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.

Low-frequency bandgap and vibration suppression mechanism of a novel square hierarchical honeycomb metamaterial

The suppression of low-frequency vibration and noise has always been an important issue in a wide range of engineering applications.To address this concern,a novel square hierarchical honeycomb metamaterial capable of reducing low-frequency noise has been developed.By combining Bloch’s theorem with the finite element method,the band structure is calculated.Numerical results indicate that this metamaterial can produce multiple low-frequency bandgaps within 500 Hz,with a bandgap ratio exceeding 50%.The first bandgap spans from 169.57 Hz to 216.42 Hz.To reveal the formation mechanism of the bandgap,a vibrational mode analysis is performed.Numerical analysis demonstrates that the bandgap is attributed to the suppression of elastic wave propagation by the vibrations of the structure’s two protruding corners and overall expansion vibrations.Additionally,detailed parametric analyses are conducted to investigate the effect ofθ,i.e.,the angle between the protruding corner of the structure and the horizontal direction,on the band structures and the total effective bandgap width.It is found that reducingθis conducive to obtaining lower frequency bandgaps.The propagation characteristics of elastic waves in the structure are explored by the group velocity,phase velocity,and wave propagation direction.Finally,the transmission characteristics of a finite periodic structure are investigated experimentally.The results indicate significant acceleration amplitude attenuation within the bandgap range,confirming the structure’s excellent low-frequency vibration suppression capability.

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.

The characteristic and prediction of scour in front of breakwaters by standing waves

In this paper bottom scours in front of vertical breakwaters by standing waves are systematically investigated, the scouring patterns, criterion for differentiating the scouring patterns and scouring mechanism are discussed ; a formula of maximum depth of scouring trough considering sediment size is given; and influence of mound foundation on bottom scours is investigated.

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.

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.

Numerical simulation of standing waves for ultrasonic purification of magnesium alloy melt

It was attempted to enhance and accelerate the separation of oxidation inclusions from magnesium alloy melt by virtue of ultrasonic agglomeration technology.In order to investigate the feasibility and effectiveness of standing waves for ultrasonic purification of magnesium alloy melt,numerical simulation and relevant experiment were carried out.The numerical simulation was broken into two main aspects.On one hand,the ultrasonic field propagations within the cells with various shapes were characterized by numerical solutions of the wave equation and with a careful choice of geometry a nearly idealized standing wave field was finally obtained.On the other hand,within such a standing wave field the agglomeration behavior of oxidation inclusions in magnesium alloy melt was analyzed and discussed.The agglomeration time and agglomeration position of oxidation inclusions were predicted with numerical simulation method.The results show that the oxidation inclusions whose apparent densities are close to the density of the melt can agglomerate at wave nodes in a short time which to a great extent enhances and accelerates the separation of oxidation inclusions from magnesium alloy melt.

A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor＇s series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.

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.

Poleward-moving recurrent auroral arcs associated with impulse-excited standing hydromagnetic waves

In Earth's high-latitude ionosphere, the poleward motion of east–west elongated auroral arcs has been attributed to standing hydromagnetic waves, especially when the auroral arcs appear quasi-periodically with a recurrence time of a few minutes. The validation of this scenario requires spacecraft observations of ultra-low-frequency hydromagnetic waves in the magnetosphere and simultaneous observations of poleward-moving auroral arcs near the spacecraft footprints. Here we present the first observational evidence from the multi-spacecraft THEMIS (Time History of Events and Macroscale Interactions during Substorms) mission and the conjugated all-sky imager to support the scenario that standing hydromagnetic waves can generate the quasi-periodic appearance of poleward-moving auroral arcs. In this specific event, the observed waves were toroidal branches of the standing hydromagnetic waves, which were excited by a pulse in the solar wind dynamic pressure. Multi-spacecraft measurements from THEMIS also suggest higher wave frequencies at lower L shells (consistent with the distribution of magnetic field line eigenfrequencies), which indicates that the phase difference across latitudes would increase with time. As time proceeds, the enlarged phase difference corresponds to a lower propagation speed of the auroral arcs, which agrees very well with the ground-based optical data.

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.

A study on the Antarctic circumpolar wave mode-A coexistence system of standing and traveling wave

The Antarctic circumpolar wave （ACW） has become a focus of the air-sea coupled Southern Ocean study since 1996, when it was discovered as an air-sea coupled interannual signal propagating eastward in the region of the Antarctic Circumpolar Current （ACC）. In order to analyze the mechanism of discontinuity along the latitudinal propagation, a new idea that ACW is a system with a traveling wave in the Southern Pacific and Atlantic Ocean and with a concurrent standing wave in the southern Indian Ocean is proposed in this paper. Based on the ideal wave principle, the average wave parameters of ACW is achieved using a non-linear approximation method, by which we find that the standing part and the traveling part possess similar radius frequency, proving their belonging to an integral system. We also give the latitudinal distribution of wave speed with which we could tell the reason for steady propagation during the same period. The spatial distribution of the propagation reveals complex process with variant spatial and temporal scales--The ENSO scale oscillation greatly impacts on the traveling process, while the result at the south of Australia indicates little connection between the Indian Ocean and the Pacific, which may be blocked by the vibration at the west of the Pacific. The advective effect of ACC on the propagation process should be examined clearly through dynamical method.

Development of a seven-cell S-band standing-wave RF-deflecting cavity for Tsinghua Thomson scattering X-ray source

A 2856-MHz,π-mode,seven-cell standingwave deflecting cavity was designed and fabricated for bunch length measurement in Tsinghua Thomson scattering X-ray source(TTX)facility.This cavity was installed in the TTX and provided a deflecting voltage of 4.2 MV with an input power of 2.5 MW.Bunch length diagnoses of electron beams with energies up to 39 MeV have been performed.In this article,the RF design of the cavity using HFSS,fabrication,and RF test processes are reviewed.High-power operation with accelerated beams and calibration of the deflecting voltage are also presented.

Standing surface acoustic wave-assisted fabrication of patterned microstructures for enhancing cell migration

Microfluidic device with patterned microstructures on the substrate surface was used to regulate cell adhesion,morphology,and functions in tissue engineering.We developed a microfluidic device which employing microscale patterned microstructures to achieve enhanced cell adhesion and migration.Biocompatible hydrogel substrates with micro-wavy and lattice-patterned microstructures were fabricated using standing surface acoustic waves and ultraviolet solidification.After seeding the L929 mouse fibroblast cells onto the patterned substrate of the microfluidic device,we determined that the viability and proliferation rate of cell migration can be greatly enhanced.Furthermore,L929 cells showed two types of gathering modes after 48 h of culturing.Cell growth was guided by the patterned substrate used in the microfluidic device and showed differences in the location distribution.Therefore,the developed microfluidic device with patterned microstructures can extend the application of in vitro cell culturing for future drug development and disease diagnosis.

Experiment on low-frequency electromagnetic waves propagating in shock-tube-generated magnetized cylindrical enveloping plasma

We propose a method of applying a static magnetic field to reduce the attenuation of the magnetic field component(SH) of low-frequency electromagnetic(LF EM) waves in dense plasma. The principle of this method is to apply a static magnetic field to limit electron movement, thereby increasing the equivalent resistance and thus reducing the induced current and SH. We consider the static magnetic field acting on the plasma of the entire induced current loop rather than on the local plasma, where the induced current is excited by the magnetic field component of LF EM waves. Analytical expressions of SH suitable for magnetized cylindrical enveloping plasma are derived by adopting an equivalent circuit approach, by which SHis calculated with respect to various plasma parameter settings. The results show that SH can be reduced under a static magnetic field and the maximum magnetic field strength that mitigates blackout is less than 0.1 T. Experiments in which LF EM waves propagate in a shock-tubegenerated magnetized cylindrical enveloping plasma are also conducted. SH measured under the magnetic field(the magnetic field strength B0 acting on the magnetic field probe was about0.06 T) reduces at f=10 MHz and f=30 MHz when ne≈1.9×1013 cm-3, which is consistent with theoretical results. The verification of the theory thus suggests that applying a static magnetic field with a weak magnetic field has the potential to improve the transmission capacity of LF EM waves in dense plasma.

Analysis of Second-Harmonic Generation of Low-Frequency Dilatational Lamb Waves in a Two-Layered Composite Plate

We analyze the effect of second-harmonic generation（SHG） of primary Lamb wave propagation in a two-layered composite plate, and then investigate the influence of interfacial properties on the said effect at low frequency. It is found that changes in the interfacial properties essentially affect the dispersion relation and then the maximum cumulative distance of the double-frequency Lamb wave generated. This will remarkably influence the efficiency of SHG. To overcome the complications arising from the inherent dispersion and multimode natures in analyzing the SHG effect of Lamb waves, the present work focuses on the analysis of the SHG effect of low-frequency dilatational Lamb wave propagation. Both the numerical analysis and finite element simulation indicate that the SHG effect of low-frequency dilatational Lamb wave propagation is found to be much more sensitive to changes in the interfacial properties than primary Lamb waves. The potential of using the SHG effect of low-frequency dilatational Lamb waves to characterize a minor change in the interfacial properties is analyzed.

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.

Defects detection in typical positions of bend pipes using low-frequency ultrasonic guided wave

In order to analyze the possibility of detecting defects in bend pipe using low-frequency ultrasonic guided wave, the propagation of T(0,1) mode and L(0,2) mode through straight-curved-straight pipe sections was studied. FE(finite element) models of bend pipe without defects and those with defects were introduced to analyze energy distribution, mode transition and defect detection of ultrasonic guided wave. FE simulation results were validated by experiments of four different bend pipes with circumferential defects in different positions. It is shown that most energy of T(0,1) mode or L(0,2) mode focuses on extrados of bend but little passes through intrados of bend, and T(0,1) mode or L(0,2) mode is converted to other possible non-axisymmetric modes when propagating through the bend and the defect after bend respectively. Furthermore, L(0,2) mode is more sensitive to circumferential notch than T(0,1) mode. The results of this work are beneficial for practical testing of pipes.