Exact solution to the sine-Gordon equation in two space dimensions

An exact doubly periodic solution to the sine-Gordon equation in two space dimensions is found. The limiting case corresponds to a two-dimensional soliton.     

On the inverse scattering transform in two spatial and one temporal dimensions

1 IlltroductionThe.goal of this paper is to develop a formalism for (a) solving inverse problem in theplane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct thepotential q(x, y)); (b) .solving the initial value problem (for appropriately decaying initial data)of certain non-linear evolution equations in two spatial and one temporal dimension (i.e. givenq(x, ys 0) find q(x, y, t)). Several results thus obtained are also formallied. This formalism hasbeen d…     

High-order solutions of three-dimensional rough-surface scattering problems at high frequencies. I: the scalar case

We present a new high-order numerical method for the solution of high-frequency scattering problems from rough surfaces in three dimensions. The method is based on the asymptotic solution of appropriate integral equations in the high-frequency regime, in a manner that bypasses the need to resolve the fields on the scale of the wavelength of radiation. Indeed, inspired by prior work in two dimensions, we seek a solution of the integral equation in the form of a slow modulation of the incoming radiation, and we choose a series expansion in inverse powers of the wavenumber to represent the unknown slowly varying envelope. As we show, this framework can be made to yield an efficiently computable recursion for the terms in the series to any arbitrary order. The resulting algorithms generally provide a very significant improvement over classical (e.g. Kirchhoff) approximations in both accuracy and applicability and they can, in fact, effectively produce results with full double-precision accuracy for configurations of practical interest and up to the resonance regime.     

An exact solution of a regularized Mekkenko-Migdal equation in two dimensions for small Wilson loops

A regularized form of the Makeenko-Migdal equation is introduced. For the particular case of two dimensions this equation can be solved for small, simple loops. The solution reveals the third-order phase transition of Gross and Witten.     

He scattering in the study of the initial stages of adsorption and phase transitions in two dimensions

Experiments designed to measure both the (00) peak and the elastic incoherent component during adsorption at low coverages, θ ≃ 0.15, yielded greater detail in the elastic incoherent component than the (00) peak, suggestive of information both on the behaviour of isolated atoms as well as subsequent phase changes as islands of adsorbate develop. For immobile adsorbates confined to a terrace the situation both experimentally and theoretically is fairly clear, i.e. the (00) peak falls exponentially and the elastic incoherent component shows a maximum. For mobile physisorbed gases the situation seems more complex as steps and surface features can play a role as to whether initial detail is observed in the elastic incoherent versus coverage scans. However, for the first time increases and maxima have been observed for Ar and Xe on Au(111) in such scans. Some gases also show similar details on desorption both at constant temperature and on heating to initiate desorption. The heights of maxima observed in the elastic incoherent component for some systems show a dependence upon the angle of scattering from the (00) peak — a feature not understood at the moment. The answer to this probably lies in the fine details of the scattering of He atoms from the adsorbate potential at the surface. The cross section for CO-Au(111) showed a slight dependence on incident energy.     

High-order solutions of three-dimensional rough-surface scattering problems at high-frequencies. Part II: The vector electromagnetic case

We introduce a new numerical scheme for three-dimensional electromagnetic rough-surface scattering simulations with the capability of delivering very accurate results from low to high frequencies at a cost that is independent of the wavelength of radiation. The method is an extension of the ideas and techniques introduced in the first paper of this series (Waves in Random and Complex Media, 15 (2005), pp. 1-16) to the vector electromagnetic case, and it is based on the solution of an integral equation formulation of the scattering problem. As in the scalar case, the solution of the integral equation (i.e. the current) is expressed as a slow modulation of an oscillatory exponential of known phase, and explicit recursive formulae are derived for the successive terms in a series expansion of the slow envelope in inverse powers of the wavenumber. As we show, and in spite of the considerably more involved nature of the derivations and resulting formulae, the performance of the method retains the favourable characteristics that were demonstrated in the treatment of acoustic scattering problems. In particular, results with full double-precision accuracy are presented for various geometries, incidences and polarizations.     

Application of the improved method of moments to the solution of three-body scattering problems

The improved method of moments procedure which was developed in an earlier paper is applied to the solution of the three-body scattering equations for separable two-body interactions. Results of these calculations further support conclusions reached previously that the improved method of moments provides a rapid and accurate calculational procedure for solving scattering problems.     

Analytical solution for the one-dimensional scattering problem

We introduce the scattering problem in the case of one-dimensional space. The transmission factor is calculated for the whole domain of the potential energy. Connection with a field in interaction with a two-level quantum system is established for which the coupling parameter is considered to be explicitly time-dependent.     

Exact results for the Potts model in two dimensions

By considering the zeros of the partition function, we establish the following results for the Potts model on the square, triangular, and honeycomb lattices: (i) We show that there exists only one phase transition; (ii) we give an exact determination of the critical point; (iii) we prove the exponential decay of the correlation functions, in one direction at least, for all temperatures above the critical point. The results are established forq 4, whereq is the number of components.Work supported by the Fond. National Suisse de la Recherche Scientifique.Work supported in part by NSF Grant No. DMR 76-20643.     

Dynamic criteria for melting in two dimensions

The two-dimensional (2D) melting transition is analyzed on the basis of the long-time behavior of a modified Lindemann parameter in 2D gamma(L)(t) and the bond-angular correlation function g(6)(t). Using video microscopy complete positional data are obtained over five decades in time for an ensemble of superparamagnetic colloidal particles confined to an air-water interface. We find that each of the three phases (solid/hexatic/isotropic liquid) is uniquely characterized by the long-time behavior of gamma(L)(t), g(6)(t), and the non-Gaussian parameter of the relative neighbor-neighbor displacement.     

Inverse scattering for geophysical problems

A method for finding wave velocity from the scattering data is given for three-dimensional problems.     

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.     

Resonant scattering of ultracold atoms in low dimensions

Low energy scattering amplitudes for two atoms in one- and two-dimensional atomic waveguides are derived for short range isotropic and resonant interactions in high partial wave channels. Taking into account the finite width of the resonance which was neglected in previous works is shown to have important implications in the properties of the confinement induced resonances. For spin polarized fermions in quasi-1D waveguides, it imposes a strong constraint on the atomic density for achieving the Fermi Tonks Girardeau gas. For a planar waveguide, the characteristics of the 2D induced scattering resonances in p- and d-waves are determined as a function of the 3D scattering parameters and of the waveguide frequency.