The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion

Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.

PERIODIC BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM OF THE GENERALIZED CUBIC DOUBLE DISPERSION EQUATION

In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.

Dispersion equation of magnetoelastic shear waves in irregular monoclinic layer

This paper studies the propagation of horizontally polarized shear waves in an internal magnetoelastic monoclinic stratum with irregularity in lower interface. The stratum is sandwiched between two magnetoelastic monoclinic semi-infinite media. Dispersion equation is obtained in a closed form. In the absence of magnetic field and irregularity of the medium, the dispersion equation agrees with the equation of classical case in three layered media. The effects of magnetic field and size of irregularity on the phase velocity are depicted by means of graphs.

G-type dispersion equation under suppressed rigid boundary:analytic approach

This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentially and periodically along the depth. The displacements of the wave are found in the individual medium followed by a dispersion equation using a suitable analytic approach and a boundary condition. The prominent effect of inhomogeneity contained in the media, the rigid boundary plane, and the initial stress on the phase and group velocities is shown graphically.

Dispersion Equation of Low-Frequency Waves Driven by Temperature Anisotropy

The plasma temperature （or the kinetic pressure） anisotropy is an intrinsic characteristic of a collisionless magnetized plasma. In this paper, based on the two-fluid model, a dispersion equation of low-frequency （ω〈〈ωci, ωci the ion gyrofrequency） waves, including the plasma temperature anisotropy effect, is presented. We investigate the properties of low-frequency waves when the parallel temperature exceeds the perpendicular temperature, and especially their dependence on the propagation angle, pressure anisotropy, and energy closures. The results show that both the instable Alfven and slow modes are purely growing. The growth rate of the Alfven wave is not affected by the propagation angle or energy closures, while that of the slow wave depends sensitively on the propagation angle and energy closures as well as pressure anisotropy. The fast wave is always stable. We also show how to elaborate the symbolic calculation of the dispersion equation performed using Mathematica Notebook.

Dispersion Equation in Non-uniform Optical Waveguide

We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we make some remarks.

GUIDED CIRCUMFERENTIAL WAVES IN DOUBLE-WALLED CARBON NANOTUBES

A model of guided circumferential waves propagating in double-walled carbon nanotubes is built by the theory of wave propagation in continuum mechanics, while the van der Waals force between the inner and outer nanotube has been taken into account in the model. The dispersion curves of the guided circumferential wave propagation are studied, and some dispersion characteristics are illustrated by comparing with those of single-walled carbon nanotubes. It is found that in double-walled carbon nanotubes, the guided circumferential waves will propagate in more dispersive ways. More interactions between neighboring wave modes may take place. In particular, it has been found that a couple of wave modes may disappear at a certain frequency and that, while a couple of wave modes disappear, another new couple of wave modes are excited at the same wave number.

Scholte wave dispersion and particle motion mode in ocean and ocean crust

The dispersion equation of the Scholte wave was reviewed using the homogeneous elastic half-space covered by a liquid layer,and the range of the Scholte wave propagation velocity was examined using the dispersion equation.The displacement expressions of the Scholte waves in liquid and solid were derived.Additionally,the mode of motion of Scholte waves in liquid and solid and their variation with depth was studied.The following results were obtained:The dispersion equation shows that the propagation velocity of the fundamental Scholte wave was greater than the P-wave in liquid and less than that of the Scholte wave in homogeneous elastic half-space.In contrast,the velocity of higher-order Scholte waves was greater than that of P waves in liquid and S-waves in solid.Only the fundamental Scholte wave has no cutoff frequency.The Scholte wave at the liquid surface moved only vertically,while the particles inside the liquid medium moved elliptically.The amplitude variation with depth in the solid medium caused the particle motion to change from a retrograde ellipse to a prograde ellipse.The above results imply the study of Scholte waves in the ocean and oceanic crust and help estimate ocean depths.

Linear theory of a dielectric-loaded rectangular Cerenkov maser with a sheet electron beam

A three-dimensional model of a dielectric-loaded rectangular Cerenkov maser with a sheet electron beam for the beam-wave interaction is proposed.Based on this model,the hybrid-mode dispersion equation is derived with the Borgnis potential function by using the field-matching method.Its approximate solution is obtained under the assumption of a dilute electron beam.By using the Ansoft high frequency structural simulator（HFSS） code,the electromagnetic field distribution in the interaction structure is given.Through numerical calculations,the effects of beam thickness,beam and dielectric-layer gap distance,beam voltage,and current density on the resonant growth rate are analysed in detail.

Theoretical dispersion curves for borehole real-valued wave modes in vertically transverse isotropic formations

The dispersion curves of real-valued modes in a fluid-filled borehole are widely used in acoustic well logging.The accurate dispersion curves are the precondition of theoretical analysis and inversion process.Generally,these curves can be obtained by solving the conventional dispersion equation for isotropic formations and most vertically transverse isotropy(VTI)formations.However,if the real-valued solutions exist when the radial wavenumbers for the formation quasi-P and quasi-S equals to each other,the existed methods based on the conventional dispersion equation could lead to incorrect results for some VTI formations.Few studies have focused on the influence of these real-valued solutions on dispersion curve extraction.To remove these real-valued solutions,we have proposed a modified dispersion equation and its corresponding solving process.When solving the dispersion equation,the Scholte wave velocity of VTI formation at high frequency is used as the initial guess.The two synthetic examples including fast and slow VTI formations validate that these real-valued solutions do not contribute to the wavefield,and the new dispersion curve extraction method is suitable for all kinds of VTI formations.Consequently,the method can provide reliable dispersion curves for both theoretical analysis and anisotropic parameters inversion in VTI formations.

Modified method of surface plasmons in metal superlattices

We present a modified method to solve the surface plasmons （SPs） of semi-infinite metal/dielectric superlattices and predicted new SP modes in physics. We find that four dispersion-equation sets and all possible SP modes are determined by them. Our analysis and numerical calculations indicate that besides the SP mode obtained in the original theory, the other two SP modes are predicted, which have either a positive group velocity or a negative group velocity. We also point out the possible defect in the previous theoretical method in accordance to the linear algebra principle.

Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space

The paper studies the propagation of Love waves in a non-homogeneous substratum over an initially stressed heterogeneous half-space. The dispersion equation of phase velocity is derived. The velocities of Love waves are calculated numerically as a function of kH and presented in a number of graphs, where k is the wave number, and H is the thickness of the layer. The case of Gibson＇s half-space is also considered. It is observed that the speed of Love waves is finite in the vicinity of the surface of the half-space and vanishes as the depth increases for a particular wave number. It is also observed that an increase in compressive initial stresses causes decreases of Love waves velocity for the same frequency, and the tensile initial stress of small magnitude in the half-space causes increase of the velocity.

STUDY ON RECTANGULAR WAVEGUIDE GRATING SLOW-WAVE STRUCTURE WITH COSINE-SHAPED GROOVES

This paper focuses on a new rectangular waveguide grating Slow-Wave Structure (SWS) with cosine-shaped grooves and studies the propagation characteristics of the wave in the SWS. By using the approximate field-matching conditions,the dispersion equation and the coupling impedance of this circuit are obtained. The dispersion curves and coupling impedances of the fundamental wave are calculated and the influences of the various geometrical dimensions are discussed. The results show that the bandwidth of the cosine-shaped groove SWS is much wider than that of rectangular-shaped groove one. And reducing the groove width can broaden the frequency-band and decrease the phase-velocity,while increment of the groove-depth can also decrease phase-velocity. For above cases,the coupling impedance is more than 16Ω. The present analysis will be helpful on further study and design of the RF systems used in millimeter wave Traveling Wave Tube (TWT).

Solute Transportin Sand Columns as Affected by Effluent Surface Tension

Transport of nonreactive solutes in soils is principally controlled by soil properties, such as particle-size distribution and pore geometry. Surface tension of soil water yields capillary forces that bind the water in the soil pores. Changes in soil water surface tension by contaminants may affect flow of soil water due to decreased capillary forces, caused by lowered soil water surface tension. This study aimed at assessing solute transport in sand columns as affected by effluent surface tension. Miscible displacement （MD） tests were conducted on sand columns repacked with sands sieved from 2.0, 1.0, 0.5 and 0.25 mm screens. The MD tests were conducted with 0.05 M bromide solutions prepared using water with surface tension adjusted to 72.8, 64, 53.5 and 42 dyne/cm2. Obtained breakthrough curves were modeled with the convection-dispersion equation （CDE） model. Coefficient of hydrodynamic dispersion and pore-water velocity responded inconsistently across decreased particle-sizes and water surface tensions and this was attributed to non-uniform effect of lowered effluent surface tension on solute transport in different pore-size distribution.

THE STABILITY OF VISCOUS LIQUID JETS IN A SWIRLING GAS

Based on the linear analysis of stability, a dispersion equation is deduced which delineates the evolution of a general 3-dimensional disturbance on the free surface of an incompressible viscous liquid jet injected into a gas with swirl. Here, the dimensionless parameter J(e) is again introduced, in the meantime, another dimensionless-parameter E called as circulation is also introduced to represent the relative swirling intensity. With respect to the spatial growing disturbance mode, the numerical results obtained from solving the dispersion equation reveal the following facts. First, at the same value of E, in pace-with the changing of J(e), the variation of disturbance and the critical disturbance mode still keep the same characters. Second, the present results are the same as that of S.P. Lin when J(e) > 1; but in the range of J(e) < 1, it's no more the case, the swirl decreases the axisymmetric disturbance, yet increases the asymmetric disturbance, furthermore the swirl may make the character of the most unstable disturbance mode changed (axisymmetric or asymmetric); the above action of the swirl becomes much Stronger when J(e) << 1.

Bright and dark optical solitons in the nonlinear Schrdinger equation with fourth-order dispersion and cubic-quintic nonlinearity

By the use of an auxiliary equation, we find bright and dark optical soliton and other soliton solutions for the higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion (FOD), cubic-quintic terms, self-steepening, and nonlinear dispersive terms. Moreover, we give the formation condition of the bright and dark solitons for this higher-order NLSE.

Comparison of the performance of traditional advection-dispersion equation and mobile-immobile model for simulating solute transport in heterogeneous soils

The traditional advection-dispersion equation(ADE) and the mobile-immobile model(MIM) are widely used to describe solute transport in heterogeneous porous media. However, the fitness of the two models is casedependent. In this paper, the transport of conservative,adsorbing and degradable solutes through a 1 m heterogeneous soil column under steady flow condition was simulated by ADE and MIM, and sensitivity analysis was conducted. Results show that MIM tends to prolong the breakthrough process and decrease peak concentration for all three solutes, and tailing and skewness are more pronounced with increasing dispersivity. Breakthrough curves of the adsorbing solute simulated by MIM are less sensitive to the retardation factor compared with the results simulated by ADE. The breakthrough curves of degradable solute obtained by MIM and ADE nearly overlap with a high degradation rate coefficient, indicating that MIM and ADE perform similarly for simulating degradable solute transport when biochemical degradation prevails over the mass exchange between mobile and immobile zones. The results suggest that the physical significance of dispersivity should be carefully considered when MIM is applied to simulate the degradable solute transport and/or ADE is applied to simulate the adsorbing solute transport in highly dispersive soils.

Dispersion equation of a double shell-fluid coupled system

The dispersion behaviour of a double shell-fluid system, which consists of two thin concentric cylindrical elastic shells coupled by the entrained annular fluid, is a fundamental for investigation of its vibroacoustical characteristics. Based on Flugge's infinite shell equations,the sound wave equation for the fluid field and boundary conditions at the fluidstructure interfaces, the dispersion equationfor the system is deduced and a corresponding numerical example is given in the paper.

Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell’s Equations

In this paper an explicit finite-difference time-domain scheme for solving the Maxwell’s equations in non-staggered grids is presented.The proposed scheme for solving the Faraday’s and Amp`ere’s equations in a theoretical manner is aimed to preserve discrete zero-divergence for the electric and magnetic fields.The inherent local conservation laws in Maxwell’s equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme.The remaining spatial derivative terms in the semi-discretized Faraday’s and Amp`ere’s equations are then discretized to provide an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumbers in two space dimensions.To achieve the goal of getting the best dispersive characteristics,we propose a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations.Through the computational exercises,the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions.

Calculation and verification of formula for the range of sprinklers based on jet breakup length

Jet breakup length is an important parameter which reflects the length of sprinkler range.Based on the linear instability theory,the dispersion equation of cylindrical jet was established and the theoretical value of jet breakup length was calculated.The jet breakup length and initial amplitude of surface wave were measured by applying the high-speed photography technology.Meanwhile,the numerical simulation was conducted by combining Level Set-VOF method for describing the jet breakup length to verify the theoretical and experimental results.Within the jet velocity and working pressure range of discussion,the results of comparison showed that the theoretical analysis gave a reasonable explanation to the influence of jet velocity,nozzle diameter and nozzle cone angle on jet breakup length.Comparing the theoretical value of jet breakup length with the experimental and simulated values,the three results accorded one another.The experimental jet breakup lengths were the lowest and the simulation values were the largest,and the relative error was less than 10%,especially the theoretical value was closer to the average value.For choosing the theoretical calculation of jet breakup length,a semi-empirical and semi-theoretical formula of range for the rotating sprinkler was concluded by the curve fitting method and the fitting formula was verified.The results showed the high accuracy of the ranges determined by this formula and the average relative error was less than 2.5%.The new formula was in good agreement with the data of different types of sprinklers comparing with other empirical formulas,and the error was only 5%.Meanwhile,the possibility of using this formula widely to determine the ranges of same series of sprinkler was confirmed.