Envelope Equation Research Articles

An analytical envelope model for space charge in cyclotrons

The statistical σ-matrix approach is used to derive general envelope equations for ellipsoidal bunches with space charge in isochronous and non-isochronous fixed-field accelerators (FFAs) These accelerators couple the radial and longitudinal phase spaces due to momentum dispersion induced by the space charge effect. This generates a rotation of the bunch around the local vertical axis which is known in the field as the vortex effect. A special invariant of the vortex is found which represents its total angular momentum with respect to its center. For the isochronous cyclotron, a special solution of the envelope equations is found which describes a circular bunch with equal radial and longitudinal RMS sizes. The existence of this solution is a direct consequence of the circular symmetry of the single particle Hamiltonian in a co-moving coordinate frame. The stationary solution of the circular bunch fixes the relation between bunch sizes and accelerated current (or the total charge per bunch) for given emittances. The general envelope equations may easily be implemented in existing numerical envelope codes.

“Failure envelops for foundation subjected to inclined and eccentric loading considering steady state and transient flow conditions in unsaturated soils”

In this paper, an analytical method and semi-empirical equations are presented for determining the bearing capacity failure envelopes for foundations subjected to inclined or eccentric loads in unsaturated soils considering the influence of steady and transient state flow conditions. Firstly, the analytical method is proposed extending the slip line theory coupled with analytical solutions for different matric suction profiles. The failure envelopes in the V - H (Vertical - Horizontal) and V - M (Vertical - Moment) loading spaces for two fine-grained soils were investigated by conducting parametric studies using the proposed method. The influence of the ground water table depth variation, internal friction angles, surface flux boundary conditions and different infiltration durations were considered. Based on the parametric studies results, semi-empirical equations were proposed for the failure envelopes in the two loading spaces. The validity of the proposed failure envelope equations is examined by providing comparisons with the proposed analytical method and published results from the literature. Good agreement is found between the fitted failure envelopes and that obtained from the analytical method and published studies.

Research on real-time quality evaluation method for intelligent compaction of soil-filling

How to evaluate the compaction quality in real-time, i.e. how to establish the relationship between acceleration and dry density, is the key issue in the field of intelligent compaction. The response of acceleration to dry density involves two different physical mechanisms, where the relationship between force and acceleration follows the laws of kinematics and the relationship between force and void ratio (i.e. dry density) follows the constitutive law of soil. The current approach is to directly correlate dry density with acceleration, by which it is difficult to distinguish the role played by these two mechanisms. Moreover, the empirical formula established based on limited field data may not be in the correct mathematical form and may lead to invalid prediction in some cases. Firstly, the formula for peak acceleration and peak impact was obtained based on the kinematic equation, and the relationship between the acceleration and peak impact stress is further acquired. Additionally, the compaction envelope equation was established by the theoretical analysis of the compaction process under lateral restriction conditions, which describes the relationship between the peak impact stress and void ratio. The real-time calculation formulation for dry density in terms of peak acceleration was gained by coupling the above two equations based on the corresponding relationship between void ratio and dry density. Considering that there is lateral deformation during the actual compaction process, the proposed dry density formulation is approximated, and the calculation deviation caused by the difference between the actual constraint conditions and the ideal conditions can be reflected by adjusting parameters. Subsequently, the proposed compaction envelope equation was verified by the agreement between the existing laboratory experimental results and the prediction results. Finally, the real-time formulation for dry density was applied to the prediction of two sets of field tests, demonstrating that the formulation proposed in this paper is capable of predicting reasonably the compaction quality of soil.

Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields

In this paper, we are concerned with capillary–gravity waves propagating on a two-dimensional conducting fluid under the effect of an electric field imposed in the direction perpendicular to the undisturbed free surface. This work aims to investigate the impact of the electric fields on the nonlinear wave interactions in this context. The full system is mathematically difficult to solve since it is a nonlinear, two-layered, free boundary problem, and the interface dynamics results from the strong coupling between the Euler equations for the lower fluid layer and an electric contribution from the upper gas layer. To investigate electrohydrodynamic wave interactions, we propose a novel numerical scheme based on a time-dependent conformal map and an interpolation technique to conduct unsteady simulations of the fully nonlinear electrified Euler equations of the two-layered problem. To gain analytical insights, we first derive weakly nonlinear envelope equations based on the method of multiple scales for the resonant triad, long-short wave interaction, and modulation of a single-mode wavetrain (a special case of resonant quartet interaction). Summative remarks are made to illustrate the transition from 3-wave interactions to 4-wave interactions and vice versa, occurring when the coefficients of the associated nonlinear Schrödinger equation become singular. The fully nonlinear results obtained by the prescribed numerical method are compared with the predictions by the weakly nonlinear theories, and good agreement is achieved.

Characteristics of in situ stress field of coalbed methane reservoir and its influence on permeability in western Guizhou coalfield, China

In-situ stress is an important indicator for the preferential selection of coalbed methane (CBM) exploration dessert zones, and is a key factor affecting the production capacity of coalbed methane wells. Coal reservoir permeability is one of the key parameters to evaluate the recoverability and modifiability of coalbed methane and reflects the seepage capacity of coal reservoirs. In this study, in situ stress data were collected from multiple injection/fall-off tests of multiple parameter wells in western Guizhou province, China The relationships among parameters such as pore pressure (Pp), closure pressure (Pc), breakdown pressure (Pb), in situ stress, coal permeability, and depth were explore. Using Anderson’s classification method, the distribution of three different in situ stress states was counted. A new simplified model diagram of triaxial principal stress and depth in the study area is proposed by linearly fitting the triaxial principal stress and burial depth. The envelope equation and median equation of the lateral pressure coefficient k-value stress ratio with depth of burial obtained by Brown and Hoek method were calculated using hyperbolic regression algorithm. The k-values were found to be discrete at shallower depths and converge at deeper depths, gradually converging to .65. The control of in situ stress on the permeability of coal reservoirs was explored, and a strong positive correlation was found between the permeability and the Z-shaped variation of the lateral pressure coefficient k-values at shallow depths of 1,000 m. Also, the distribution pattern of vertical permeability basically corresponds to the stress transition zone from the strike-slip fault mode to the normal fault mode. The coal seam permeability has a strong sensitivity to effective in situ stress (EIS). In this study, the least squares method with multiple fitting of power exponents is applied to analyze the control mechanism of EIS on permeability in depth and reveal a new relationship between permeability and EIS that is different from that considered by previous authors. Summarizing the above research results, the vertical CBM in western Guizhou is divided into three development potential zones, and 400–1,000 m burial depth is the most favorable vertical development zone.

Dissipative solitons stabilized by nonlinear gradient terms: Time-dependent behavior and generic properties.

We study the time-dependent behavior of dissipative solitons (DSs) stabilized by nonlinear gradient terms. Two cases are investigated: first, the case of the presence of a Raman term, and second, the simultaneous presence of two nonlinear gradient terms, the Raman term and the dispersion of nonlinear gain. As possible types of time-dependence, we find a number of different possibilities including periodic behavior, quasi-periodic behavior, and also chaos. These different types of time-dependence are found to form quite frequently from a window structure of alternating behavior, for example, of periodic and quasi-periodic behaviors. To analyze the time dependence, we exploit extensively time series and Fourier transforms. We discuss in detail quantitatively the question whether all the DSs found for the cubic complex Ginzburg-Landau equation with nonlinear gradient terms are generic, meaning whether they are stable against structural perturbations, for example, to the additions of a small quintic perturbation as it arises naturally in an envelope equation framework. Finally, we examine to what extent it is possible to have different types of DSs for fixed parameter values in the equation by just varying the initial conditions, for example, by using narrow and high vs broad and low amplitudes. These results indicate an overlapping multi-basin structure in parameter space.

Design of quasi-phase-matching nonlinear crystals based on quantum computing

Quasi-phase-matching (QPM) makes it possible to design domain engineered nonlinear crystals for highly efficient and multitasking nonlinear frequency conversion. However, finding the optimal crystal domain arrangement in a meaningful time is very challenging sometimes impossible by classical computing. In this paper, we proposed a quantum annealing computing method and used D-Wave superconducting quantum computer to design aperiodically poled lithium niobate (APPLN) for coupled third harmonic generation (CTHG). We converted the optical transformation efficiency function to an Ising model which can be solved by D-Wave quantum computer. The crystal design results were simulated by using nonlinear envelope equation (NEE), which showed very similar conversion efficiencies to the crystals designed by using simulated annealing (SA) method, demonstrating that quantum annealing computing is a powerful method for QPM crystal design.

On the Universality of Microwave Envelope Equations for Narrowband Optoelectronic Oscillators

Narrowband optoelectronic oscillators are time-delayed system that can generate ultrapure microwave signals. In order to study these oscillators, one approach is to derive a time-delayed envelope equation. In this article, we show that this approach is universal and applies to a wide variety of OEO families. By studying different oscillators, it is possible to show that their dynamics is governed by an envelope equation that can be rewritten under a generalized universal form. In this article, this universal microwave envelope equation is proposed and investigated analytically for optoelectronic microwave oscillators, and the bifurcation analysis of the deterministic envelope equation is performed. The universal model is further generalized and written in a stochastic form in order to predict the phase noise performance of the oscillators.

Predicting the transverse emittance of space charge dominated beams using the phase advance scan technique and a fully connected neural network

The transverse emittance of a charged particle beam is an important figure of merit for many accelerator applications, such as ultra-fast electron diffraction, free electron lasers and the operation of new compact accelerator concepts in general. One of the easiest to implement methods to determine the transverse emittance is the phase advance scan method using a focusing element and a screen. This method has been shown to work well in the thermal regime. In the space charge dominated laminar flow regime, however, the scheme becomes difficult to apply, because of the lack of a closed description of the beam envelope including space charge effects. Furthermore, certain mathematical, as well as beamline design criteria must be met in order to ensure accurate results. In this work we show that it is possible to analyze phase advance scan data using a fully connected neural network (FCNN), even in setups, which do not meet these criteria. In a simulation study, we evaluate the perfomance of the FCNN by comparing it to a traditional fit routine, based on the beam envelope equation. Subsequently, we use a pre-trained FCNN to evaluate measured phase advance scan data, which ultimately yields much better agreement with numerical simulations. To tackle the confirmation bias problem, we employ additional mask-based emittance measurement techniques.

Eckhaus instability of stationary patterns in hyperbolic reaction–diffusion models on large finite domains

We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic–quintic Ginzburg–Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.

General formulation of Coulomb explosion dynamics of highly symmetric charge distributions

We present a theoretical approach to study the dynamics of spherical, cylindrical and ellipsoidal charge distributions under their self-Coulomb field and a stochastic force due to collisions and random motions of charged particles. The approach is based on finding the current density of the charge distribution from the charge-current continuity equation and determining the drift velocities of the particles. The latter can be used either to derive the Lagrangian of the system, or to write Newton’s equation of motion with the Lorentz force. We develop a kinetic theory to include the stochastic force due to random motions of electrons in our model. To demonstrate the efficacy of our method, we apply it to various charge distributions and compare our results to N-body simulations. We show that our method reproduces the well-known emittance term in the envelope equation of uniform spherical and cylindrical charge distributions with correct coefficients. We use our model for the gravitational collapse of an ideal gas as well as the cyclotron dynamics of a cylindrical charge distribution in a uniform magnetic field and propose a method to measure the emittance of electron beams.

Hybrid Zakharov-kinetic simulation of nonlinear stimulated Raman scattering

We present a novel 2D reduced numerical model for stimulated Raman scattering (SRS) in laser fusion plasmas in which envelope equations for the electromagnetic fields are coupled to a hybrid description of the electron species. Specifically, the electron distribution is split between a bulk part described by a Zakharov-like linear model and a kinetic tail discretized using a particle-in-cell-like (PIC) scheme. By avoiding to sample the bulk-electron distribution, this approach greatly reduces the numerical cost of SRS simulations compared with PIC codes, while still being able to describe the nonlinear evolution of the electron tail and trapping-related kinetic phenomena. First, our model is shown to reproduce accurately the linear Landau damping of an infinitesimal electron plasma wave (EPW) whose phase velocity falls into the tail of the electron distribution. Then, applying it to the simulation of the trapped-particle modulational instability of a large-amplitude EPW, results comparable to those of previously published 2D Vlasov simulations are obtained. Finally, we simulate the excitation of kinetic backward SRS from a single strong laser speckle (λ=0.527 μm, I=1016 W cm−2) in an underdense (ne=0.036 nc) plasma, which drives an EPW with wavenumber kλD≈0.34. The model predictions fairly agree with the results of a PIC simulation regarding the kinetic saturation mechanisms (i.e., trapped-particle instabilities), and with experimental data and Vlasov simulations related to the frequency shift of nonlinear EPWs. For this SRS simulation, we estimate that our hybrid model is over an order of magnitude less costly than an equivalent PIC simulation due to the lower particle count.

A novel exploration for traveling wave solutions to the integrable equation of wave packet envelope

In this paper, with the aid of symbolic computation, different types of traveling wave solutions to a model involving an integrable equation for wave packet envelope have been presented. The exp ( − Φ ( ξ ) ) -expansion and modified Kudryashov procedures have been adopted and finally 3D and 2D graphics of the obtained solutions have been plotted. The obtained results are including dark, breather, bright, and periodic soliton solutions.

Spatio–temporal evolution dynamics of ultrashort Laguerre–Gauss vortices in a dispersive and nonlinear medium

The additional degree of freedom as introduced by the orbital angular momentum (OAM) of light has revolutionized the various technological applications. Optical pulses with OAM have applications in the generation of twisted attosecond pulses, ultrafast spectroscopy, telecommunication, high harmonic generation and in many other areas of physics as well. In this paper, we present a numerical investigation of the propagation dynamics of ultrashort Laguerre Gauss (LG) vortices using nonlinear envelope equation in a dispersive and nonlinear medium. Asymmetric splitting of ultrashort LG vortices in time at its bright caustic are observed above a certain pulse power. The asymmetric splitting owes its origin to space- time focussing. We also observe that spatial evolution at pulse center (t = 0 in pulse frame) and temporal evolution at bright caustic of the ultrashort LG vortex are quite similar. In the spectral domain, oscillatory structures are formed and above a certain peak power, we observe the generation of new frequency components with more intense lower frequency components.

Self-channeling of spatially modulated femtosecond laser beams in the post-filamentation region

The propagation of high-intensity femtosecond laser pulses in air under conditions of superposed spatial phase modulation is considered theoretically. The numerical simulations are carried out on the basis of the reduced form of a nonlinear Schrödinger equation for a time-averaged electric field envelope. Initial spatial modulations are applied to pulse wavefront profiling by a staggered ( T E M 22 ) phase plate, which is simulated numerically. The dynamics of laser pulse self-focusing, filamentation, and post-filamentation self-channeling after the phase plates with variable phase jumps is studied. We show that, with specific phase modulations, the pulse filamentation region in air can be markedly shifted further and elongated compared with a nonmodulated pulse. Moreover, during the post-filamentation propagation of spatially structured radiation, the highly localized light channels are formed, possessing enhanced intensity and reduced angular divergence, which enables post-filamentation pulse self-channeling on the distance multiple exceeding the Rayleigh range.

On the use of a Pseudo-spectral method in the Asymptotic Numerical Method for the resolution of the Ginzburg–Landau envelope equation

The present work deals with the resolution of the Ginzburg–Landau envelope equation. It is a nonlinear partial differential equation that requires a robust solver. Nowadays, Newton methods are available in many existing commercial codes. However, the computation of the associated tangent operator and its factorization at each incremental step requires a computational cost. In order to reduce this computational cost, we focus on the use of a high-order solver. This algorithm, named in this work (ANM-SM), is made by associating the Asymptotic Numerical Method (ANM) with the Spectral Method (SM). The efficiency and robustness of the used algorithm are illustrated by numerical results of a simple example of a beam resting on a non-linear Winkler foundation.

New Model for Predicting the Bearing Capacity of Large Strip Foundations on Soil under Combined Loading

The bearing capacity of large strip foundations under combined loading is an important issue in geotechnical engineering, which is related to the design and stability analysis of foundations such as gravity dams, retaining walls, and ground anchorages of bridges. Herein, a new model for predicting the bearing capacity of large strip foundations under combined loading was proposed. First, a series of numerical simulation analyses of a large strip foundation resting on the surface of the soil were conducted to investigate the shape of the failure envelope in the V–M–H (vertical load, overturning moment, and horizontal load) loading space, and an improved form of failure equation was proposed. Second, the main factors influencing Vmax (the vertical ultimate bearing capacity) and the shape of the failure envelope were determined. Last, an applicable empirical equation of the failure envelope in the V–M–H loading space was presented. The results show that the deflection angle of the ellipse (θ), which is considered as a certain constant by previous studies, varies with the vertical load (V); the width of the foundation (B), the depth of the foundation (D), the cohesion of the soil (c), and the internal friction angle of the soil (φ) are the main factors influencing Vmax and the shape of the failure envelope; only parameters B, D, c, φ, and γ are needed to determine, in a unique way, the empirical equation of the failure envelope in the V–M–H loading space. The proposed model provides a convenient means of calculating the bearing capacity of large strip foundations on soil under combined loading.

Nonlinear Propagation of Gaussian Laser Beam in Magnetized Plasma: Effect of Oblique Magnetic Field

In this paper, the propagation of Gaussian laser beam in underdense and magnetized plasma has been studied. The effect of oblique external magnetic field on self-focusing of laser is specifically explored. An envelope equation governing dependence of beam-width parameter with dimensionless distance of propagation of laser is derived by adopting parabolic equation approach under WKB and paraxial approximations. It is found that increasing the angle between the laser beam magnetic field and external magnetic field decreases the self-focusing of laser propagating through plasma.

Nonlinear adiabatic electron plasma waves. II. Applications

In this article, we use the general theory derived in Paper I [M. Tacu and D. Bénisti, Phys. Plasmas 29, 052108 (2022)] in order to address several long-standing issues regarding nonlinear electron plasma waves (EPWs). First, we discuss the relevance and practical usefulness of stationary solutions to the Vlasov–Poisson system, the so-called Bernstein–Greene–Kruskal modes, to model slowly varying waves. Second, we derive an upper bound for the wave breaking limit of an EPW growing in an initially Maxwellian plasma. Moreover, we show a simple dependence of this limit as a function of kλD, with k being the wavenumber and λD the Debye length. Third, we explicitly derive the envelope equation ruling the evolution of a slowly growing plasma wave, up to an amplitude close to the wave breaking limit. Fourth, we estimate the growth of the transverse wavenumbers resulting from wavefront bowing by solving the nonlinear, nonstationary, ray tracing equations for the EPW, together with a simple model for stimulated Raman scattering.

Strength criterion for frozen silty clay considering the effect of initial water content

The initial water content and confining pressure significantly affect the deformation and strength behaviors of frozen soils. To investigate the influence of the initial state of water on the mechanical properties of frozen silty clay (FSC), triaxial compression tests are conducted on artificial FSC with five initial water contents (12.5%, 14.0%, 16.0%, 18.0%, and 20.0%) at the temperature of −6 °C. The results show that when the initial water content is 12.5%, the frozen soil samples present strain softening and strain hardening successively with increasing confining pressure. As the initial water content increases, the degree of strain hardening under the same confining pressure gradually weakens. The frozen soil samples mainly present strain softening when the initial water content is more than or equal to 18%. According to the test results, when the initial water contents ranging from 12.5% to 20%, the relationship between the strength and mean stress has three forms: increases nonlinearly, increases first and then decreases, and maintains an approximately constant value. Linear and nonlinear strength envelope equations based on different types of strength envelopes are proposed. Additionally, the relationship between the initial water content and internal friction angle is analyzed. In the p-q plane, based on the nonlinear strength characteristics of FSC with initial water contents of 12.5%, 14.0%, and 16.0%, a modified mean stress expression p* is derived to represent the nonlinear strength criterion equation according to the modified Cam-Clay model. Furthermore, a shape function in the deviatoric plane is proposed by modifying the Lade–Duncan model, and a strength criterion is established by integrating the strength functions in the p-q and deviatoric planes. The proposed strength criterion can effectively represent the nonlinear strength behaviors of FSC, including the effects of the initial water content, ice melting, and crushing.