Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L2-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.

Helical streamers guided by surface electromagnetic standing waves

The streamer that is driven by the specific pulse DC discharge parameters can stably form a three-dimensional helical plasma channel in a long dielectric tube in the low-temperature plasma experiment,in cases when there were neither external background magnetic field or other factors that destroyed the poloidal symmetry of the tube.The formation mechanism and chirality of helical streamers are discussed according to the surface electromagnetic standing wave theory.The shape of the helical streamers and the characteristics of helical branches are quantitatively analyzed to further expand the application of plasma and streamer theory in the helix problem and chiral catalytic synthesis.

Contact Analysis and Modeling of Standing Wave Linear Ultrasonic Motor

A contact model for describing the contact mechanics between the stator and slider of the standing wave linear ultrasonic motor was presented. The proposed model starts from the assumption that the vibration characteristics of the stator is not affected by the contact process. A modified friction models was used to analyze the contact problems. Firstly, the dynamic normal contact force, interface friction force, and steady-state characteristics were analyzed. Secondly, the influences of the contact layer material, the dynamic characteristics of the stator, and the pre-load on motor performance were simulated. Finally, to validate the contact model, a linear ultrasonic motor based on in-plane modes was used as an example. The corresponding results show that a set of simulation of motor performances based on the proposed contact mechanism is in good agreement with experimental results. This model is helpful to understanding the operation principle of the standing wave linear motor and thus contributes to the design of these types of motor.

Numerical Modelling of Standing Waves with Three-Dimensional Non-Linear Wave Propagation Model

Based on the theoretical high-order model with a dissipative term for non-linear and dispersive wave in water of varying depth, a 3-D mathematical model of non-linear wave propagation is presented. The model, which can be used to calculate the wave particle velocity and wave pressure, is suitable to the complicated topography whose relative depth (d/lambda(0), ratio of the characteristic water depth to the characteristic wavelength in deep-water) is equal to or smaller than one. The governing equations are discretized with the improved 2-D Crank-Nicolson method in which the first-order derivatives are corrected by Taylor series expansion, And the general boundary conditions with an arbitrary reflection coefficient and phase shift are adopted in the model. The surface elevation, horizontal and vertical velocity components and wave pressure of standing waves are numerically calculated. The results show that the numerical model can effectively simulate the complicated standing waves, and the general boundary conditions possess good adaptability.

Physical design and cooling test of C-band standing wave accelerating tube

The physical design and cooling test of a C-band 2MeV standing wave （SW） accelerating tube are described in this paper. The designed accelerating structure consists of 3-cell buncher and 4-cell accelerating section with a total length of about 163mm, excited with 1MW magnetron. Dynamic simulation presents that about 150mA beam pulse current and 30% capture efficiency can be achieved. By means of nonlinear Gauss fit on electron transverse distribution, the diameter of beam spot FWHM （full width at half maximum of density distribution） is about 0.55mm. Cooling test results of the accelerating tube show that frequencies of cavities are tuned to 5527MHz and the field distribution of bunching section is about 3：9：10.

Relationship of mode transitions and standing waves in helicon plasmas

Helicon wave plasma sources have the well-known advantages of high efficiency and high plasma density, with broad applications in many areas. The crucial mechanism lies with mode transitions, which has been an outstanding issue for years. We have built a fluid simulation model and further developed the Peking University Helicon Discharge code. The mode transitions, also known as density jumps, of a single-loop antenna discharge are reproduced in simulations for the first time. It is found that large-amplitude standing helicon waves(SHWs) are responsible for the mode transitions, similar to those of a resonant cavity for laser generation.This paper intends to give a complete and quantitative SHW resonance theory to explain the relationship of the mode transitions and the SHWs. The SHW resonance theory reasonably explains several key questions in helicon plasmas, such as mode transition and efficient power absorption, and helps to improve future plasma generation methods.

Influence of the Phase of the Antenna Current Standing Wave on the Power Flux in Ion Cyclotron Heating

For ion cyclotron resonance heating, the current on the antenna surface exists in a form of standing wave, and the phase of the poloidal current standing wave affects significantly on the performance of the coupling. In this paper, a coupling calculation is carried out based on a practical model for the loop antenna. The ion cyclotron wave coupling performance depends greatly on the antenna current propagation constant and the phase of standing wave. For a small antenna-current-propagation constant, the antenna coupling performance is more sensitive to a π/2 change in the phase of standing wave.

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHRDINGER SYSTEM

We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the SchrSdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

Standing Waves for Quasilinear Schrödinger Equations with Indefinite Nonlinearity

In this article, we consider quasilinear Schrödinger equations of the form Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Unlike all known results in the literature, the nonlinearity is allowed to be indefinite. It is very interesting from physical and mathematical viewpoint. By mountain pass theorem and some special techniques, we prove the existence of solutions for the quasilinear Schrödinger equations with indefinite nonlinearity. This indefinite problem had never been considered so far. So our main results can be regarded as complementary work in the literature.

Modeling of kinetic characteristics of alkaline-surfactant-polymer-strengthened foams decay under ultrasonic standing wave

Foaming issues are encountered at the stages in crude oil production, transportation, processing, especially in chemical flooding enhanced oil recovery(EOR) oilfields. These accumulated foams would cause a lot of trouble for downstream operation. The destruction of foams under ultrasonic has been increasingly paying attention in the background of green oilfield development. This study focuses on the decay kinetic characteristics of alkaline-surfactant-polymer-strengthened foams under the ultrasonic standing wave.The performance of the diverse foams was characterized. A decay kinetic model incorporating the energy correlation was developed and validated. The factors that affect the decay kinetic characteristics were discussed. The results indicated that the collapse rate and the collapse volume fraction decreased when the foam size decreased, the gas-liquid ratio decreased and the surface tension increased. Ultrasonic standing wave parameters have a significant impact on the decay behavior of the foam. Both the ultrasonic frequency and ultrasonic amplitude were increased by 50%, the collapse volume fraction of foams increased by about 1.25 times in the identical irradiation time. The relative deviation between the measured results and the model prediction was less than 10%. The potential collapse mechanism was also explained using the principle of energy correlation of foam surface. This study is not only beneficial to provide a robust and rigorous way to defoam of produced liquid in the alkaline/surfactant/polymer(ASP)flooding EOR process but also meaningful to well understand the decay process of oil-based foams.

Radiation force and torque on a two-dimensional circular cross-section of a non-viscous eccentric layered compressible cylinder in acoustical standing waves

The purpose of this study is to develop an analytical formalism and derive series expansions for the time-averaged force and torque exerted on a compound coated compressible liquid-like cylinder,insonified by acoustic standing waves having an arbitrary angle of incidence in the polar(transverse)plane.The host medium of wave propagation and the eccentric liquid-like cylinder are non-viscous.Numerical computations illustrate the theoretical analysis with particular emphases on the eccentricity of the cylinder,the angle of incidence and the dimensionless size parameters of the inner and coating cylindrical fluid materials.The method to derive the acoustical scattering,and radiation force and torque components conjointly uses modal matching with the addition theorem,which adequately account for the multiple wave interaction effects between the layer and core fluid materials.The results demonstrate that longitudinal and lateral radiation force components arise.Moreover,an axial radiation torque component is quantified and computed for the non-absorptive compound cylinder,arising from geometrical asymmetry considerations as the eccentricity increases.The computational results reveal the emergence of neutral,positive,and negative radiation force and torque depending on the size parameter of the cylinder,the eccentricity,and the angle of incidence of the insonifying field.Moreover,based on the law of energy conservation applied to scattering,numerical verification is accomplished by computing the extinction/scattering energy efficiency.The results may find some related applications in fluid dynamics,particle trapping,mixing and manipulation using acoustical standing waves.

The influence of divergence angle on the deposition of neutral chromium atoms using a laser standing wave

The characteristics of neutral chromium atoms in the standing wave field are discussed. Based on a semi-classical model, the motion equation of neutral atoms in the laser standing wave field is analyzed, and the trajectories of the atoms are obtained by simulations with the different divergence angles of the atomic beam. The simulation results show that the full width at half maximum （FWHM） of the stripe is 2.75 nm and the contrast is 38.5 ： 1 when the divergence angle equals 0 mrad, the FWHM is 24.1 nm and the contrast is 6.8：1 when the divergence angle equals 0.2 mrad and the FWHMs are 58.6 and 137.8 nm, and the contrasts are 3.3 ： 1 and 1.6 ： i when the divergence angles equal 0.5 and 1.0 mrad, respectively.

Motion and acceleration of electrons in high-intensity laser standing waves

The motion and the energy of electrons driven by the ponderomotive force in linearly polarized high-intensity laser standing wave fields are considered. The results show that there exists a threshold laser intensity, above which the motion of electrons incident parallel to the electric field of the laser standing waves undergoes a transition from regulation to chaos. We propose that the huge energy exchange between the electrons and the strong laser standing waves is triggered by inelastic scattering, which is related to the chaos patterns. It is shown that an electron＇s energy gain of tens of MeV can be realized for a laser intensity of 10^20 W/cm^2.

Magnetic probe diagnostics of nonlinear standing waves and bulk ohmic electron power absorption in capacitive discharges

It is recognized that standing wave effects appearing in large-area,very-high-frequency capacitively coupled plasma(CCP)reactors cause center-high plasma non-uniformity.Using a high-frequency magnetic probe,we present a direct experimental diagnostic of the nonlinear standing waves and bulk ohmic electron power absorption dynamics in low pressure CCP discharges for different driving frequencies of 13.56,30,and 60 MHz.The design,principle,calibration,and validation of the probe are described in detail.Spatial structures of the harmonics of the magnetic field,determined by the magnetic probe,were used to calculate the distributions of the harmonic current and the corresponding ohmic electron power deposition,providing insights into the behavior of nonlinear harmonics.At a low driving frequency,i.e.13.56 MHz,no remarkable nonlinear standing waves were identified and the bulk ohmic electron power absorption was observed to be negligible.The harmonic magnetic field/current was found to increase dramatically with the driving frequency,due to decreased sheath reactance and more remarkable nonlinear standing waves at a higher driving frequency,leading to the enhancements of the ohmic heating and the plasma density in the bulk,specifically at the electrode center.At a high driving frequency,i.e.60 MHz,the high-order harmonic current density and the corresponding ohmic electron power absorption exhibited a similar node structure,with the main peak on axis,and one or more minor peaks between the electrode center and the edge,contributing to the center-high profile of the plasma density.

A ring type standing wave ultrasonic motor using cantilever type longitudinal transducers

The disadvantages of classic ring type ultrasonic motor were analyzed.To obtain large mechanical output power,a ring type standing wave ultrasonic motor using cantilever type longitudinal transducers was proposed.There are four pairs of cantilevers on one side of the ring,and four PZT ceramics are set between each pair of cantilevers.A screw fastened the PZT ceramics and cantilevers together to form a longitudinal transducer.The bending vibration mode of the ring is excited by the stretching vibration of the PZT ceramics.Thus,linear simple harmonic motions are achieved at the particle on the teeth.And the driving force is the frictional force between rotor and stator.The working principle of the proposed motor was analyzed.The ring and the longitudinal transducer were designed with FEM.The sensitive parameters of resonant frequency of the transducer and ring could be gained with modal analysis.The longitudinal vibration modal of transducer and bending vibration modal of ring were degenerated,and the motion trajectories of nodes on the teeth were analyzed.The results of this paper could guide the development of this new type of motor.

ORBITAL INSTABILITY OF STANDING WAVES FOR THE GENERALIZED 3D NONLOCAL NONLINEAR SCHR?DINGER EQUATIONS

The existence and orbital instability of standing waves for the generalized three- dimensional nonlocal nonlinear SchrSdinger equations is studied. By defining some suitable functionals and a constrained variational problem, we first establish the existence of standing waves, which relys on the inner structure of the equations under consideration to overcome the drawback that nonlocal terms violate the space-scale invariance. We then show the orbital instability of standing waves. The arguments depend upon the conservation laws of the mass and of the energy.

Coherence of SH-waves near a semi-circular inclusion-the role of interference and standing waves

We present examples of a controlled numerical experiment that contribute towards understanding of the physical phenomena that lead to the reduction of coherency of strong earthquake ground motion.We show examples for separation distance of 100 m between the two points on the ground surface,which is in the range of engineering interest.Our examples illustrate the consequences of:(a)standing waves that result from interference of the incident and reflected waves from a near vertical contrast in material properties,(b)standing waves within a concave inhomogeneity(a semi-circular valley in our examples),and(c)smaller motions in the diffraction zone,behind the inhomogeneity.We show that it is possible to reduce coherency,to the extent observed for recorded strong earthquake ground motion,even by a single inclusion in a half space,for incident ground motion that is coherent.We also illustrate the combined effects of geometric spreading and finite fault width,superimposed on the otherwise dominating effects caused by interference.Our examples show reduction of coherence for specific angles of incident waves,while,for other angles of incidence,the coherence remains essentially equal to one.

EFFECTS OF BOTTOM TOPOGRAPHY ON THE STANDING WAVES IN CIRCULAR BASINS

Standing waves in the cylinder hasins with inhomogeneous bottom are considered in this paper. We assume that the inviscid, incompressible fluid is in irrotational undulatory motion. For convenience sake, cylindrical coordinates are chosen. The velocitv potentials, the wave profiles and the modified frequencies are determined (to the third order) as power series in terms of the amplitude divided by the wavelength. Axisymmetrical analytical solutions are worked out. When w=0 , the second order frequency are gained.As an example, we assume that cylinder bottom is an axisymmetricat paraboloid. We find out that the uneven bottom has influences on standing waves. In the end. we go into detail on geometric factors.

Existence and Stability of Standing Waves with Prescribed L2-Norm for a Class of Schrödinger-Bopp-Podolsky System

In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L2-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .

The Image Conversion in Optical Echo-Holography by External Electromagnetic Standing Waves

The recording and playback of information using a reverse stimulated photon—echo hologram when exposed to the recording medium pulse of non-resonant electromagnetic standing wave was considered. It was shown that the spatial intensity distribution in stimulated echo hologram response depended on the electric field intensity of non-resonant standing wave that allowed controlling by a reproducible image.