The response of an interacting many-particle system to a time-dependent external field can usually be treated within linear response theory. Due to rapid experimental progress in the field of laser physics, however, ultra-short laser pulses of very high intensity have become available in recent years. The electric field produced in such pulses can reach the strength of the electric field caused by atomic nuclei. If an atomic system is placed in the focus of such a laser pulse one observes a wealth of new phenomena [1] which cannot be explained by traditional perturbation theory. The non-perturbative quantum mechanical description of interacting particles moving in a very strong time-dependent external field therefore has become a prominent problem of theoretical physics. In principle, it requires a full solution of the time-dependent Schrodinger equation for the interacting many-body system, which is an exceedingly difficult task. In view of the success of density functional methods in the treatment of stationary many-body systems and in view of their numerical simplicity, a time-dependent version of density functional theory appears highly desirable, both within and beyond the regime of linear response.

Density-Functional Theory for Time-Dependent Systems

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

Although the theories of thermionic and field emission of electrons from metals are very well understood, the two types of emission have usually been studied separately by first specifying the range of temperature and field and then constructing the appropriate expression for the current. In this paper the emission is treated from a unified point of view in order to establish the ranges of temperature and field for the two types of emission and to investigate the current in the region intermediate between thermionic and field emission. A general expression for the emitted current as a function of field, temperature, and work function is set up in the form of a definite integral. Each type of emission is then associated with a technique for approximating the integral and with a characteristic dependence on the three parameters. An approximation for low fields and high temperatures leads to an extension of the Richardson-Schottky formula for thermionic emission. The values of temperature and field for which it applies are established by considering the validity of the approximation. An analogous treatment of the integral, for high fields and low temperatures, gives an extension of the Fowler-Nordheim formula for field emission, and establishes the region of temperature and field in which it applies. Also another approximate method for evaluating the integral is given which leads to a new type of dependence of the emitted current on temperature and field and which applies in a narrow region of temperature and field intermediate between the field and thermionic emission regions.

Thermionic Emission, Field Emission, and the Transition Region

An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.

/pdf/extended-finite-element-method-for-three-dimensional-crack-1fm01px0sf.pdf

Extended finite element method for three-dimensional crack modelling

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.

Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping

Handbook of Elliptic Integrals for Engineers and Physicists

Definitions and Fundamental Relations

In the foregoing sections, where elliptic integrals having diverse algebraic, trigonometric, and hyperbolic integrands were reduced to those involving elliptic functions, it is seen that certain standard integrals constantly recur. The following tables of integrals1 give explicit evaluation to those Jacobian normal forms, to which specific reference was made in each formula of Item Nos. 200–299.

Table of Integrals of Jacobian Elliptic Functions

The incomplete elliptic integral of the third kind in Legendre’s canonical form is defined by 

$${400.01^*}\left\{ {\begin{array}{*{20}{l}} {\Pi \left( {\varphi ,{\alpha ^2},k} \right) \equiv \int\limits_0^y {\frac{{dt}}{{\left( {1 - {\alpha ^2}{t^2}} \right)\sqrt {\left( {1 - {t^2}} \right)\left( {1 - {k^2}{t^{^2}}} \right)} }}} = \int\limits_0^\varphi {\frac{{d\vartheta }}{{\left( {1 - {\alpha ^2}{{\sin }^2}\vartheta } \right)\sqrt {1 - {k^2}{{\sin }^2}\vartheta } }}} } \\ {\quad \quad \quad \quad \;\; = \int\limits_0^{{u_1}} {\frac{{du}}{{1 - {\alpha ^2}s{n^2}u}} \equiv \Pi \left( {{u_1},{\alpha ^2}} \right)} ,} \\ {\quad \quad \quad \quad \;\;\left[ {y = \sin \varphi = sn\;{u_1},\quad t = \sin \vartheta = sn\;u;\quad {\alpha ^2} \ne 1\;or\;{k^2}} \right].} \end{array}} \right.$$

. When ϕ=π/2, y-1, u 1=K, the integral is said to be complete.

Elliptic Integrals of the Third Kind

Expansions in Series

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