Dimensional Integrable Equations Research Articles

Kinetic Theory of the Lorentz Gas

AbstractThe steady state kinetic equation for the Lorentz gas, where μ=ex · v/v, has been solved numerically. Analytic approximations have been developed for f(z, v, 0) in the limits of mean‐free‐path long and short compared with the system size. These have been shown to be good, and to provide a useful starting point for the numerical procedure employed, which iterates on f(x, v, 0). Their success suggests that counterpart approximations may work well for more complicated steady state kinetic problems of interest in the physics of the edge in proposed controlled fusion reactors, the diffusion of neutrons, the Brownian motion of heavy molecules, and the structure of shocks. It has also been demonstrated that the determination of f(z, v, 0) can be reduced to the solution of a one dimensional integral equation.

マリーナ用浮桟橋の波浪動揺特性

Recently, many types of floating pier have been developed for Marina. It is important for a planning of the floating pier to know the characteristics of motions in waves, because the motions of the floating pier and the mooring force are influenced by the mooring condition. The characteristics of motions of two floating piers moored each other are investigated by using three dimensional boundary integral equation method. The obtained results are summarized as follows ; a) The agreements between the theoretical and experimental results of the motions are fairly well. b) The motions become large when the natural period of motions is closed to the wave period. So it is important to decide the mooring condition by considering the predominant waves in Marina. c) On design of the connecting device between two floating piers, it is important to consider the characteristics of the relative motions and the restoring force.

Resonant phenomena in conductor-backed coplanar waveguides (CBCPW's)

A thorough and systematic investigation of resonant phenomena in conductor-backed coplanar waveguides (CBCPWs) is reported. A rigorous three dimensional full-wave space-domain integral equation method accompanied by an S-parameter extraction technique is used. A series of measurements has been conducted to confirm the theoretical results. A patch-resonator model and a microstriplike (MSL) model are employed to understand the origin of resonance existing in one of the test circuits. The 3-D program is also invoked by displaying the current distributions to help visualize how the CBCPW test circuits work at both through and resonant states. The results obtained validate the conclusions drawn for the patch-resonator model and the MSL model. >

不規則波中における船体周りの波高分布について : 小型船の接近の容易さに関する考察

Motions of a small boat in the vicinity of a large ship are complicated because of three types of waves: radiation, diffraction and incident waves. In this paper, the authors propose an evaluation method for the approaching course of a small boat approaching alongside a large ship in irregular waves. The method consists of two steps. Firstly, the estimation of frequency response functions for the ship motions and the free surface elevation using the three dimensional integral equation method. Secondly, calculation of the distribution of significant wave height around the ship. The results of the numerical calculations are given and the optimum course for the small boat to the large ship is discussed.

A boundary element analysis of helically symmetric potential fields

This paper presents a boundary element analysis of helically symmetric potential fields. A two dimensional boundary integral equation is introduced in general curvilinear coordinates. On the basis of this formulation, a boundary integral equation for helically symmetric potential problems is obtained using twisted coordinates. Moreover, the boundary element solution by the present scheme is compared with perturbation and finite element solutions for a simple potential problem and it is shown that the boundary element solution is in good agreement with the finite element solution for arbitrary helical pitch values while the perturbed solution becomes inaccurate for large pitch values.

岸壁前面に任意の角度で係留された浮体式係船岸の動揺特性の計算

Some feasibility studies of a floating pier for large vessels are carried out by the Ministry of Transportation in Japan. It is very important for a planning of the floating pier to know the motions in waves. In this paper, the motions of the floating body moored obliquely in front of a quay wall are investigated by using three dimensional boundary integral equation method. The obtained results are summarized as follows; a) The motions of the floating pier moored obliquely in front of a quay wall can be calculated by using the boundary integral equation method, and the calculated results agree fairly well with the experimental ones. b) The motions of the floating body considerably varies with the mooring angle which is given as the angle between the axes of the quay wall and the floating body. c) On design of floating pier, it is necessary to select the best mooring angle from the standpoint of the motions caused by the predominant waves at the considering sea.

2浮体動揺よりみた浮体式係船岸の荷役性能について : クレーン吊り貨物の振れ回りの影響

When cargo handling is carried out in a harbour using a crane on the floating pier, the operation can be interrupted as a result of cargo swing caused by severe motions of floating pier. In this paper, the motions of two floating bodies moored along a slope are investigated by using two dimensional boundary integral equation method. A floating container terminal in port Valdez, Alaska, is treated as an example. The cargo swing and relative motions between ship and cargo are calculated.

Pomeron and odderon in QCD and a two dimensional conformal field theory

The problem of solving the Bethe-Salpeter equations in LLA for t-channel partial waves corresponding to Feynman diagrams with many reggeized gluons is simplified significantly by using their conformal invariance in the impact parameter representation and the separability property of their integral kernels. In particular, for the three gluon system with the odderon quantum numbers we obtain a one dimensional integral equation.

On 1/f noise in nonlinear physical systems described by infinite dimensional integrable equations

The nonlinear physical systems which are described by infinite dimensional integrable equations have two types of solutions: solitons which can be viewed as nonlinear localized ‘‘normal modes’’ and radiation solution which can be viewed as linear modes (termed as phonons, photons, spin waves, etc., in condensed matter physical systems). These linear modes are modulationally unstable to nonlinear normal modes and this instability is expressed by a time domain decay of their amplitude as t−1/2, which corresponds to 1/f spectrum. This mechanism is explored for three types of nonlinear equations and an analogy with 1/f noise generated as a result of low-frequency bremsstrahlung is revealed.

The non-local delta problem and (2+1)-dimensional soliton equations

A method of constructing (2+1)-dimensional non-linear integrable equations and their solutions by means of the non-local delta problem is developed. A 'basic set' of equations is obtained by using different normalisations of the non-local delta problem and the Lagrangian of the set is found. Other integrable equations, which are degenerate cases of the basic set, are also Lagrangian.

On the infinite-dimensional symmetry group of the Davey–Stewartson equations

The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)-dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.

A surface impedance boundary condition for 3-D non destructive testing modeling

In this paper, the authors present an application of the surface impedance boundary condition to eddy current non destructive testing (NDT) modeling. When the skin depth is small, the electromagnetic field problem is reduced to an exterior field problem taking into account the influence of a ferromagnetic material (permeability,conductivity) by means of the boundary condition on the interface air-metal. The surface impedance model is implemented in a three dimensional boundary integral equation package and is compared with an axisymmetrical calculation. The comparison shows a good agreement between the two calculations. The application of the method is for NDT on surface cracks scanned with a circular coil fed with high frequency alternating current. In this case, the skin depth is small and the problem is three dimensional so that 2-D programs can't be used. For this special problem, the surface impedance boundary condition is particularly well adapted and the proposed method shows encouraging results.

Simplifications in the stochastic Fourier transform approach to random surface scattering

Further developments in the application of the stochastic Fourier transform approach (SFTA) to random surface scattering are presented. It is first shown that the infinite dimensional integral equation for the stochastic Fourier transform of the surface current can be reduced to the three dimensions associated with the random surface height and slopes. A three-dimensional integral equation of the second kind is developed for the average scattered field in stochastic Fourier transform space using conditional probability density functions. Various techniques for determining the transformed current (and, subsequently, the incoherent scattered power) from the average scattered field in stochastic Fourier transform space are developed and studied from the point of view of computational suitability. The case of vanishingly small surface correlation length is reexamined and the SFTA is found to provide erroneous results for the average scattered field due to the basic failure of the magnetic field integral equation (MFIE) in this limit.

Lie Symmetry, Rational Solution and Bilinear Operator Structure for One and Two Dimensional Nonlinear Equations

We have systematized and extended the technique of Sym for obtaining the rational solutions of nonlinear equations in one and two space dimension through the use of Lie Symmetry and Hirota's bilinear operator, our method is easily extensible for higher dimensional integrable equations.

On approximation methods for the solution of one dimensional singular integral equations

In the present paper we give a survey on some new result obtained in the last few years in the theory of approximation methods for one dimensional singular equations (among others to the methods of collocation and of mechanical quadratures). Conditions for convergence and stability of these methods in the spaces L p (1<p<∞) and H α (=Lipα)(0<α<1) are formulated and their rate of convergence is determined. simultaneously here for the first time the necessitgy of the conditions for the convergence of the methods mentioned above is investigated. In the first section a general and uniform methods for the treatment of projection methods for linear operator equations in Banach spaces in presented. The Core of this methods with unbounded projectors which is closely connected with a far-reaching generalization of the stability concept of S.G. Mikhlin.

The use of three dimensional boundary integral equation method for determining stresses at tunnel interactions : Brown, ET Hocking, G Paper to SympoSium, Investigation of Stress in Rock-Advances Shafts Conference, Melbournem Aug. 1976, P55–64

The use of three dimensional boundary integral equation method for determining stresses at tunnel interactions : Brown, ET Hocking, G Paper to SympoSium, Investigation of Stress in Rock-Advances Shafts Conference, Melbournem Aug. 1976, P55–64

The Analysis of Circular Waveguide Phased Arrays*

In this work, a planar phased array of circular waveguides arranged in an equilateral triangular grid is analyzed. The boundary value problem is first formulated rather generally in terms of a vector two dimensional integral equation for an array of elements that are arranged in a doubly periodic grid along two skewed (nonorthogonal) coordinates. Dielectric plugs, covers, and loading, as well as thin irises at the aperture, are accounted for in the formulation. Numerical solutions are obtained by using the Ritz-Galerkin method to solve the integral equation. Excellent agreement with experimental measurements using a waveguide simulator is observed. The existence of forced surface wave phenomena in equilateral triangular grid arrays and their strong dependence upon the mode of excitation is also demonstrated. These phenomena are shown to exist at isolated points in the scan coordinates. Reflection characteristics as well as the polarization characteristics of the radiation pattern are illustrated at selected planes of scan for both linear and circular polarization excitation.

Molecular Collisions. III. Symmetric Top Molecules

The question of rotational energy transfer in molecular collisions is treated for the case of symmetric top molecules. The Schrödinger equation for this problem, which can be taken as an integral equation over nine variables, is reduced to a set of coupled one dimensional integral equations by means of expansions over the irreducible representations of the three-dimensional rotation group. Various selection rules are found, of which two, in indices (in the matrix elements of the potential) related to the quantum number k, have no counterpart in the previously developed theory for diatomic molecules. The results are exhibited as cross sections for the excitation of rotational states, given the states before the collision. The rotational states for the symmetric top rotors are specified by the usual quantum numbers k, l, m, and the translational states by the velocity and direction of motion. The angular dependence of the cross section is treated by means of an expansion in spherical harmonics which is new in this context and which promises to be extremely useful in the application to transport processes.

Solutions of a Bethe-Salpeter Equation for Scattering States

A complete set of scattering-state solutions of a Bethe-Salpeter equation is obtained in this paper. This equation is the one for which Wick and Cutkosky succeeded to obtain the bound-state solutions. For bound-state solutions, they have found the same degeneracy as that of the non-relativistic hydrogen atom. This result is very suggestive for the present case, and indeed the scattering-state solutions are obtained without decomposing them into partial waves as is the case for Rutherford scattering. Adopting the integral transformation method, the four dimensional integral equation is reduced to a simple ordinary differential equation. It is interesting to see that this equation is identical with Cutkosky's one for n = 0. Finally by a close inspection of the resulting equation and boundary condition, it is shown that the virtual-state solutions, if any, can be obtained by modifying the inhomogeneous boundary condition for scattering states into a homogeneous one. Furthermore, it is proved that an isolated virtual level, if it really exists, gives rise to a resonance scattering.