On integrable systems close to the toda lattice

We study the generalized discrete self-trapping (DST) system formulated in terms of the u(n) Lie-Poisson algebra as well as its noncompact analog given on the gl(n) algebra. The Hamiltonian is a quadratic-linear function of the algebra generators where the quadratic part consists of the squared generators of the Cartan subalgebra only: $$H = \sum\limits_{i = 1}^n {\frac{{\gamma _i }}{2}A_{ii}^2 + } \sum\limits_{i,j = 1}^n {m_{ij} } A_{ij} $$ Two integrable cases are discovered: one for the u(n) case and the other for the gl(n) case. The correspondingL-operators (2 × 2 andn ×n) are found which give the Lax representation for these systems. The integrable model on the gl(n) algebra looks like the Toda lattice because in this case,m _ij=c _iδ_ij-1. The corresponding 2 × 2L-operator satisfies the Sklyanin algebra.     

On the convergence of reflectionless approximations to confining potentials

The construction of reflectionless potentials supporting a prescribed spectrum of Schrödinger bound states is discussed and related to the inverse problem for confining potentials. A simple formula is derived for the Jost solution in a one-dimensional reflection-less potential with N bound states. This leads to compact expressions for the potential and the bound-state wavefunctions in terms of the bound-state energies. For symmetric potentials, N-fold product formulas are obtained for bound-state wavefunctions and their slopes at the origin. Corresponding quantities in a confining potential are given by infinite products. Comparison of the finite-product and infinite-product expressions allows a demonstration of the convergence of the reflectionless results to the confining potential results as N → ∞. Several sum rules satisfied by the reflectionless potential at the origin are applied to numerical studies of convergence.     

An analytical dispersion model for sources in the surface layer

Summary A two-dimensional model, based on an analytical solution of the diffusion equation in a shear flow, is validated by comparison with cross-wind integrated concentrations of Prairie-Grass experiment. The wind velocity and eddy diffusivity power law profiles, defined respectively by exponent α and β, are determined by fitting log-linear profiles using both Businger and Dyer flux-profile relationships. They result to depend on the vertical plume spread which is a function both of the stability and of the distance from the source. The model, with Dyer's parametrization, fits the data quite well, except for very unstable conditions. Paper presented at the 1^o Congresso del Gruppo Nazionale per la Fisica dell'Atmosfera e dell'Oceano, June 19–22, 1984, Rome.     

低马赫数射流混合层中的相干声源

通过二维射流混合层中相干时间尺度的测量,对声激励不稳定波增长过程及破碎点后的相干特性进行研究。实验结果表明,相干时间尺度在基波和第一亚谐波增长区较大,在破碎点后突然下降并在下游呈逐渐减小趋势。增加声激励强度可使不稳定波增长区的相干时间尺度增大,并对下游的流动结构有明显影响。     

Ferroelectric sources and their application to pulsed power: areview

We present in this review an account of recent research into the use of ferroelectrics as electron beam sources for pulsed power applications. The work is reviewed according to the ferroelectric material used and the switching process employed. Most of the current research uses PLZT or PZT, which can be ferroelectric, antiferroelectric, or paraelectric depending on the stoichiometry. Switching is accomplished by the application of a fast-rising electric field to the ferroelectric material. The most commonly used materials are PLZT 2/95/5 and PLZT 8/65/35 or PZT (with no lanthanum): in the former case, the applied field switches the material from the antiferroelectric state to the ferroelectric state, and in the latter cases, around a hysteresis loop. Results have been reported with ferroelectric cathodes where current densities of up to a few hundred amperes per square centimeter have been achieved, with pulse durations of several microseconds. With shorter duration pulses and PLZT cathodes, repetition rates of up to 2 MHz have been achieved. In this paper, we focus on the results reported in the literature, and include a brief account of the physical interpretation of the data. The possible use of ferroelectric sources for pulsed power applications is indicated     

On the description of classical reflectionless potentials

A repulsive potential is called reflectionless in the case of classical mechanics if limit velocities of one-dimensional particles as t → ∞ arise from the analogous velocities as t → ? ∞ by changing order. The class reflectionless potentials is investigated inside the classes of potentials with the finite range and potentials decaying as a power function of the distance.     

On the thermodynamic potentials of elastic dielectrics

It is shown that four of the eight reduced densities of thermodynamic potentials as defined in a previous paper are not invariant under rigid rotation. Another pair of mechanical variables is used, and the definition of the potentials is properly changed. The relations obtained are analogous to the well-known thermodynamic equations of nonlinear thermo-elasticity.In conclusion, the author would like to express his sincere thanks to Assist. Prof. RNDr. J. Tichý for many helpful discussions.     

On the generalization of the Enskog equation to the case of steep repulsive potentials

For a classical homogeneous system of particles interacting via steeply repulsive potentials a generalization of the Enskog equation is proposed. This kinetic equation has the properties that it reduces to the usual Enskog equation in the limit of hard-sphere potentials and that the total instead of the kinetic energy is conserved in the system. The expression for the potential energy obtained is correct at arbitrary densities in equilibrium.     

On the physical reality of edge sources

In the celebrated Sommerfeld's rigorous solution to the half-plane problem, a condition fully demonstrated states that the edge neither radiates nor absorb energy. So, the well known Grimaldi's phenomenon, the shining of the edge, is claimed as an “optical illusion” (literally optische Täuschung) in his original paper. This paper deals with an apparent contradiction about this matter showing experimentally that the edge acts as a real source.     

Surface potentials and their measurement by the diode method

This review deals with the precise meaning of work functions and surface potentials for single crystal and polycrystalline surfaces. The methods of measuring surface potentials using a diode valve are described together with a number of the experimental diode cells used for these measurements. Particular attention is paid to the value of surface potentials measured by the diode method on polycrystalline surfaces.     

Interatomic interaction potentials and their application to the description of particle scattering by a surface

The data on interatomic interaction potentials obtained from experiments are compared with the well-known theoretical models of potentials. A new analytical form of a potential that provides the best fit for experiments is proposed. The accuracy of its application is estimated as a function of the internuclear distance under consideration. Individual potentials for systems consisting of inert gases are discussed, and an algorithm for constructing potentials is proposed for cases not yet studied.     

On the theory of scattering by complex potentials

The scattering problem for a non-relativistic spinless particle under the influence of a complex effective potential, which is spherically symmetric and tends to zero faster than 1/r at infinity, is considered. Certain general relations, which illuminate the influence of the imaginary part of the potential on the scattering process, are derived with the use of the expression for the probability current density. The rigorous phase-integral method developed by N. Fröman and P. O. Fröman is used for obtaining an exact, general formula for the scattering matrix, or, equivalently, for the phase shift. The formula is expressed in terms of phase-integral approximations of an arbitrary order and certain quantities defined by convergent series. Estimating the latter quantities and omitting small corrections, an approximate formula is derived for the phase shift, valid for the case that only one complex turning point contributes essentially to the phase shift. Criteria for classifying a scattering problem as such a one-turning-point problem are given. The treatment is made general enough to also cover situations of interest in Regge-pole or complex angular momentum theory.     

On integrable rational potentials of the Dirac equation

The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.