM) and the modified orientation-dependent monomer-monomer interaction(ODI) model.Attractive channels(ε cp = 2.0,1.0,0.5),repulsive channels(ε cp = 0.5,1.0,2.0),and a neutral channel(ε cp = 0) are discussed.The results of the chain dimension and the energy show that Z0 = 1.0 is an important case to distinguish the types of the channels.For the strong attractive channel,more contacts form during the process of translocation.It is also found that an external force is needed to drive the chain outside of the channel with the strong attraction.While for the neutral,the repulsive,and the weak attractive channels,the translocation is spontaneous.     

Stability of the ground state of a harmonic oscillator in a monochromatic wave

The stability of the ground state of a harmonic oscillator in a monochromatic wave is studied. This model describes, in particular, the dynamics of a cold ion in a linear ion trap, interacting with two laser fields with close frequencies. The stability of the "classical ground state"-the vicinity of the point (x=0,p=0)-is analyzed analytically and numerically. For the quantum case, a method for studying a stability of the quantum ground state is developed, based on the quasienergy representation. It is demonstrated that stability of the ground state may be substantially improved by increasing the resonance number, l, where l=Omega/omega+delta, Omega and omega are, respectively, the wave frequency and the oscillator frequency, l=1,2, em leader, mid R:deltamid R:<1; or by detuning the system from exact resonance, so that delta not equal 0. The influence of a large-amplitude wave (in the presence of chaos) on the stability of the ground state is analyzed for different parameters of the model in both the quantum and classical cases. (c) 2001 American Institute of Physics.     

Kinetics of a Migration-Driven Aggregation-Fragmentation Process

We propose a reversible model of the migration-driven aggregation-fragmentation process with the sym-metric migration rate kernels K(k;j) = K‘(k;j) = λkjv and the constant aggregation rates I1, I2 and fragmentationrates J1, J2. Based on the mean-field theory, we investigate the evolution behavior of the aggregate size distributions inseveral cases with different values of index v. We find that the fragmentation reaction plays a more important role in the kinetic behaviors of the system than the aggregation and migration. When J1 = 0 and J2 = 0, the aggregate sizedistributions ak(t) and bk(t) obey the conventional scaling law, while when J1 ＞ 0 and J2 ＞ 0, they obey the modifiedscaling law with an exponential scaling function. The total mass of either species remains conserved.     

Measurement of neutron fluence with CR-39 using a UV spectrophotometer

Measurement of the correct neutron fluence in various energy intervals in and around the neutron sources is important for the purpose of personnel and environmental neutron dosimetry. In the present paper we present a method for the measurement of neutron fluence using a CR-39 plastic track detector. The samples exposed to fission neutrons from a reactor with 10¹²– fluence and to fast neutrons from a ²⁴¹Am–Be source with a fluence of were etched for 14 time intervals starting from 10 to 570 min in 6 N NaOH at . The etched samples were exposed to the monochromatic light signal from an iodine–tungsten lamp and the transmittance was measured after each etching time interval using a UV Spectrophotometer. An exponential decay of the transmittance has been observed with increasing etching time. The behaviour of the fission neutrons having four different values of fluence has been observed to be distinctly different from that of the fast neutrons. It has also been observed that there is a linear relationship between the transmittance decay constant and fission neutron fluence.     

The resonant mechanism of subdivision of a gas bubble in a fluid

Small nonlinear oscillations of a bubble in a fluid at the resonance of the frequencies of the radial mode and an arbitrary deformational mode 2 : 1 are considered. The deformational mode is determined by the associated Legendre polynomial with indices n = 2, 3, ??, m = 0, 1, ??, n. The energy transfer from the radial mode into the Legendre deformational mode is described by the method of invariant normalization. An analogy is established with oscillations of the material point on the string with the frequency ratio of the vertical mode to the horizontal one of 2. During the transfer, the amplitude of the Legendre mode with indices n, m exceeds the amplitude of radial oscillations by a factor of 3n at m = 0. As index m increases, the transfer time increases considerably and the maximal amplitude of the Legendre mode increases insignificantly in this case. From here, it is concluded that the deformational Legendre mode with indices n, m = n has the greatest probability to rise. The considered effect can serve as a mechanism of subdivision of gas bubbles under varying the external pressure in the fluid.     

Shape of a wave front in a heterogenous medium

Wave propagation in a heterogeneous medium, characterized by a distribution of local elastic moduli, is studied. Both acoustic and elastic waves are considered, as are spatially random and power-law correlated distributions of the elastic moduli with nondecaying correlations. Three models--a continuum scalar model, and two discrete models--are utilized. Numerical simulations indicate the existence, at all times, of the relation, alpha = H, where alpha is the roughness exponent of the wave front in the medium, and H is the Hurst exponent that characterizes the spatial correlations in the distribution of the local elastic moduli. Hence, a direct relation between the static morphology of an inhomogeneous correlated medium and its dynamical properties is established. In contrast, for a wave front in random media, alpha = 0 (logarithmic growth) at short times, followed by a crossover to the classical value, alpha = 1/2, at long times.     

Design, construct and test of a calibration radon chamber

A new radon chamber has been designed, constructed, and tested. The design was chosen to meet the needs of several different radon detection techniques, both active and passive. The chamber is a cubic shape made of pixy glass with volume about . On top, it has a circular opening with double lock system for entering and removing samples without significant disturbance of the inside concentration. It also has the possibility to expose passive radon detectors at different levels inside the chamber. In addition, two fans, with variable speeds, were fitted in two opposite corners to maintain the uniformity of radon gas inside the volume. This calibration chamber can be used to perform researches and studies on radon behavior, in addition to calibrating different types of detectors (passive and active) which are used for measuring radon and its daughters. Solid radium-226 source with activity of 122 kBq was placed at the bottom of the chamber to generate radon.     

Observation of a continuous phase transition in a shape-memory alloy

Elastic neutron-scattering, inelastic x-ray scattering, specific-heat, and pressure-dependent electrical transport measurements have been made on single crystals of AuZn and Au0.52Zn0.48. Elastic neutron scattering detects new commensurate Bragg peaks (modulation) appearing at Q =(1.33,0.67,0) at temperatures corresponding to each sample's transition temperature (TM = 64 and 45 K, respectively). Although the new Bragg peaks appear in a discontinuous manner in the Au0.52Zn0.48 sample, they appear in a continuous manner in AuZn. Surprising us, the temperature dependence of the AuZn Bragg peak intensity and the specific-heat jump near TM are in favorable accord with a continuous transition. A fit to the pressure dependence of TM suggests the presence of a critical end point in the AuZn phase diagram located at TM* = 2.7 K and p* = 3.1 GPa.     

Zero-field satellites of a zero-bias anomaly

Spin-orbit (SO) splitting, +/-omega(SO), of the electron Fermi surface in two-dimensional systems manifests itself in the interaction-induced corrections to the tunneling density of states, nu(epsilon). Namely, in the case of a smooth disorder, it gives rise to the satellites of a zero-bias anomaly at energies epsilon = +/-2 omega(SO). Zeeman splitting, +/-omega(Z), in a weak parallel magnetic field causes a narrow plateau of a width delta epsilon = 2 omega(Z) at the top of each sharp satellite peak. As omega(Z) exceeds omega(SO), the SO satellites cross over to the conventional narrow maxima at epsilon = +/-2 omega(Z) with SO-induced plateaus delta epsilon = 2 omega(SO) at the tops.     

Pseudomorphic growth of a single element quasiperiodic ultrathin film on a quasicrystal substrate

An ultrathin film with a periodic interlayer spacing was grown by the deposition of Cu atoms on the fivefold surface of the icosahedral Al70Pd21Mn9 quasicrystal. For coverages from 5 to 25 monolayers, a distinctive quasiperiodic low-energy electron diffraction pattern is observed. Scanning tunneling microscopy images show that the in-plane structure comprises rows having separations of S=4.5+/-0.2 A and L=7.3+/-0.3 A, whose ratio equals tau=1.618... within experimental error. The sequences of such row separations form segments of terms of the Fibonacci sequence, indicative of the formation of a pseudomorphic Cu film.     

Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate

We have studied the roughness and the dynamics of the contact line of a viscous liquid on a disordered substrate. We have used photolithographic techniques to obtain a controlled disorder with a correlation length ξ = 10μm. Liquids with different viscosity were used: water and aqueous glycerol solution. We have found that the roughness W of the contact line depends neither on the viscosity nor on the velocity v of the contact line for v in the range 0.2-20μm/s. W is found to scale with the length L of the line as L ^ζ with a roughness exponent ζ = 0.51±0.03. This value is similar to the one obtained with superfluid helium. In the present experiment, we have checked that the motion of the contact line is actually overdamped, so that the phenomenological equation first proposed by Ertas and Kardar should be relevant. However, our measurement of ζ is in disagreement with the predicted value ζ = 0.39. We have also analyzed the avalanche-like motion of the contact line. We find that the size distribution does not follow a power law dependence. Received 18 April 2002