Equations with a Small Parameter for Surface-Current Synthesis

We obtain general equations with a small parameter, which allow one, using a given radiation pattern, to synthesize the corresponding surface current on a nonclosed surface that is a part of the coordinate surface in an orthogonal curvilinear coordinate system. These equations are uniquely solvable and their solutions, in contrast to the previously known solutions, satisfy the Meixner edge condition.     

Numerical on-the-fly implementation of the action of the kinetic energy operator on a vibrational wave function: application to methanol

ABSTRACT

In quantum dynamics, physically well-adapted curvilinear coordinates are coordinates that lead to a Hamiltonian operator as separable as possible, in order to simplify the resolution of the corresponding time-independent or time-dependent Schrödinger equations. Various equivalent curvilinear expressions of the kinetic energy operator (KEO) are well known. They can be used in either an analytical or a numerical approach. The latter has the feature of allowing to straightforwardly compute the KEO in terms of sophisticated (yet easy to define) physically well-adapted curvilinear coordinates. Nevertheless, the number of terms to be computed on a full grid, scales as n²/2 (n being the number of degrees of freedom), so that, for systems with n?>?10, the memory storage of the KEO's becomes extremely demanding and therefore often unrealistic. We show here that it is possible, starting from the basic quantum expression of the KEO as a curvilinear Laplacian operator, to reduce the memory storage bottleneck by numerically computing the KEO on-the-fly, i.e. each time it is required, without computing the extrapotential term. This new approach opens the way to rigorous quantum studies of systems with many degrees of freedom. The comparison of torsional levels of methanol obtained by the present on-the-fly method with our previous results shows excellent agreement.     

Mathieu Progressive Waves

A new family of exact solutions to the wave equation representing relatively undistorted progressive waves is constructed using separation of variables in the elliptic cylindrical coordinates and one of the Bateman transforms. The general form of this Bateman transform in an orthogonal curvilinear cylindrical coordinate system is discussed and a specific problem of physical feasibility of the obtained solutions, connected with their dependence on the cyclic coordinate, is addressed. The limiting case of zero eccentricity, in which the elliptic cylindrical coordinates turn into their circular cylindrical counterparts, is shown to correspond to the focused wave modes of the Bessel-Gauss type.     

New interpolation technique for the CIP method on curvilinear coordinates

We propose a new interpolation technique for the CIP method applied to curvilinear coordinates. The CIP method can hardly maintain third-order accuracy on curvilinear coordinates. The reason for the degeneracy in accuracies has not been discussed in detail. This paper reveals the problems of the CIP method on curvilinear coordinates and presents an improved CIP method to solve the advection equation accurately. The features of the presented method are: (1) the metric computation on the upwind stencil is defined in the same manner as in the advection phase of the CIP method; and (2) gradient values in the physical domain in the computation on the curvilinear coordinates are used. Various test problems show that the improved CIP method has approximate third-order accuracy.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

New classes of exact solutions for magnetic reconnective annihilation

Analytical solutions for reconnective annihilation in curvilinear geometry are presented. These solutions are characterized by current density distributions varying both along the radial and azimuthal coordinates. They represent an extension of previously proposed models, based on purely radially dependent current densities. Possible applications of these solutions to the modeling of solar flares are also discussed.     

Factor structure of the Tomimatsu‐Sato solutions and their recurrence relations

We consider stationary axisymmetric vacuum solutions of Einstein's equations for which the Ernst potential is rational in prolate spheroidal coordinates. Extending an earlier study we show that there are several new expressions which are factorizable. In particular, we concentrate on the Tomimatsu‐Sato solutions and their recurrence relations. Various continuum limits of the recurrence relations will be discussed.     

Curvilinear coordinates in the scaling theory of tricritical points

We demonstrate the existence of homogeneity properties of the singular part of the free energy at the tricritical point of a soluble magneto-elastic model. The free energy is only homogeneous when viewed as a function of the appropriate curvilinear coordinates in the space of independent thermodynamic variables. It is not homogeneous as a function of the linear coordinates of either Griffiths of Riedel. The implications of this result for the general scaling theory of tricritical points are discussed.     

Instantons in non-Cartesian coordinates

The multi-instanton solutions of ’t Hooft and of Jackiw, Nohl, and Rebbi are generalized to the case of curvilinear coordinates. The resulting formulas are considerably simplified if the transformation of coordinates is supplemented with a gauge transformation. As a result, the gauge field develops a term that has the same form as in Cartesian coordinates and which describes pseudoparticles and a compensating addition of a geometric origin (it is determined by the coordinate frame used). The singularities of the compensating field are irrelevant to physical quantities, but they can affect gauge-dependent quantities.     

On general-relativistic and gauge field theories

The fundamental open questions of general relativity theory are the unification of the gravitational field with other fields, aiming at a unified geometrization of physics, as well as the renormalization of relativistic gravitational theory in order to obtain their self-consistent solutions. These solutions are to furnish field-theoretic particle models—a problem first discussed by Einstein. In addition, we are confronted with the issue of a coupling between gravitational and matter fields determined (not only) by Einstein's principle of equivalence, and also with the question of the geometric meaning of a gravitational quantum theory. In our view, all these problems are so closely related that they warrant a general solution. We treat mainly the concepts suggested by Einstein and Weyl.     

Gravitational spin-spin interaction can balance two positive masses

We show that a recently published paper by Ghosal and Nandi contains an unusual number of mistakes about the role of coordinates in interpreting exact solutions of Einstein's equations.     

On Performance of Methods with Third- and Fifth-Order Compact Upwind Differencing

The difference schemes for fluid dynamics type of equations based on third- and fifth-order Compact Upwind Differencing (CUD) are considered. To validate their properties following from a linear analysis, calculations were carried out using the inviscid and viscous Burgers' equation as well as the compressible Navier–Stokes equation written in the conservative form for curvilinear coordinates. In the latter case, transonic cascade flow was chosen as a representative example. The performance of the CUD methods was estimated by investigating mesh convergence of the solutions and comparing with the results of second-order schemes. It is demonstrated that the oscillation-free steep gradients solutions obtained without using smoothing techniques can provide considerable increase of accuracy even when exploiting coarse meshes.     

Numerical simulation of planar Bragg waveguide bends

An analytical form was found and transparency boundary conditions were numerically approximated for the parabolic wave equation in curvilinear coordinates. It was shown that the solutions obtained by the parabolic equation method are in agreement with the solutions to the spectral problem defining the Bragg waveguide modes. The field amplitude and bending loss were numerically simulated depending on the curvature radius and parameters of the Bragg waveguide.     

Momentum and Hamiltonian operators in generalized coordinates

The self-adjointness of momentum operators in generalized coordinates, questioned by Domingos and Caldeira is shown. The momentum operators of a particle and the kinetic part of its Hamiltonian operator constructed from them are characterized as self-adjoint operators and geometrical objects in coordinate-free form. Local coordinates of ann-dimensional Riemannian manifold are taken as the generalized coordinates of the particle. As an example the curvilinear coordinates of Euclidean space are treated. The coefficients of connection and curvature are given on the manifold for which the assumed momentum operators exist. It is found that if our momentum operators form a complete set of mutually commuting observables, the manifold is locally Euclidean, i.e., there exists a local coordinate system such that we obtain the usual Schrödinger correspondence rule.     

О ДВИжЕНИИ ЧАстц В сИММЕтРИЧНых пОльх

The paper is based on the Hamiltonian of a charged particle in a general electromagnetic field transformed to time dependent curvilinear coordinates. From the assumption of the independence of the Hamiltonian on one of the coordinates the author derives the inequalities which the particle must satisfy in its motion. The sets of points accessible to the given particle are thus determined in the space of the remaining two coordinates. The problem of the absolute maintenance of particles is paid special attention. The necessary and sufficient conditions fulfilled by a field intended for maintaining the particles are derived.     

Three-dimensional finite element for thick shells of general shape

When the thickness of a shell increases in comparison with other dimensions as seen in arch dams, pressure vessels and shelter roofs, the effect of the unit deformations and shear deformations along the thickness have to be taken into account for achieving an acceptable approximation. This however, may entail the use of three dimensional elasticity solutions in developing suitable finite elements. In the present study, a twenty node isoparametric finite element is developed for analysis of thick shells having a general shape by using the three-dimensional theory of elasticity in orthogonal curvilinear coordinates. Shell geometric properties and their derivatives, which are necessary for obtaining a displacement-strain matrix, are determined by means of the shape functions. The accuracy of the presented formulation is assessed by comparing the numerical results with those given in the three dimensional elasticity solutions as well as with those obtained by the use of three dimensional finite elements.     

The Hadamard Construction of Green's Functions on a Curved Space-Time with Symmetries

The Hadamard constituents of Green's functions for a ζ-parametrized generalization of the massless scalar d'Alembert equation to a curved space-time including the conformally invariant wave equation: the world function of space-time, the transport scalar, and the tail-term coefficients, being simultaneously coefficients in the Schwinger-DeWitt expansion of the Feynman propagator for the corresponding invariant Klein-Gordon equation, are considered on a general static spherically symmetric and (2,2)-decomposable metric. The construction equations determining the Hadamard building elements are cast into a symmetry-adapted form and used to obtain, on a specific model metric, exact explicit solutions.     

Natural curvilinear coordinates for ideal MHD equations. Non-stationary flows with constant total pressure

Equations of magnetohydrodynamics (MHD) in the natural curvilinear system of coordinates where trajectories and magnetic lines play a role of coordinate curves are reduced to the non-linear vector wave equation coupled with the incompressibility condition in the form of the generalized Cauchy integral. The symmetry group of obtained equation, equivalence transformation, and group classification with respect to the constitutive equation are calculated. New exact solutions with functional arbitrariness describing non-stationary incompressible flows with constant total pressure are given by explicit formulae. The corresponding magnetic surfaces have the shape of deformed nested cylinders, tori, or knotted tubes.     

An approximate global solution of Einstein’s equations for a rotating finite body

We obtain an approximate global stationary and axisymmetric solution of Einstein’s equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find second-order approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to check that Kerr’s metric cannot be the exterior part of that metric.     

Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials

In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schrödinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.