Wave Envelope Research Articles

Solitons in multi-body interactions for a fully modulated cubic–quintic Gross–Pitaevskii equation

• Dynamics of solitons in a fully modulated cubic–quintic Gross–Pitaevskii equation is analyzed. • Role of quintic parameter on the size of wave numbers of carrier and envelope is analyzed. • Lax-pair for the model with varying coefficients and external potential is given. • Regions of modulational instability and linear and nonlinear stabilities are investigated. A phase imprint approach is applied to a cubic Gross–Pitaevskii equation (GPE) in order to obtain a modified non-autonomous derivative cubic–quintic nonlinear GPE with fully variable coefficients. This model describes the dynamics of condensates in Bose–Einstein condensates, with both two- and three-body interatomic interactions with an external potential when the coefficient of the dispersion term is constant. This model is also applicable to fiber optics media in the absence of external potential. We show that this modified GPE model has a Lax-pair when all of its coefficients depend on time. However, the external potential depends on both the time and space variables. We obtain two classes of exact analytical solutions. These classes of solutions contain four different types of solitary-like wave solutions in the form of kink, anti-kink, bright, dark solitary, and periodic wave solutions. We reveal the effects of a quintic term on wave numbers, for both the carrier and envelope waves. Stability analysis is carried out, and conditions on parameters that determine regions with linear stability are discussed. Graphical analysis of some solutions are presented.

Effects of group velocity and multiplasmon resonances on the modulation of Langmuir waves in a degenerate plasma.

We study the nonlinear wave modulation of Langmuir waves (LWs) in a fully degenerate plasma. Using the Wigner-Moyal equation coupled to the Poisson equation and the multiple scale expansion technique, a modified nonlocal nonlinear Schrödinger (NLS) equation is derived which governs the evolution of LW envelopes in degenerate plasmas. The nonlocal nonlinearity in the NLS equation appears due to the group velocity and multiplasmon resonances, i.e., resonances induced by the simultaneous particle absorption of multiple wave quanta. We focus on the regime where the resonant velocity of electrons is larger than the Fermi velocity and thereby the linear Landau damping is forbidden. As a result, the nonlinear wave-particle resonances due to the group velocity and multiplasmon processes are the dominant mechanisms for wave-particle interaction. It is found that in contrast to classical or semiclassical plasmas, the group velocity resonance does not necessarily give rise the wave damping in the strong quantum regime where ℏk∼mv_{F} with ℏ denoting the reduced Planck's constant, m the electron mass, and v_{F} the Fermi velocity; however, the three-plasmon process plays a dominant role in the nonlinear Landau damping of wave envelopes. In this regime, the decay rate of the wave amplitude is also found to be higher compared to that in the modest quantum regime where the multiplasmon effects are forbidden.

A coupled interpolating meshfree method for computing sound radiation in infinite domain

SummaryIn this paper, the coupling of the improved interpolating element‐free Galerkin (IIEFG) method and the variable‐order infinite acoustic wave envelope element (WEE) method is studied. A coupled IIEFG‐WEE method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages. The coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the IIEFG and WEE domains. In the IIEFG domain, the improved interpolating moving least‐squares (IIMLS) method is used to form the shape functions satisfying the Kronecker delta condition while nonsingular weight functions can be used. The impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated. The numerical results show that the coupled IIEFG‐WEE method can take full advantage of both the IIEFG and WEE methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the WEE. The experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain.

Effect of electric field on the ratio between the rashba and dresselhaus parameters in III–V heterostructures

The size-quantized energy subbands and envelope wave functions for [001] quantum wells based on zinc-blende III–V semiconductors are numerically calculated using the eight-band Kane model and finite-difference discretization scheme in coordinate space. The effect of the quantum-well band parameters and external electric field oriented along the structure growth direction on the ratio between the Rashba and Dresselhaus spin–orbit coupling parameters is studied. It is demonstrated that at certain electric-field values the spin–orbit coupling parameters in GaAs/InGaAs structures can be equal, which ensures the condition for forming stable spin helices. In addition, it is established that the spin–orbit coupling linear in wave vector in symmetric GaAs/InGaAs wells can disappear under certain well widths and barrier chemical compositions.

Dark solitons, modulation instability and breathers in a chain of weakly nonlinear oscillators with cyclic symmetry

In the aerospace industry the trend for light-weight structures and the resulting complex dynamic behaviours currently challenge vibration engineers. In many cases, these light-weight structures deviate from linear behaviour, and complex nonlinear phenomena can be expected. We consider a cyclically symmetric system of coupled weakly nonlinear undamped oscillators that could be considered a minimal model for different cyclic and symmetric aerospace structures experiencing large deformations. The focus is on localised vibrations that arise from wave envelope modulation of travelling waves. For the defocussing parameter range of the approximative nonlinear evolution equation, we show the possible existence of dark solitons and discuss their characteristics. For the focussing parameter range, we characterise modulation instability and illustrate corresponding nonlinear breather dynamics. Furthermore, we show that for stronger nonlinearity or randomness in initial conditions, transient breather-type dynamics and decay into bright solitons appear. The findings suggest that significant vibration localisation may arise due to mechanisms of nonlinear modulation dynamics.

Using direct and coda wave envelopes to resolve the scattering and intrinsic attenuation structure of Southern California

AbstractCharacterizing scattering and absorbing properties and the power spectrum of crustal heterogeneity is a fundamental problem for informing strong ground motion estimates at high frequencies, where scattering and attenuation effects are critical. We perform a comprehensive study of local earthquake coda waves in Southern California to constrain scattering and intrinsic attenuation structure. We analyze data from 1195 spatially distributed earthquakes from 1981 to 2013 at source depths of 10 to 15 km and epicentral distances from 0 to 250 km with magnitudes larger than 1.8. We stack envelope functions from 28,127 vertical component and 27,521 transverse component seismograms, filtered from 2 to 4 Hz. We model these observations using a particle‐based Monte Carlo algorithm that includes intrinsic attenuation as well as both P and S wave scattering and both single and multiple scattering events. We find that spatially averaged coda wave behavior for Southern California can be explained only with models containing an increase in scattering strength and intrinsic attenuation within the uppermost crust, i.e., they are poorly fit with half‐space models of constant scattering strength. A reasonable fit to our data is obtained with a two‐layer model, composed of a shallow crustal layer with strong wide‐angle scattering and high P and S intrinsic attenuation and a deeper layer with weaker scattering and lower intrinsic attenuation (top 5.5 km: αQI=250, βQI=125, heterogeneity correlation length a = 50 m, fractional velocity heterogeneity ε = 0.4; lower crust: αQI=900, βQI=400, a = 2 km, ε = 0.05).

The influence of non-uniform blockages on transient wave behavior and blockage detection in pressurized water pipelines

Blockages in piping systems are formed from potentially complex combinations of bio-film build up, corrosion by-products, and sediment deposition. Transient-based methods seek to detect blockages by analyzing the evolution of small amplitude pressure waves. In theory, such methods can be efficient, nearly non-intrusive and economical but, thus far, studies have only considered symmetrical blockages, uniform in both the radial and longitudinal directions. Laboratory experiments are described here that involve pipe blockages with various levels of irregularity and severity; the way the transient response is affected by a non-uniform blockage is investigated. The differences between uniform and non-uniform blockages are quantified in terms of the rate that wave envelopes attenuate and the degree that phases are shifted. Two different methods for modeling these impacts are compared, namely through an increase in pipe roughness and through a wave scattering model. Wave scattering is shown to play a dominant role in explaining both wave envelope attenuation and phase shift. The accuracy of existing transient-based methods of blockage detection in the frequency domain is also examined, and is found that the predictions of rough blockage locations and sizes by current method are in good agreement with data, with relatively larger discrepancies for rough blockage lengths.

Ion-acoustic rogue waves and breathers in relativistically degenerate electron-positron plasmas

In this paper, we employ a weakly relativistic fluid model to study the nonlinear amplitude modulation of electrostatic waves in an unmagnetized electron-positron-ion plasma. It is assumed that the degeneracy pressure law for electrons and positrons follows the Chandrasekhar limit of state whereas ions are warm and classical. The hydrodynamic approach along with the perturbation method have been applied to obtain the corresponding nonlinear Schrodinger equation (NLSE) in which nonlinearity is in balance with the dispersive terms. Using the NLSE, we could evaluate the modulational instability to show that various types of localized ion acoustic excitations exist in the form of either bright-type envelope solitons or dark-type envelope solitons. The regions of the stable and unstable envelope wave have been confined punctually for various regimes. Furthermore, it is proposed that the exact solutions of the NLSE for breather waves are the rogue waves (RWs), Akhmediev breather (AB), and Kuznetsov-Ma breather (KM) soliton. In order to show that the characteristics of breather structures is influenced by the plasma parameters (namely, relativistic parameter, positron concentration, and ionic temperature), the relevant numerical analysis of the NLSE is examined. In particular, it is observed that by increasing the values of the mentioned plasma parameters, the amplitude of the RWs will be decreased. Our results help researchers to explain the formation and dynamics of nonlinear electrostatic excitations in super dense astrophysical regimes.

Characterisation of waves via the Lax–Wendroff method

In recent years, the problem of rogue waves has been a topical issue among oceanographers because of their dangerous and unpredictable properties, leading to safety demands for ships and offshore platforms. Even with efforts to physically quantify the mechanism of rogue wave generation within the last two decades, rogue wave events are still unpredictable, as the rogue wave mechanism has not yet been fully explained, both mathematically and physically. This study investigates whether a Lax–Wendroff method-based algorithm with the wave action equation can characterise a 2-dimensional regular wave group over deep water. The Lax–Wendroff method up to a second-order approximation was applied to the wave action equation and the formulated algorithm was validated by means of experimental data, which consisted of a measured local regular wave group (wave frequency: 1.6 Hz, wave amplitude: 0.54 cm). The model consists of a computational algorithm which can describe a spatial evolution of wave frequencies (radian frequencies) and envelopes in the time domain with second-order accuracy, when the initial values at the first position are provided. As a result, the Lax–Wendroff method-based algorithm characterised an evolution of the regular wave group up to 4.5 m, implying that the Lax–Wendroff method-based algorithm could be a promising mathematical formulation in the understanding and characterisation of rogue wave formation, providing a further insight into a linear focusing mechanism. In the future, the results of this study will be extended to investigate the validity of the Lax–Wendroff method-based algorithm in finite water depth.

Local Wave Activity and the Onset of Blocking along a Potential Vorticity Front

Abstract Interaction between a train of transient waves and a diffluent westerly jet is examined using a regional quasigeostrophic equivalent barotropic model with a (nearly) binary potential vorticity (PV) distribution. Unlike most previous studies, but consistent with the observed extratropical tropopause, cross-stream variation in the layer thickness is allowed to contribute to the discontinuity in PV. In all cases examined, short (i.e., barotropic) edge waves are continuously forced in the upstream, then migrate downstream, and eventually exit the domain. A quasilinear 1D theory based on the conservation of local wave activity predicts that no steady wave train can be maintained where the westerly zonal flow is decelerated below one-half of the initial value, at which point the wave envelope develops a migratory shock analogous to the Lighthill–Whitham–Richards traffic flow problem. Fully nonlinear high-resolution 2D calculations show that the wave train indeed undergoes a significant transformation once the zonal flow along the jet axis is decelerated below the threshold. The subsequent flow evolution depends on the nature of the discontinuity in the basic-state PV. When the discontinuity is entirely due to the vorticity profile, waves are compressed and partially deflected sideways but no complete blocking occurs. When the discontinuity in PV is augmented by the layer thickness variation, the incident wave train is blocked and split into two tracks at the stagnation point, eventually leading to a formation of a modon-like vortex pair, reminiscent of an atmospheric blocking. Implications for low-frequency variability of the atmosphere are discussed.

Universal behavior of electron g -factors in semiconductor nanostructures

We combine analytic developments and numerical tight-binding calculations to study the evolution of the electron $g$-factors in homogeneous nanostructures of III-V and II-VI semiconductors. We demonstrate that the $g$-factor can be always written as a sum of bulk and surface terms. The bulk term, the dominant one, just depends on the energy gap of the nanostructure but is otherwise isotropic and independent of size, shape, and dimensionality. At the same time, the magnetic moment density at the origin of the bulk term is anisotropic and strongly dependents on the nanostructure shape. The physical origin of these seemingly contradictory findings is explained by the relation between the spin-orbit-induced currents and the spatial derivatives of the electron envelope wave function. The tight-binding calculations show that the $g$-factor versus energy gap for spherical nanocrystals can be used as a reference curve. In quantum wells, nanoplatelets, nanorods, and nanowires, the $g$-factor along the rotational symmetry axis can be predicted from the reference curve with a good accuracy. The $g$-factors along nonsymmetric axes exhibit more important deviations due to surface contributions but the energy gap remains the main quantity determining their evolution. The importance of surface-induced anisotropies of the $g$-factors is discussed.

Modelling and observations of oceanic nonlinear internal wave packets affected by the Earth’s rotation

The large-amplitude internal solitary waves commonly observed in the coastal ocean can propagate for long distances for long times, so that it may be necessary to take account of the effects of the Earth’s background rotation. In this case an appropriate model wave evolution equation is the Ostrovsky equation, whose typical solutions indicate that internal solitary waves will evolve into envelope wave packets. Unlike the more usual Korteweg-de Vries solutions which are typically rank-ordered wave packets, these are centred with the largest waves in the middle. This qualitative feature, together with certain key quantitative parameters such as the envelope carrier wavenumber and speed, can be sought in oceanic observations. Hence we have examined many SAR images of internal solitary waves with the general aim of finding features indicating that rotational effects have become significant. From these we report in detail on six typical cases of which four give indications of rotational effects. In addition we use a two-layer fluid model to estimate how the rotational parameters depend on the background stratification and topography.

Spontaneous brightening of dark excitons in GaAs/AlGaAs quantum dots near a cleaved facet

Dark excitons (DEs) confined in epitaxial quantum dots (QDs) are interesting because of their long lifetime compared to bright excitons (BEs). For the same reason they are usually difficult to access in optical experiments. Here we report on the observation of vertically polarized light emission from DEs confined in high-quality epitaxial GaAs/AlGaAs QDs located in proximity of a cleaved facet of the QD specimen. Calculations based on the eight-band k\ifmmode\cdot\else\textperiodcentered\fi{}p method and configuration interaction allow us to attribute the brightening of the DE to the anisotropic strain present at the sample edge, which breaks the symmetry of the system and enhances valence-band mixing. The mechanism of DE brightening is discussed in detail by inspecting both the Bloch and envelope wave functions of the involved hole states. In addition, by investigating experimentally and theoretically QDs with different sizes, we find that the energy separation between DE and BEs tends to decrease with increasing QD height. Finally, the presence of a cleaved facet is found also to enhance the BE fine structure splitting. This work provides a simple method to optically probe dark excitonic states in QDs and shows that the properties of QDs can be significantly affected by the presence of nearby edges.

Stability, diffusion and interactions of nonlinear excitations in a many body system

When repelling particles are confined in a quasi-one-dimensional trap by a transverse potential, a configurational phase transition happens. All particles are aligned along the trap axis at large confinement, but below a critical transverse confinement they adopt a staggered row configuration (zigzag phase). This zigzag transition is a subcritical pitchfork bifurcation in extended systems and in systems with cyclic boundary conditions in the longitudinal direction. Among many evidences, phase coexistence is exhibited by localized nonlinear patterns made of a zigzag phase embedded in otherwise aligned particles. We give the normal form at the bifurcation and we show that these patterns can be described as solitary wave envelopes that we call bubbles. They are stable in a large temperature range and can diffuse as quasi-particles, with a diffusion coefficient that may be deduced from the normal form. The potential energy of a bubble is found to be lower than that of the homogeneous bifurcated phase, which explains their stability. We observe also metastable states, that are pairs of solitary wave envelopes which spontaneously evolve toward a stable single bubble. We evidence a strong effect of the discreteness of the underlying particles system and introduce the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive so that the bubbles come closer and eventually merge as a single bubble. In contrast, the bubbles interaction is found to be repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: it is attractive for NF-systems, repulsive for F-systems, and decreases exponentially with the bubbles distance.

Observations of nearshore infragravity wave dynamics under high energy swell and wind-wave conditions

Infragravity waves (0.005–0.04Hz) can dominate the water motion close to shore on low sloping beaches and play a significant role in beach and dune erosion. A new field data set of water surface elevation at 15 cross-shore locations on a dissipative, fetch-unlimited beach is analysed to investigate the forcing and surf zone behaviour of infragravity waves during a wide range of offshore wave conditions (Ho=0.38–3.88m; Tp=6–20s). Infragravity waves approach the shore as bound waves lagging slightly (~4s) behind the short wave (0.04–0.33Hz) envelope and are released in the surf zone as free waves. Infragravity wave heights of up to 1m are measured close to shore and are best predicted using an offshore forcing parameter that represents the short wave energy flux (Ho2Tp). Considerable infragravity dissipation is observed in the surf zone and dissipation increases with offshore wave energy. Dissipation is highly frequency-dependant and a frequency-domain Complex Empirical Orthogonal Function analysis reveals (quasi-)standing waves at frequencies <0.017Hz, but an increasingly progressive wave pattern at higher frequencies with reflection coefficients <0.1, indicative of more than 90% dissipation. Much of the observed dissipation occurs very close to shore and the dependence of the reflection coefficient on a normalised bed slope parameter implies that energy at high infragravity frequencies is dissipated by wave breaking, since these frequencies fit into a mild sloping regime. This is supported by the results of bispectral analysis which show predominantly infragravity-infragravity interactions in shallow water and the development of infragravity harmonics indicative of steepening and eventual breaking of the infragravity waves.

Wave propagation in strongly dispersive superthermal dusty plasma

The attributes of acoustic envelope waves in a collisionless dust ion unmagnetized plasmas model composed of cold ions, superthermal electrons and positive-negative dust grains have been studied. Using the derivative expansion technique in a strong dispersive medium, the system model is reduced to a nonlinearly form of Schrodinger equation (NLSE). Rational solution of NLSE in unstable region is responsible for the creation of large shape waves; namely rogue waves. The subjection of instability regions upon electron superthermality (via κ), carrier wave number and dusty grains charge is discussed.

Interaction, coalescence, and collapse of localized patterns in a quasi-one-dimensional system of interacting particles.

We study the path toward equilibrium of pairs of solitary wave envelopes (bubbles) that modulate a regular zigzag pattern in an annular channel. We evidence that bubble pairs are metastable states, which spontaneously evolve toward a stable single bubble. We exhibit the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive, whereas it is repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: It is attractive for NF systems and repulsive for F systems and decreases exponentially with the bubbles distance. Moreover, for NF systems, the bubbles come closer and eventually merge as a single bubble, in a coalescence process. We also evidence a collapse process, in which one bubble shrinks in favor of the other one, overcoming an energetic barrier in phase space. This process is relevant for both NF systems and F systems. In NF systems, the coalescence prevails at low temperature, whereas thermally activated jumps make the collapse prevail at high temperature. In F systems, the path toward equilibrium involves a collapse process regardless of the temperature.

Modulation and nonlinear evolution of multi-dimensional Langmuir wave envelopes in a relativistic plasma

The modulational instability (MI) and the evolution of weakly nonlinear two-dimensional (2D) Langmuir wave (LW) packets are studied in an unmagnetized collisionless plasma with weakly relativistic electron flow. By using a 2D self-consistent relativistic fluid model and employing the standard multiple-scale technique, a coupled set of Davey-Stewartson (DS)-like equations is derived which governs the slow modulation and the evolution of LW packets in relativistic plasmas. It is found that the relativistic effects favor the instability of LW envelopes in the k{\theta} plane, where k is the wave number and {\theta} the angle of modulation. It is also found that as the electron thermal velocity or {\theta} increases, the growth rate of MI increases with cutoffs at higher wave numbers of modulation. Furthermore, in the nonlinear evolution of the DS-like equations, it is seen that with an effect of the relativistic flow, a Gaussian wave beam collapses in a finite time, and the collapse can be arrested when the effect of the thermal pressure or the relativistic flow is slightly relaxed. The present results may be useful to the MI and the formation of localized LW envelopes in cosmic plasmas with a relativistic flow of electrons.

Application of non-extensive entropy to study of decoherence of RbCl quantum dot qubit: Tsallis entropy

In this work, an electron which is strongly coupled to the LO-phonon in triangular quantum dots with Coulomb impurity is considered. The eigenenergies and eigenfunctions of the ground and the first-excited states of the electron are obtained using the Pekar variational method. We have studied decoherence of RbCl quantum dot qubit using the non-extensive entropy (Tsallis entropy) for different values of Coulomb impurity parameter, polaronic radius and electron-LO phonon coupling strength. Numerical analysis shows that the entropy has the oscillatory periodic evolution as function of the time due to the triangular form of the confinement. It is found that entropy oscillates under a standing wave envelope with increasing the Coulomb impurity parameter, electron-LO phonon coupling strength and polaronic radius. With reducing the non-extensive parameter q, the entropy increases and thereby we can miss information about the system.

Influence of thin AlAs layer insertion on intersubband optical transitions in GaAs/AlGaAs quantum- well structures

In this work, we demonstrate the thin AlAs layer insertion into GaAs/AlGaAs quantum well (QW) structures and its influence in energy transition in the frequency range of mid-infrared. To realize the more accurate calculation, the graded interface model of QW structures is integrated into our self-consistent solving of Schrodinger and Poisson equations to obtain the energy level and envelope wave functions of QW. We find the thin AlAs layer inserted at various positions in the well can obviously tune intersubband optical transitions. The corresponding tuning range can be 50meV. We find that the thicker AlAs layer (2 monolayers) can provide wider tuning range and larger oscillator strength between subbands 1 and 3, compared with the thinner one (1 monolayer). Our results suggest that thin semiconductor layer may be an idea optimization design for the quantum well terahertz lasers which are based on optical pumping with mid-infrared lasers.