Higher-order Nonlinear Interaction Research Articles

Sonic horizon formation for two-dimensional Bose–Einstein condensates with higher-order nonlinear interaction

Based on the Gross–Pitaevskii equation model that incorporates higher-order nonlinear interaction, we studied sonic black hole and sonic horizon formation involved in nonlinear dynamical evolution for Bose–Einstein condensates with higher-order nonlinear interaction. On the basis of the modified variational method and the scenario where the system starts dynamic evolution from ground state, we derived the typical system distribution width function, which is analytically formulated as periodic oscillation solution and monotonically damped variational oscillation solution under different parametric settings. We also calculated the criteria formula for sonic black hole horizon formation with regard to the two evolution modes: oscillation mode and monotonically decay mode, pictorially demonstrating the time interval of sonic horizon appearance. The theoretical results obtained here can be used to guide relevant experimental studies of sonic horizon and sonic black hole formation for Bose–Einstein condensates incorporating the higher-order nonlinear interaction effects.

Kink soliton dynamics in one-dimensional Bose–Einstein condensate with higher-order nonlinear interactions

We investigated the soliton behavior in one-dimensional Bose–Einstein condensates. Based on the modified Gross–Pitaevskii equation (GPE) model with higher-order nonlinear interaction and through the F-expansion method, we derived the analytical kink soliton solution of the one-dimensional GPE. The typical kink soliton features under the specific experimental setting are identified, and the physical implication of the analytical results is also demonstrated. The derived theoretical results can be used to guide the experimental study of kink soliton in ultracold atomic systems with higher-order nonlinearity.

Multi-Informant Ratings of Childhood Limited Prosocial Emotions: Mother, Father, and Teacher Perspectives

ABSTRACT Objective Modest agreement between mothers’, fathers’, and teachers’ reports of child psychopathology can cause diagnostic ambiguity. Despite this, there is little research on informant perspectives of youth’s limited prosocial emotions (LPEs). We examined the relationship between mother-, father-, and teacher-reported LPE in a clinical sample of elementary school-aged children. Method The sample included 207 primarily Caucasian (n = 175, 84.5%) children (136 boys; 65.7%) aged 6–13 years (M = 8.35, SD = 2.04) referred to an outpatient child diagnostic clinic focused on externalizing problems. We report the percentage of youth meeting LPE criteria as a function of informant perspective(s). Utilizing standard scores, we report distributions of informant dyads in agreement/disagreement regarding child LPE, followed up by polynomial regressions to further interrogate the relationship between mother-, father-, and teacher-reported LPE as it relates to conduct problems (CPs). Results The prevalence of child LPE was approximately twice as large when compared to those reported in community samples; mothers and fathers generally agreed on their child’s LPE symptoms (55% agreement). Higher-order nonlinear interactions between mothers and fathers, as well as parents and teachers, emerged; discrepancies between informants, characterized by low levels of LPE reported by the child’s mother, were predictive of youth at the highest risk for CPs. Conclusions Our findings emphasize the clinical utility of gathering multiple reports of LPE when serious CPs are suspected. It may be beneficial for clinicians to give significant consideration to teacher reported LPE when interpreting multiple-informant reports of LPE.

Spike prediction on primary motor cortex from medial prefrontal cortex during task learning

Objectives. Brain–machine interfaces (BMIs) aim to help people with motor disabilities by interpreting brain signals into motor intentions using advanced signal processing methods. Currently, BMI users require intensive training to perform a pre-defined task, not to mention learning a new task. Thus, it is essential to understand neural information pathways among the cortical areas in task learning to provide principles for designing BMIs with learning abilities. We propose to investigate the relationship between the medial prefrontal cortex (mPFC) and primary motor cortex (M1), which are actively involved in motor control and task learning, and show how information is conveyed in spikes between the two regions on a single-trial basis by computational models. Approach. We are interested in modeling the functional relationship between mPFC and M1 activities during task learning. Six Sprague Dawley rats were trained to learn a new behavioral task. Neural spike data was recorded from mPFC and M1 during learning. We then implement the generalized linear model, the second-order generalized Laguerre–Volterra model, and the staged point-process model to predict M1 spikes from mPFC spikes across multiple days during task learning. The prediction performance is compared across different models or learning stages to reveal the relationship between mPFC and M1 spike activities. Main results. We find that M1 neural spikes can be well predicted from mPFC spikes on the single-trial level, which indicates a highly correlated relationship between mPFC and M1 activities during task learning. By comparing the performance across models, we find that models with higher nonlinear capacity perform significantly better than linear models. This indicates that predicting M1 activity from mPFC activity requires the model to consider higher-order nonlinear interactions beyond pairwise interactions. We also find that the correlation coefficient between the mPFC and M1 spikes increases during task learning. The spike prediction models perform the best when the subjects become well trained on the new task compared with the early and middle stages. The results suggest that the co-activation between mPFC and M1 activities evolves during task learning, and becomes stronger as subjects become well trained. Significance. This study demonstrates that the dynamic patterns of M1 spikes can be predicted from mPFC spikes during task learning, and this will further help in the design of adaptive BMI decoders for task learning.

Dark soliton dynamics for quantum systems with higher-order dispersion and higher-order nonlinearity

In this work, we investigated one-dimensional and two-dimensional quantum systems with higher-order dispersions and higher-order nonlinear interactions. Based on the high-order nonlinear Schrödinger equation (NLSE) and via the [Formula: see text]-expansion method, we derived the analytical dark soliton solution for the one-dimensional system first. By applying the self-similar method and using the results of the one-dimensional case, the analytical dark soliton solution of the system in the two-dimensional case was derived. The dynamic evolution pattern of the two-dimensional dark soliton is pictorially demonstrated. The theoretical results of our work can be used to guide the detection and experimental study of dark soliton in a two-dimensional quantum system, using high-order dispersion and higher-order nonlinear interactions.

Sonic black hole horizon dynamics for one dimensional Bose-Einstein condensate with quintic-order nonlinearity

We study the dynamics of sonic black hole horizon formation of quasi-one-dimensional (1D) Bose-Einstein condensate incorporating higher-order quintic nonlinear interaction. Based on the one dimensional Gross-Pitaevskii equation with nonlinearity up to the quintic order, we derived a novel analytical formula for the key dynamical variables of sonic horizon formation using the modified variational method and exact F-expansion method. We obtained good agreement between the key dynamical variables from the two different methods. The stabilization effects of higher-order nonlinear interaction along with the more precise location of the sonic horizon boundary were illustrated. The theoretical results obtained in this work can be used to guide relevant experimental observations of sonic black hole-related dynamics incorporating the effects of higher-order nonlinear interaction.

Experimental Implementation of a Raman-Assisted Eight-Wave Mixing Process

Engineering higher-order nonlinear interactions is vital in autonomous protection of quantum systems against errors. Such interactions are often not directly available, though, or are slow compared to error rates of the system. The authors present a nonlinear eight-wave mixing process that exchanges four photons of a harmonic oscillator with two excitations of a transmon-qubit mode and two pump photons, by combining more accessible lower-order interactions via a sort of Raman transition. Surprisingly, this technique produces a stronger interaction than a six-wave mixing process in the same system. This eight-wave mixing process is expected to become a key component of autonomous continuous-variable quantum error correction.

Sonic black hole horizon formation for Bose-Einstein condensates with higher-order nonlinear effects

We study sonic horizon formation dynamics for Bose-Einstein condensate systems with higher-order nonlinear interaction. Based on the Gross-Pitaevskii equation incorporating higher-order nonlinear effects and through a variational method, we derived the criteria formula for sonic horizon occurrence. The key features of the sonic horizon are pictorially demonstrated, and we identified the stabilization and widening metastable effects of the higher-order nonlinear interaction, from which the quantitative results can be used to guide relevant experimental observations of sonic black holes with higher-order nonlinear effects.

Solitary vortex dynamics of two-dimensional harmonically trapped Bose-Einstein condensates with higher-order nonlinear interactions

We studied the evolutionary patterns of two-dimensional Bose-Einstein condensates incorporating higher-order nonlinear interactions in harmonic potential. Using the Gross-Pitaevskii equation model with higher-order nonlinear corrections, we derived the analytical solitary vortex solutions via the variational method. The impact of the higher-order nonlinear interaction on the vortex dynamics is quantitatively analyzed, showing its key nonlinear feature contribution in the asymmetric vortex evolution with more precise evolutionary pattern generated. We found that, for the circular symmetric solution, if the nonlinear strength is not high, the higher-order nonlinear corrections essentially have only a tiny perturbative effect on the system’s quasi-static oscillation state, whereas for asymmetric evolution of the solitary vortex, incorporating higher-order corrections will generate an evolution pattern that better matches the results of numerical simulation. The theoretical results derived here can be used to guide relevant experimental studies of higher-order nonlinear effects in ultracold atomic systems.

Soliton dynamics for one dimensional quantum system incorporating higher-order dispersion effect and nonlinear interactions

We investigate (1+1) dimensional higher order nonlinear Schrödinger equation with higher order nonlinear interaction which models systems incorporating higher order dispersion and nonlinear effects. Through the F-expansion method, we analytically solved the higher order nonlinear Schrödinger equation model andidentify bright soliton type solution under certain parametric setting, demonstrating the practical significance of the theoretical treatment presented in this study.

Soliton dynamics for trapped Bose-Einstein condensate with higher-order interaction

We model the three-dimensional (3D) trapped Bose-Einstein condensate with the Gross-Pitaevskii equation (GPE) incorporating higher-order nonlinear interaction effects. Based on the F-expansion method and self-similar approach, we analytically derive the exact soliton solutions of the 3D GPE, identifying the oscillatory mode under certain experimental settings. The oscillatory kinetic parameter values we obtained coincide very well with those reported in previous work, which is based on the variational approach, showing the applicability of the theoretical treatment presented in this work.

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue (J. Fluid Mech., vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

A simple analytical method for evaluation of limit-cycle oscillation amplitude in boiling water reactors

An analytical method to obtain the amplitude of limit-cycle inphase oscillation in boiling water reactors is proposed, which is applicable to small-amplitude limit cycles actually observed in various stability tests. The expression of a zero-power transfer function including the limit-cycle amplitude can be derived by considering higher-order nonlinear interaction between neutron density and feedback reactivity. In a limit-cycle oscillation, the gainand phase-margin of the open-loop transfer function, which is defined by the product of the zero-power and a feedback transfer function, must be zero. From this requirement the oscillation amplitude and resonance frequency can be determined. A numerical example indicates that the fifth-order nonlinear approximation is adequate for the evaluation of the amplitude and that the resonance frequency is only slightly dependent on the amplitude. Assuming the independence of the resonance frequency from the amplitude, a simple formula can also be derived. When a resonance frequency is obtained by linear stability analysis, the oscillation amplitude can be readily determined from the formula expressed as a function of the amplitude by a polynomial.

Statistical models for predicting a beneficial response to interferon-alpha in patients with chronic hepatitis B.

Therapy with interferon-alpha has been reported to induce remissions in 35% of patients with chronic hepatitis B. The ability to identify patients likely to respond would be helpful in making recommendations for treatment. In this statistical analysis we included 82 patients with chronic hepatitis B who received interferon-alpha in clinical trials at the National Institutes of Health between 1984 and 1991. A response was defined as the loss of hepatitis B virus (HBV) DNA and hepatitis B e antigen (HBeAg) within 1 year of therapy. Multiple clinical parameters measured at pretreatment (month 0) and after the first month (month 1) of therapy were selected by stepwise regression to support the development of the prognostic models: the two-stage logistic regression model and a neural network that utilized higher-order non-linear interactions between variables. Among the 82 patients, 24 (29%) were responders. The two-stage logistic model using pretreatment variables: sex, hepatic fibrosis and alanine aminotransferase (ALT) levels correctly identified 61% of responders and 76% of non-responders. When HBV DNA at month 1 along with sex, initial ALT and fibrosis was included, the resultant model correctly identified 69% of responders and 77% of non-responders. The neural network, by incorporating interactions between variables, correctly identified 77% and 86% of responders, and 87% and 92% of non-responders, using pretreatment factors alone and the combination of pretreatment and month 1 factors respectively. Hence, the neural network was more accurate than the simple logistic regression model in predicting a response to interferon-alpha in chronic hepatitis B. The universality of these models needs to be further verified.

A higher-order nonlinear interaction among spectral components

A higher-order nonlinear interaction was measured in the wake behind a thin flat plate. In the experiment, a sound with two frequencies was used to specify the frequencies of interacting components. A new spectrum with many spectral components was found.

Explosive Plasma Instabilities

The interaction of three monochromatic waves (two positive energy longitudinal waves and one negative energy wave) is investigated. When only the first-order nonlinear interactions are considered, an explosive instability occurs. However, this instability is stabilized by the higher order nonlinear interactions. The matrix elements of the three-wave and the four-wave interactions in a magnetized plasma are calculated.