Wave Envelope Research Articles

The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.

Oblique modulation of electron acoustic waves in nonextensive plasma

In this paper, we have studied the oblique modulation of electron acoustic (EA) waves and formation and properties of envelope solitons in three component plasma consisting of cold electrons, nonextensive hot electrons, and stationary ions. To study the stability criteria of waves, the nonlinear Schrödinger equation (NLSE) is derived. The numerical calculations of the NLSE show the variation of the stability region with different plasma parameters. It reveals that the stability criteria of EA waves depend on the modulation angle θ. The effect of nonextensive parameter q and electron temperature ratio σ on the formation of EA wave envelope excitations has been studied in detail, and the findings obtained here will be beneficial for further astrophysical investigations.

Quasioptical modeling of wave beams with and without mode conversion. I. Basic theory

This work opens a series of papers where we develop a general quasi-optical theory for mode-converting electromagnetic beams in plasma and implement it in a numerical algorithm. Here, the basic theory is introduced. We consider a general quasimonochromatic multicomponent wave in a weakly inhomogeneous linear medium with no sources. For any given dispersion operator that governs the wave field, we explicitly calculate the approximate operator that governs the wave envelope ψ to the second order in the geometrical-optics parameter. Then, we further simplify this envelope operator by assuming that the gradient of ψ transverse to the local group velocity is much larger than the corresponding parallel gradient. This leads to a parabolic differential equation for ψ (“quasioptical equation”) on the basis of the geometrical-optics polarization vectors. Scalar and mode-converting vector beams are described on the same footing. We also explain how to apply this model to electromagnetic waves in general. In the next papers of this series, we report successful quasioptical modeling of radio frequency wave beams in magnetized plasma based on this theory.

Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

The coupled nonlinear Schrödinger equation (CNLSE) is a wave envelope evolution equation applicable to two crossing, narrow-banded wave systems. Modulational instability (MI), a feature of the nonlinear Schrödinger wave equation, is characterized (to first order) by an exponential growth of sideband components and the formation of distinct wave pulses, often containing extreme waves. Linear stability analysis of the CNLSE shows the effect of crossing angle, θ , on MI, and reveals instabilities between 0 ∘ < θ < 35 ∘ , 46 ∘ < θ < 143 ∘ , and 145 ∘ < θ < 180 ∘ . Herein, the modulational stability of crossing wavetrains seeded with symmetrical sidebands is determined experimentally from tests in a circular wave basin. Experiments were carried out at 12 crossing angles between 0 ∘ ≤ θ ≤ 88 ∘ , and strong unidirectional sideband growth was observed. This growth reduced significantly at angles beyond θ ≈ 20 ∘ , reaching complete stability at θ = 30–40 ∘ . We find satisfactory agreement between numerical predictions (using a time-marching CNLSE solver) and experimental measurements for all crossing angles.

An Element-Free Galerkin Coupled with Improved Infinite Element Method for Exterior Acoustic Problem

It is known that the variable-order infinite acoustic wave envelope element (WEE) must be coupled with finite element method (FEM) by element matching for computing acoustic field radiated from radiators with complex geometric shapes. Therefore, the WEEs have to be reconstructed when the finite elements are refined or changed. To overcome the shortcoming, the element-free Galerkin (EFG) coupled with improved WEE (IWEE) method is presented to compute acoustic problems in the infinite domain. The continuity and compatibility of the acoustic pressure are maintained by IWEE that is composed of a standard WEE and a fictitious finite mesh. A key feature of the method is the introduction of the EFG method which is employed to eliminate the element matching and improve the accuracy of predicted acoustic pressure. The factors that affect the performance of the method are investigated by numerical examples, which include shape function construction, the weight function and the size of the influence domain. The numerical results show that the present method provides more accurate results compared to the coupled FEM-WEE method. The experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain.

Time-fractional effect on dust-acoustic double-layer waves in four-component dusty plasma

Abstract The Gardener equation (GE) is resulting by using reductive perturbation technique (RPT) for a system of dusty plasma of opposite polarity and kappa-distributed electrons and ion. Also, the time fractional (TF) Gardenr's equation obtained by Euler-Lagrange equation (ELe). It has been discover that the effect of the ions (electrons) kappa parameter ki(ke) and the ratio between temperatures of positive dust grains σ and that for ions studied, where the effect of ions kappa parameter k i on the amplitude and steepness of the double layers is more effective that the effect of both electrons kappa parameter k e and the ratio between temperatures of positive dust grains σ . The dependence of the envelope wave on the TF parameter α studied. The current inspection is founded in Jupiter, Cometary tails, and Earth's mesosphere.

Dynamic Source Model for the 2011 Tohoku Earthquake in a Wide Period Range Combining Slip Reactivation with the Short-Period Ground Motion Generation Process

This paper describes a validated dynamic rupture model of the 2011 Tohoku earthquake that reproduces both long-period (20–100 s) and short-period (3–20 s) ground motions. In order to reproduce the observed large slip area (slip asperity), we assign a large Dc (slip critical distance) area following kinematic source inversion results. Sufficiently large slip is achieved through rupture reactivation by the double-slip-weakening friction model. In order to reproduce the strong-motion generation areas (SMGAs), we assign short Dc and large stress-drop areas following empirical Green’s function (EGF) simulation results, which indicate that, although more distant from the hypocenter, SMGA1 ruptured earlier than SMGA2 or SMGA3, which are closer to the hypocenter. This observation is confirmed by the backprojection method. In order to reproduce this important feature in dynamic simulation results, we introduce a chain of small high stress-drop patches between the hypocenter and SMGA1. By systematic adjustment of stress drops and Dc, the rupture reproduces the observed sequence and timing of SMGA ruptures and the final slip derived by kinematic models. This model also reproduces the multiseismic wavefront observed from strong ground motion data recorded along the Pacific coast of the Tohoku region. We compare the velocity waveforms recorded at rock sites along the coastline with one-dimensional (1D) synthetic seismograms for periods of 20–100 s. The fit is very good at stations in the northern and central areas of Tohoku. We also perform finite-difference method (FDM) simulations for periods of 3–20 s, and confirm that our dynamic model also reproduces wave envelopes. Overall, we are able to validate the rupture process of the Tohoku earthquake.

Dispersion and Self-Modulation of Waves Propagating in a Solid with Dislocations

In the paper, the basic equations describing ultrasonic wave propagation in a medium with dislocations are derived. Dispersion relations are given under the assumption that dislocations oscillate without damping, i.e., the dislocation component of the general system is conservative. It is shown that ultrasonic wave propagation is characterized by two dispersion branches (acoustic and optical). As the wave number increases, the phase velocity of the wave belonging to the acoustic branch decreases asymptotically from a finite value to zero, while the velocity of the wave belonging to the optical branch decreases asymptotically from infinity to a finite value corresponding to the longitudinal wave velocity. The Nighthill criterion is applied to study modulation instability. The form of the wave packets into which a quasi-harmonic wave is divided due to modulation instability is determined. There can be both periodic stationary wave envelopes and a solitary stationary wave envelope. It is found how the height and width of the wave packet formed due to self-modulation of a quasi-harmonic wave correlate with the basic characteristics of the dislocation structure.

Quantitative View on the Processes Governing the Upscale Error Growth up to the Planetary Scale Using a Stochastic Convection Scheme

Abstract Two diagnostics based on potential vorticity and the envelope of Rossby waves are used to investigate upscale error growth from a dynamical perspective. The diagnostics are applied to several cases of global, real-case ensemble simulations, in which the only difference between the ensemble members lies in the random seed of the stochastic convection scheme. Based on a tendency equation for the enstrophy error, the relative importance of individual processes to enstrophy-error growth near the tropopause is quantified. After the enstrophy error is saturated on the synoptic scale, the envelope diagnostic is used to investigate error growth up to the planetary scale. The diagnostics reveal distinct stages of the error growth: in the first 12 h, error growth is dominated by differences in the convection scheme. Differences in the upper-tropospheric divergent wind then project these diabatic errors into the tropopause region (day 0.5–2). The subsequent error growth (day 2–14.5) is governed by differences in the nonlinear near-tropopause dynamics. A fourth stage of the error growth is found up to 18 days when the envelope diagnostic indicates error growth from the synoptic up to the planetary scale. Previous ideas of the multiscale nature of upscale error growth are confirmed in general. However, a novel interpretation of the governing processes is provided. The insight obtained into the dynamics of upscale error growth may help to design representations of uncertainty in operational forecast models and to identify atmospheric conditions that are intrinsically prone to large error amplification.

Maximum wave heights from global model reanalysis

Very large waves populate world oceans and challenge seafarers and offshore structures, but their long-term and global assessment is uneasy because of the scarcity of observations and their narrow time-coverage. Modern model reanalysis datasets with high spatio-temporal extent and resolution represent a valuable tool for this scope. In this paper, we use for the first time reanalysis datasets to provide a long-term and global statistical assessment of the maximum wave parameters, namely crest, crest-to-trough and envelope heights. In particular, we rely on the ERA-Interim directional wave spectra that are used to estimate the parameters of the probability distributions of wave maxima. To represent the customary single-point observations we use time extreme statistical models, while to account for the three-dimensional geometry and short-crestedness of stormy ocean waves, the statistical models are extended to space-time. In order to evaluate the accuracy of the reanalysis-based wave maxima estimates we verify them against buoy and stereo-video wave observations gathered in the North Pacific Ocean. We then provide the global assessment of maximum crest, crest-to-trough and envelope heights during typical and extreme conditions, showing the regions attaining the largest values, which we show are located in the mid-latitude storm belts, in particular in the North Atlantic Ocean. With respect to previous wave climate studies that focused on the significant wave height only, in this study we quantify the maximum wave height extent, also highlighting the role of mean wave steepness and kurtosis (measures of nonlinearity) and spectral bandwidth (measure of irregularity). Beside this, we show that the contribution of the short-crestedness is significant and that taking it into account may be relevant for the safety of navigation, ship routing and marine structural design.

The nonlinear optical properties of GaAs-based quantum wells with Kratzer–Fues confining potential: Role of applied static fields and non-resonant laser radiation

In the present paper, we report a theoretical investigations on the linear and nonlinear optical properties of a GaAs-based quantum well with the Kratzer-Fues confining potential, including the effects of external applied static electric, magnetic, and intense THz laser fields. The electronic structure of this system is obtained by using the effective-mass and envelope wave function approaches. Then, the coefficients of linear, third-order nonlinear, and total optical absorption (OACs) and relative refractive index changes (RICs) are calculated by using the iterative solutions of the compact-density matrix approach (CDMA). The obtained numerical results are reported as functions of the incident photon energy taking into account several values of static electric, static magnetic, and intense THz laser fields. These results show that the nonlinear optical properties of this structure are significantly affected by the magnitude of the external applied fields.

Nonlinear optical properties of a semi-exponential quantum wells: Effect of high-frequency intense laser field

In this present work, the effect of non-resonant, mono-chromatic intense laser field (ILF) on the linear and nonlinear optical properties of a semi-exponential quantum well (SEQW) is investigated theoretically within the effective-mass and envelope wave function approach. The bound subband energy levels and their envelope wave functions of the structure are calculated by using the diagonalization method. The optical properties of the system are obtained by using the compact-density matrix approach. The obtained numerical results show that the applied ILF and structure parameters have a significant impact on the optical properties of these structures, such as the linear, third-order nonlinear and total absorption coefficients and relative refractive index changes. Furthermore, from the findings of this study, it has been concluded that the linear and nonlinear optical properties in a SEQW under the ILF can be tuned by changing structure parameters, such as the effective range and depth of the system.

Envelope solitons in a nonlinear string with mirror nonlocality

The monoatomic nonlinear string in the presence of a mirror-type nonlocal interaction is considered. Using the method of multiscale expansions, a nonlocal equation describing the envelope wave dynamics in such medium is obtained. Properties of derived equation are studied, and exact soliton-like solutions are constructed by using Darboux method.

Generation of Wave Groups by Shear Layer Instability

The linear stability theory of wind-wave generation is revisited with an emphasis on the generation of wave groups. The outcome is the fundamental requirement that the group move with a real-valued group velocity. This implies that both the wave frequency and the wavenumber should be complex-valued, and in turn this then leads to a growth rate in the reference frame moving with the group velocity which is in general different from the temporal growth rate. In the weakly nonlinear regime, the amplitude envelope of the wave group is governed by a forced nonlinear Schrödinger equation. The effect of the wind forcing term is to enhance modulation instability both in terms of the wave growth and in terms of the domain of instability in the modulation wavenumber space. Also, the soliton solution for the wave envelope grows in amplitude at twice the linear growth rate.

On the quintic time-dependent coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics

Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painleve integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

Generalized time delay, velocity, and absorption in dispersive and absorbing media

The time delay and absorption of ultrashort electromagnetic pulses are defined from the local maximum of the wave's electric field and envelope. A time-domain framework is derived to obtain, in dispersive and absorbing linear media, the expressions of the phase and group time delays, and of the absorption undergone by the wave. These expressions are related to the dielectric properties of the medium but also to the local curvature of the wave's electric field. The result provides a unified and rational picture of the propagation, in dispersive and absorbing media, of ultrashort electromagnetic pulses in transmission and attenuated total reflection (ATR) geometries.

The Binding Energy of Impurities in the Asymmetrical Semi-Exponential Quantum Wells

Within the framework of effective-mass and envelope wave function approach, using a variational method, we have investigated the binding energy of shallow-donor impurities in an asymmetrical semi-exponential quantum wells (ASEQW). The obtained numerical results show that the structure parameters (σ and, V0) lead to significant changes in the impurity binding energy. Furthermore, these results show that the hydrogenic donor impurity binding energy may be increased or decreased as a function of the impurity position in quantum well.

Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress

In this paper we focus our attention on physical parameters of so-called envelope solitary waves beneath an ice cover. The form and propagation of waves in water basins under the ice cover are described by the 2D Euler equations. The ice cover is modelled by an elastic Kirchhoff–Love plate and is assumed to be of considerable thickness so that the inertia of the plate is taken into account in the formulation of the model. The Euler equations involve the additional pressure from the plate that freely floats at the surface of the fluid. We consider both the self-focusing case, when envelope solitary waves exist, for which the envelope speed (group speed) is equal to the velocity of filling (phase velocity) and the defocusing case, when they are replaced by so-called dark solitons. The indicated families of waves are parameterized by the velocity of the waves, and their existence is proved earlier for velocities lying in some neighbourhood of the critical value corresponding to the quiescent state. The waves, in turn, bifurcate from the quiescent state and lie in some neighbourhood of it. Analysing the critical parameters for above mentioned waves we determine characteristic values of the length and velocity of the wave. These physical parameters can be compared with possible observations detecting such waves in practice.

Efficient Modeling of Optical Excitations of Colloidal Core-Shell Semiconductor Quantum Dots by Using Symmetrized Orbitals.

An efficient method for the theoretical investigation of optical properties of semiconductor core-shell quantum dots (CSQDs) is introduced within the multiband k·p approach, which takes the advantage of the symmetry of the system. The heteroepitaxial strain and excitonic effect are included in the calculation of energy levels, envelope wave functions, exciton binding energy, and linear absorption coefficient. The adoption of symmetrized orbitals allows improvement of the computation time significantly. To avoid appearance of spurious solutions caused by imbalance of basis functions adopted, we consider an 8-band k·p model which is block-diagonalized into two conduction bands and six valence bands, that we call the 2 + 6-band model. The band nonparabolicity effect is modeled by an energy-dependent k·p term, such that the density of states obtained can mimic the actual density of states of a full-band model. The simulated absorption spectra of ZnTe/ZnSe CSQD are in good agreement with those observed experimentally, including the high rise of absorption at energies far above the absorption edge.

Electron-related optical responses in triple δ-doped quantum wells

ABSTRACTA detailed theoretical study on the electron-related optical responses in triple δ-doped GaAs quantum wells in the presence of non-resonant, monochromatic intense laser field is presented. For this purpose, we first obtained the bound subband energy levels and their corresponding envelope wave functions of the structure for different central doping concentrations within the effective-mass approximation. Then, we calculate the effect of the non-resonant intense laser field on the optical properties of this structure using the compact-density-matrix approach via the iterative method. We found that the optical absorption coefficients and refractive index changes in the triple δ-doped GaAs quantum well can be modulated by changing the central doping concentration and the intensity of the non-resonant, monochromatic laser field. In addition, it is shown that a sufficiently intense laser field suppresses the multiple quantum well configuration towards a single potential well one and the optical response becomes practically independent of the δ-doping concentration.