Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Third-order superintegrable systems with potentials satisfying nonlinear equations

Abstract: The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions Systems separating in parabolic or elliptic coordinated are shown to have potentials with only non-movable singularities

Conversion of the high-mode solitons in strongly nonlocal nonlinear media

Abstract: The conversion of high-mode solitons propagating in Strongly Nonlocal Nonlinear Media (SNNM) in three coordinate systems, namely, the elliptic coordinate system, the rectangular coordinate system and the cylindrical coordinate system, based on the Snyder–Mitchell Model that describes the paraxial beam propagating in SNNM, is discussed. Through constituting the trial solution with modulating the Gaussian beam by Ince polynomials, the closed-solution of Gaussian beams in elliptic coordinate is accessed. The Ince–Gaussian (IG) beams constitute the exact and continuous transition modes between Hermite–Gaussian beams and Laguerre–Gaussian (LG) beams, which is controlled by the elliptic parameter. The conditions of conversion in the three types of solitons are given in relation to the Gouy phase invariability in stable propagation. The profiles of the IG breather at a different propagating distance are numerically obtained, and the conversions of a few IG solitons are illustrated. The difference between the IG ...

A numerical integration scheme for the evaluation of correlation energy functionals

Abstract: A numerical integration scheme is presented for three-dimensional integrals occurring in electronic structure calculations, concentrating attention on the evaluation of the correlation energy through a density-functional expression. The scheme is based on the choice of density-based weight functions that naturally partition the space into “atomic” volumes (in which the integration is performed in terms of spherical coordinates) and “diatomic” volumes (in which the integration is performed in terms of confocal elliptical coordinates). Such a choice is justified on the basis of the analytical behavior of the integrand. The attainable accuracy and the required computational effort within the proposed scheme are discussed in detail in a test application on the C60 molecule in the symmetrical configuration. Finally, a comparison with previously proposed schemes is presented. © 1993 John Wiley & Sons, Inc.

The geometrical description of electromagnetic radiation

Abstract: The paper describes the relationship between the solutions of Maxwell’s equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics; it generalizes the results achieved by Luneburg, concerning the evolution of surfaces of electromagnetic fields discontinuities If vectors and are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G, then under some conditions there is an orthogonal coordinate system in which -lines represent rays of geometrical optics, -lines point out -direction and, -lines point out -direction This coordinate system will be called phase-ray coordinate system In the article, it will be proved that field under study can be represented by two scalar functions The article will also specify the necessary and sufficient conditions for the existence of a coordinate system, generated by the solution of Maxwell’s equations with the holonomic field of the Poynting ve

Null-field boundary integral equation approach for hydrodynamic scattering by multiple circular and elliptical cylinders

Abstract: In this paper, the null-field boundary integral formulation in conjunction with the degenerate kernel and the Mathieu function in the elliptic coordinates is proposed to solve the hydrodynamic scattering problem by multiple circular and elliptical cylinders. Based on the adaptive observer system, the present method can solve the water wave problem containing circular and elliptical cylinders at the same time in a semi-analytical manner. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the polar and elliptic coordinates for circular and elliptical cylinders, respectively. Several examples are demonstrated to see the validity of the semi-analytical approach. Keyword: null-field boundary integral equation, degenerate kernel, Mathieu function, hydrodynamic scattering, elliptical cylinders.