Asymptotic Amplitude Research Articles

Harmonic linearized Navier-Stokes equation on describing the effect of surface roughness on hypersonic boundary-layer transition

Laminar-turbulent transition is crucially influenced by wall roughness. This paper develops a numerical approach based on the harmonic linearized Navier-Stokes (HLNS) equations to accommodate the scattering effect of the rapidly distorted mean flow induced by a two-dimensional hump or indentation at the wall on the oncoming instability modes (including the Mack first and second modes) in a hypersonic boundary layer. Due to the ellipticity of the scattering system when the roughness width is comparable with the instability wavelength, the traditional linear stability theory and the linear parabolized stability equation do not apply, and therefore, the HLNS approach has advantages in both accuracy and efficiency. The impact of a roughness is characterized by a transmission coefficient, which is the ratio of the asymptotic amplitude downstream of the roughness to that upstream. At a Mach number of 5.92, the dependence of the transmission coefficient on the frequency and the oblique angle of the oncoming mode and the size and location of the hump/indentation is studied systematically. It is confirmed that the synchronization frequency appears as a critical frequency, above and below which the oncoming instability modes are suppressed and enhanced by the roughness, respectively, which provides fundamental basis to the laminar-flow control in hypersonic boundary layers.

Induced versus intrinsic magnetic moments in ultrafast magnetization dynamics

One of the ultimate goals for future spintronic applications is to manipulate magnetic states of matter on the shortest of timescales. Here, the authors investigate the optically induced ultrafast demagnetization and remagnetization dynamics of ferromagnetic alloys based on either intrinsic or induced magnetic moments. Using pump-probe spectroscopy in combination with dynamic spin model simulations, they find that a CoFeB alloy with intrinsic magnetic moments on the Fe and Co sites reveals a distinct asymptotic amplitude limit of the remagnetization process, while in FePt the induced moments of the Pt sites stay always proportional to those of Fe.

Long-wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes

Properties of solitary waves in pre-compressed Hertzian chains of particles are studied in the long wavelength limit using a well-known continuum model. Several main results are obtained by parameterizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating pulses in weakly compressed systems, and slightly supersonic solitary waves correspond to weakly nonlinear propagating pulses in strongly compressed systems. Third, explicit formulas for the physical height, width, impulse and energy of the solitary waves are obtained in both the strongly nonlinear regime and the weakly nonlinear regime. Asymptotic expansions are used to show that in the strongly nonlinear regime, solitary waves are well-approximated by Nesterenko's compacton (having the same wave speed), while in the weakly nonlinear regime, solitary waves coincide with solitons of the Korteweg-de Vries (KdV) equation, with the same wave speed. All of these results are illustrated by means of exact solitary wave solutions, including the physically important case that models a chain of spherical particles.

Effects of viscosity and elasticity on the Richtmyer-Meshkov instability

The impulsively accelerated Richtmyer-Meshkov instability (RMI) of a fluid-solid interface is theoretically studied with a decomposition method, and the least stable mode is shown to oscillate but decay due to the combined effects of elasticity and viscosity. The dispersion relation of RMI in viscous fluids is obtained as well, and the predicted interface amplitude agrees with numerical simulations better than the previous linear and weakly nonlinear theories. As time tends to infinity, the asymptotic amplitude of the disturbed interface is found to be inversely proportional to the wave number in viscous fluids but to be independent of the wave number for the interface between elastic solids.

Scattering of Tollmien-Schlichting waves as they pass over forward-/backward-facing steps

Forward-/backward-facing steps in boundary-layer flows are often seen in engineering applications, and they have potential impacts on laminar-turbulent transition through scattering of the oncoming instability modes (e.g., Tollmien-Schlichting (T-S) waves). This issue is studied in the present paper by applying a local scattering framework, which is a rather generic mathematical framework on describing the mode scattering process. In this framework, a high-Reynolds-number triple-deck formalism is employed, and a transmission coefficient, defined as the ratio of the asymptotic amplitude of the instability mode downstream of the step to that upstream, is introduced. Through the systematical study, it has been found that both the forward- and backward-facing steps have a destabilizing effect on the oncoming T-S waves in subsonic boundary layers, this effect increases with the height of the step and/or the frequency of the T-S wave, and a backward-facing step (BFS) always has a greater impact than a forward-facing step (FFS). These facts agree with most of the previous investigations. However, one numerical study (WORNER, A., RIST, U., andWAGNER, S. Humps/steps influence on stability characteristics of two-dimensional laminar boundary layer. AIAA Journal, 41, 192–197 (2003)), which was based on an ad-hoc configuration, showed an opposite impact of an FFS. Through the investigation on the specific configuration, it is revealed that the wrong conclusion was drawn by misinterpreting the numerical results.

Black disk, maximal Odderon and unitarity

We argue that the so-called maximal Odderon contribution breaks the ‘black disk’ behavior of the asymptotic amplitude, since the cross section of the events with Large Rapidity Gaps grows faster than the total cross section. That is the ‘maximal Odderon’ is not consistent with unitarity.

Nonlinear theory of transverse beam echoes

Transverse beam echoes can be excited with a single dipole kick followed by a single quadrupole kick. They have been used to measure diffusion in hadron beams and have other diagnostic capabilities. Here we develop theories of the transverse echo nonlinear in both the dipole and quadrupole kick strengths. The theories predict the maximum echo amplitudes and the optimum strength parameters. We find that the echo amplitude increases with smaller beam emittance and the asymptotic echo amplitude can exceed half the initial dipole kick amplitude. We show that multiple echoes can be observed provided the dipole kick is large enough. The spectrum of the echo pulse can be used to determine the nonlinear detuning parameter with small amplitude dipole kicks. Simulations are performed to check the theoretical predictions. In the useful ranges of dipole and quadrupole strengths, they are shown to be in reasonable agreement.

Exponential Modeling of Frequency-Following Responses in American Neonates and Adults.

The scalp-recorded frequency-following response (FFR) has been widely accepted in assessing the brain's processing of speech stimuli for people who speak tonal and nontonal languages. Characteristics of scalp-recorded FFRs with increasing number of sweeps have been delineated through the use of an exponential curve-fitting model in Chinese adults; however, characteristics of speech processing for people who speak a nontonal language remain unclear. This study had two specific aims. The first was to examine the characteristics of speech processing in neonates and adults who speak a nontonal language, to evaluate the goodness of fit of an exponential model on neonatal and adult FFRs, and to determine the differences, if any, between the two groups of participants. The second aim was to assess effective recording parameters for American neonates and adults. This investigation employed a prospective between-subject study design. A total of 12 American neonates (1-3 days old) and 12 American adults (24.1 ± 2.5 yr old) were recruited. Each neonate passed an automated hearing screening at birth and all adult participants had normal hearing and were native English speakers. The English vowel /i/ with a rising pitch contour (117-166 Hz) was used to elicit the FFR. A total of 8,000 accepted sweeps were recorded from each participant. Three objective indices (Frequency Error, Tracking Accuracy, and Pitch Strength) were computed to estimate the frequency-tracking acuity and neural phase-locking magnitude when progressively more sweeps were included in the averaged waveform. For each objective index, the FFR trends were fit to an exponential curve-fitting model that included estimates of asymptotic amplitude, noise amplitude, and a time constant. Significant differences were observed between groups for Frequency Error, Tracking Accuracy, and Pitch Strength of the FFR trends. The adult participants had significantly smaller Frequency Error (p < 0.001), better Tracking Accuracy (p = 0.001), and larger Pitch Strength (p = 0.003) values than the neonate participants. The adult participants also demonstrated a faster rate of improvement (i.e., a smaller time constant) in all three objective indices compared to the neonate participants. The smaller time constants observed in adults indicate that a larger number of sweeps will be needed to adequately assess the FFR for neonates. Furthermore, the exponential curve-fitting model provided a good fit to the FFR trends with increasing number of sweeps for American neonates (mean r2 = 0.89) and adults (mean r2 = 0.96). Significant differences were noted between the neonatal and adult participants for Frequency Error, Tracking Accuracy, and Pitch Strength. These differences have important clinical implications in determining when to stop a recording and the number of sweeps needed to adequately assess the frequency-encoding acuity and neural phase-locking magnitude in neonates and adults. These findings lay an important foundation for establishing a normative database for American neonates and adults, and may prove to be useful in the development of diagnostic and therapeutic paradigms for neonates and adults who speak a nontonal language.

Dissipative waves in real gases

In this paper, we characterize a class of solutions to the unsteady 2-dimensional flow of a van der Waals fluid involving shock waves, and derive an asymptotic amplitude equation exhibiting quadratic and cubic nonlinearities including dissipation and diffraction. We exploit the theory of nonclassical symmetry reduction to obtain some exact solutions in the limit of vanishing dissipation. Because of the nonlinearities present in the evolution equation, one expects that the wave profile will eventually encounter distortion and steepening which in the limit of vanishing dissipation culminates into a shock wave; and once shock is formed, it will propagate by separating the portions of the continuous region. Here we have shown how the real gas effects, which manifest themselves through the van der Waals parameters ã and b̃ influence the wave characteristics, namely the shape, strength, and decay behavior of shocks.

Pion distribution amplitude and quasidistributions

We extend our analysis of quasidistributions onto the pion distribution amplitude. Using the formalism of parton virtuality distribution amplitudes, we establish a connection between the pion transverse momentum dependent distribution amplitude $\mathrm{\ensuremath{\Psi}}(x,{k}_{\ensuremath{\perp}}^{2})$ and the pion quasidistribution amplitude (QDA) ${Q}_{\ensuremath{\pi}}(y,{p}_{3})$. We build models for the QDAs from the virtuality-distribution-amplitude-based models for soft transverse momentum dependent distribution amplitudes, and analyze the ${p}_{3}$ dependence of the resulting QDAs. As there are many models claimed to describe the primordial shape of the pion distribution amplitude, we present the ${p}_{3}$-evolution patterns for models producing some popular proposals: Chernyak-Zhitnitsky, flat, and asymptotic distribution amplitude. Our results may be used as a guide for future studies of the pion distribution amplitude on the lattice using the quasidistribution approach.

Systematic variation of acquisition rate in delay eyelid conditioning.

Averaging artifacts inherent in group acquisition curves can mask behavioral phenomena that are potentially revealing in terms of underlying neural mechanisms. To address this, we implemented a behavioral analysis of 106 rabbits trained over 4 sessions using delay eyelid conditioning. Group results showed the typical monotonic increase in conditioned responses (CRs). For most subjects CRs first appeared (as indexed by the criterion of 8 CRs in 9 trials) during the first 18 trials of the second training session. Subdividing subjects according to the training block at which they met criterion revealed systematic differences in the subsequent rate that CR amplitudes increased, but not in asymptotic CR amplitudes. Subjects meeting criterion early in sessions showed more rapid increases in CR amplitude than those meeting criterion later in sessions. This effect was solely dependent on how early within a session criterion was met, as subjects meeting criterion at the beginning of the third and fourth sessions showed more rapid increases in CR amplitude than those meeting criterion after the first 18 trials of the second session. The exceptions were the 7% of the subjects that met criterion late in the first session. Their CR amplitudes increased at a rate similar to subjects meeting criterion early in sessions. These results suggest an interplay between consolidation processes and a previously reported short-term plasticity process that makes CR acquisition a nonmonotonic and complex function of the point during training sessions at which CRs first appear. (PsycINFO Database Record

High-precision estimate of the hydrodynamic radius for self-avoiding walks.

The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to N=2^{25}≈34×10^{6} monomers, we find that the ratio takes the value R_{G}/R_{H}=1.5803940(45), which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of ν=0.58759700(40).

B →Vℓ+ℓ− in the Standard Model from light-cone sum rules

We present $B_q\to\rho$, $B_q\to\omega$, $B_q\to K^*$, $B_s\to K^*$ and $B_s\to \phi$ form factors from light-cone sum rules (LCSR) at $\mathcal{O}(\alpha_s)$ for twist-2 and 3 and $\mathcal{O}(\alpha_s^0)$ for twist-4 with updated hadronic input parameters. Three asymptotic light-cone distribution amplitudes of twist-$4$ (and $5$) are determined, necessary for the form factors to obey the equations of motion. It is argued that the latter constrain the uncertainty of tensor-to-vector form factor ratios thereby improving the prediction of zeros of helicity amplitudes of major importance for $B\to K^*\ell\ell$ angular observables. We provide easy-to-use fits to the LCSR results, including the full error correlation matrix, in all modes at low $q^2$ as well as combined fits to LCSR and lattice results covering the entire kinematic range for $B_q\to K^*$, $B_s\to K^*$ and $B_s\to \phi$. The error correlation matrix avoids the problem of overestimating the uncertainty in phenomenological applications. Using the new form factors and recent computations of non-factorisable contributions we provide Standard Model predictions for $B\to K^*\gamma$ as well as $B\to K^*\ell^+\ell^-$ and $B_s\to\phi\mu^+\mu^-$ at low dilepton invariant mass. Employing our $B \to (\rho,\omega) $ form factor results we extract the CKM element $|V_\mathrm{ub}|$ from the semileptonic decays $B\to(\rho,\omega) \ell\nu$ and find good agreement with other exclusive determinations.

Bilinear approach to quasi-periodic wave solutions of the Kersten-Krasil’shchik coupled KdV-mKdV system

The Hirato bilinear method is extended to construct quasi-periodic wave solutions for the Kersten-Krasil’shchik coupled KdV-mKdV system. One- and two-periodic wave solutions are obtained by means of a multidimensional Riemann theta function. The asymptotic property of the quasi-periodic wave solutions is proved. It is shown that the quasi-periodic wave solutions reduce to the soliton solutions in an asymptotic small amplitude limit.

Gibbs phenomenon in the Hermite interpolation on the circle

Hermite–Fejér interpolation problems on the unit circle and bounded interval are usually studied in relation with continuous functions. There are few references concerning these problems for functions with discontinuities. Thus the aim of this paper is to describe the behavior of the Hermite–Fejér and Hermite interpolants for piecewise continuous functions on the unit circle, analyzing the corresponding Gibbs phenomenon near the discontinuities and providing the asymptotic amplitude of the Gibbs height.

What binds quarks together at different momentum scales? A conceptual scenario

The binding effects of quarks within hadrons are discussed in terms of the pion distribution amplitude over longitudinal momentum fractions. To understand the behavior of this quantity at different momentum scales, the concept of synchronization in complex systems has been employed. It is argued that at low momentum scales, the quarks get correlated by nonlocal quark/gluon condensates that cause an endpoint-suppressed, mainly bimodal structure of the pion distribution amplitude inferred from a sum-rule analysis. The mass generation mechanism, within the framework of Dyson–Schwinger equations, and evolution effects pull these two peaks back to the center to form at Q2→∞ the asymptotic distribution amplitude which represents the most synchronized q¯q state.

The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values \documentclass[12pt]{minimal}\begin{document}$q_{l/r}(t)\equiv A_{l/r} e^{-2iA_{l/r}^2t+i\theta _{l/r}}$\end{document}ql/r(t)≡Al/re−2iAl/r2t+iθl/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl ≠ θr. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that \documentclass[12pt]{minimal}\begin{document}$\left(q(x,t)-q_{l/r}(t)\right)\in L^{1,1}(\mathbb {R^\mp })$\end{document}q(x,t)−ql/r(t)∈L1,1(R∓) with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables \documentclass[12pt]{minimal}\begin{document}$\lambda _{l/r}=\sqrt{k^2+A^2_{l/r}}$\end{document}λl/r=k2+Al/r2, where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.

Asymptotic behavior of tidal damping in alluvial estuaries

Tidal wave propagation can be described analytically by a set of four implicit equations, i.e., the phase lag equation, the scaling equation, the damping equation, and the celerity equation. It is demonstrated that this system of equations has an asymptotic solution for an infinite channel, reflecting the balance between friction and channel convergence. Subsequently, explicit expressions for the tidal amplitude and velocity amplitude are derived, which are different from the generally assumed exponential damping equation that follows from linearizing the friction term. Analysis of the asymptotic behavior demonstrates that exponential damping of the tidal amplitude is only correct for a frictionless wave or an ideal estuary (no damping). However, in estuaries with modest damping (near ideal) it provides a reasonable approximation. In natural estuaries, there is generally a need to take account of local variability of, e.g., depth and friction, by subdividing the estuary into multiple reaches. This is illustrated with an example of the Scheldt estuary, which has been gradually deepened for navigation purpose over the last half century. The analytical model is used to study the effect of this deepening on the tidal dynamics in the main navigation channel, demonstrating that the navigation channel will become “overamplified” when it reaches a depth larger than the critical depth. In the case of overamplification, a further increase of the depth reduces the amplification until critical convergence (condition for a frictionless standing wave) is reached asymptotically. Finally, based on the ratio between the tidal amplitude at the seaward boundary and the asymptotic tidal amplitude, estuaries can be classified into damped, amplified, or ideal estuaries, which is illustrated with 23 real estuaries.

Multi-soliton and double Wronskian solutions of a ([formula omitted])-dimensional modified Heisenberg ferromagnetic system

A (2+1)-dimensional modified Heisenberg ferromagnetic system, which arises in the motion of magnetization vector of the isotropic ferromagnet and biological pattern formation, is investigated. Via the Hirota bilinear method, multi-soliton solutions of such a system are derived. It is proved that the system possesses the N-soliton solutions expressed in terms of the double Wronskian determinant. Head-on and overtaking elastic interactions are exhibited. Elastic interaction behavior between the two solitons has been interpreted through the asymptotic analysis, namely, amplitude and velocity of each soliton remain unchanged except for the phase shift after the interaction. Inelastic interactions including the soliton fusion and fission between two solitons are shown. During the soliton propagation, for the product of two fields, the soliton with the smaller amplitude can travel faster than with the larger, while for the third field, the soliton with the larger amplitude can travel faster than with the smaller. On the other hand, the soliton for the third field may exhibit the solitoff-like property. With respect to the three solitons, head-on elastic interaction can be found.

Optical variability of quasars: a damped random walk

AbstractA damped random walk is a stochastic process, defined by an exponential covariance matrix that behaves as a random walk for short time scales and asymptotically achieves a finite variability amplitude at long time scales. Over the last few years, it has been demonstrated, mostly but not exclusively using SDSS data, that a damped random walk model provides a satisfactory statistical description of observed quasar variability in the optical wavelength range, for rest-frame timescales from 5 days to 2000 days. The best-fit characteristic timescale and asymptotic variability amplitude scale with the luminosity, black hole mass, and rest wavelength, and appear independent of redshift. In addition to providing insights into the physics of quasar variability, the best-fit model parameters can be used to efficiently separate quasars from stars in imaging surveys with adequate long-term multi-epoch data, such as expected from LSST.