Initial conditions in non-equilibrium statistical mechanics: I. The generation of solitary waves from discontinuities

Using an iterative projection technique method, a time-evolution equation is derived for the single-particle distribution function. An initial condition with built-in temperature and density discontinuities is used. It is shown that four soliton pairs traveling from the discontinuities may be generated. For the case of a density discontinuity alone, the classical shock tube result is recovered. It is also shown that with a temperature discontinuity, such as may be found in two metals which are brought together, it is possible to generate charged solitary pulses, in addition to the expected flow of electrons across the boundary.     

Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations with Mixed Sign Reductions and Nonzero Boundary Conditions

The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrödinger-type systems whose reductions include two equations that model certain hyperfine spin F = 1 spinor Bose-Einstein condensates, and two novel equations that were recently shown to be integrable, and that have applications in nonlinear optics and four-component fermionic condensates. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows us to develop the IST on the standard complex plane instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity, symmetries and asymptotics of the scattering eigenfunctions and scattering data are derived, and properties of the discrete spectrum are analyzed in detail. In addition, the general behavior of the soliton solutions for all four reductions is discussed, and some novel soliton solutions are presented.     

Boundary conditions for an electromagnetic field on the surface of linear and nonlinear crystals: Allowance for weak spatial dispersion and near-surface inhomogeneity of optical properties at the intermedium boundary

Boundary conditions for an electromagnetic field in the sharp-boundary model are derived for an arbitrary angle of incidence of the light with allowance made for the near-surface inhomogeneous layer actually existing and the spatial dispersion of the nonlinear medium. It is shown that the discontinuity of the tangential component of both the magnetic-field induction and electric-field intensity vectors should be taken into account. Formulas are derived for performing calculations of these discontinuities in nonlinear media.     

Determination of the boundary losses in lasers

A breakdown of the detrimental loss factor of a laser into losses within the rod (due to scattering and inactive absorption) and boundary losses (the losses in the resonator due to divergence of the laser beam) is suggested. Formulas are derived for taking account of the boundary loss factor by introducing the effective reflection coefficient of the laser resonator mirrors. The theoretical and experimental relations for variations in the boundary losses resulting from increases in resonator length are compared. It is shown that the boundary losses represent the major portion of the total detrimental losses in a laser.     

New selection rules and symmetry relations for Regge-Regge cuts

Using SU(3) and duality constraints, Worden has shown that the Finkelstein cut selection rule requires that all odd signature Regge-Regge cuts are strongly suppressed in meson-baryon scattering. We have extended Worden's analysis by constructing the SU(3) coefficients, for the Gribov vertices, that are appropriate for dual resonance models. By combining the coefficients for two Gribov vertices and preserving the Finkelstein rule, a complete set of symmetry relations is obtained. Examples of these relations are that Regge-Regge cuts do not contribute to meson-meson scattering when the external mesons have overall odd C parity, and that the I = 2 part of Δ photoproduction cannot arise from Regge-Regge cuts.     

Inequalities on left-hand discontinuities of pion-pion partial wave amplitudes

The partial wave crossing relations derived by Balachandran and Nuyts are used to derive inequalities among integrals over left-hand discontinuities of elastic pionpion scattering, weighted by Jacobi functions of the second kind. The only other input for these inequalities are — besides isospin — positivity and boundedness of the right-hand imaginary parts. It is pointed out that similar inequalities follow directly from recent inequalities of Martin.     

Exchange forces in dispersion relations investigated using circuit relations

We propose a novel method to compute in an exact manner the left-hand cut discontinuity of the electron-atom partial wave scattering amplitude in the complex energy plane within the static exchange approximation. Zero energy dispersion relations for electron-hydrogen scattering are computed numerically for illustration.     

A model for the initiation of reaction sites during the uranium–hydrogen reaction assuming enhanced hydrogen transport through linear oxide discontinuities

A model for the initiation of hydride sites on uranium metal is described for hydride attack in the region of linear discontinuities in the surface oxide film. The model considers the effect of variations in hydrogen permeation through such discontinuities due to intrinsic oxide and metal parameters and operational variables. Expressions are derived for the time dependence of the hydrogen concentration in the metal and the maximum attained hydrogen concentration in terms of these parameters. The derived expressions therefore relate hydride precipitation time in the metal underlying any oxide discontinuity to the physical width of the discontinuity, its length, the hydrogen diffusion coefficient down the oxide discontinuity, the hydrogen diffusion coefficient in the underlying metal lattice and the hydrogen concentration in the oxide discontinuity at the gas-oxide interface. The model can therefore account for how changes in operational conditions such as hydrogen pressure may change hydride attack times in the region of such oxide discontinuities. The model can also explain why, after any given time, hydride attack may occur at some but not all of the oxide discontinuities surrounding any individual metal grain. The model also considers how hydride attack times may, or may not, be modified by the intersection of linear discontinuities in the surface oxide film. Finally, the model is able to explain the reported experimental observation that oxide discontinuities having only a short rather than a long length are less preferred sites of hydride attack.     

On the influence of the nuclear quadrupole moment on the β-decay of deformed nuclei

It is pointed out that in none of the previous treatments of the quadrupole interaction in β-decay have the relevant boundary conditions on the final state been recognized or observed. The discrepancy between a purely perturbative treatment and some slightly more exact treatments is shown to stem, ultimately, from this circumstance. Similarly, the basic argument behind the prediction of discontinuities in the β-spectra of deformed nuclei is incompatible with the pertaining boundary conditions on the final state. A straightforward adherence to the proper boundary conditions results in discontinuities which, in general, are much smaller than previously anticipated. The failure to see a predicted discontinuity in a reported preliminary experiment on the β-decay of ¹⁷⁰Tm is entirely understandable, without any assumption of some kind of further hindrance in the decay.     

Plasma heating by discontinuous MHD flows in the vicinity of a magnetic reconnection region

An expression that explicitly describes variations in the internal energy of the plasma that flows through a discontinuity is derived based on the complete system of boundary conditions for the MHD equations on the discontinuity surface. The dependence of the plasma heating on the magnetic field density and configuration in the vicinity of the discontinuity surface (i.e., on the MHD flow type) is studied. The conditions of plasma heating at discontinuities in a self-consistent analytical model of magnetic reconnection are discussed.     

The method of the boundary diffraction wave for impedance surfaces

The line integral of the boundary diffraction wave theory is extended for the diffraction process of waves by the impedance surfaces with edge discontinuities. With this aim, the exact diffraction field expression of Maliuzhinets is transformed into a line integral. The method is applied to the scattering problems of waves by a spherical reflector with edge discontinuity and the diffracted fields are evaluated asymptotically. The resultant expressions of the waves are examined numerically.     

“Weighted dispersion relations” and cut to cut extrapolations

Using error-affected data for the scattering amplitude on a part of the cuts, we construct a stable extrapolation procedure for it to the remaining parts of the cuts (higher energies or crossed reactions), using Carleman-weighted dispersion relations. To this end, it is assumed that the amplitude satisfies on the cut some smoothness condition of the Hölder type.     

On reflection from a suddenly created plasma half-space: transientsolution

The reflection by a suddenly created plasma half-space of a time-harmonic plane electromagnetic wave propagating in free space is considered. The problem involves a temporal discontinuity, a spatial discontinuity, and a dispersive medium. The steady-state solution is obtained by considering the basic features of the scattering processes due to each of the discontinuities in terms of analogous transmission-line models. The electric field of the reflected wave consists of two components. One component (called component A) is of the same frequency as the incident wave frequency and is due to the spatial discontinuity. The other component (called component B) is of a different frequency and arises because of the temporal discontinuity. The B component is damped out even if the plasma is only slightly lossy. The damping rate of the B component is calculated. The transient solution is obtained through the use of Laplace transforms. The solution is given in terms of Bessel-like functions. The limiting value of this solution is shown to agree with the steady-state solution. Numerical results illustrating the transient effects are for two typical cases     

Electronuclear sum rules

Generalized sum rules are derived by integrating the electromagnetic structure functions along lines of constant ratio of momentum and energy transfer. For non-relativistic systems these sum rules are related to the conventional photonuclear sum rules by a scaling transformation. The generalized sum rules are connected with the absorptive part of the forward scattering amplitude of virtual photons. The analytic structure of the scattering amplitudes and the possible existence of dispersion relations have been investigated in schematic relativistic and non-relativistic models. While for the non-relativistic case analyticity does not hold, the relativistic scattering amplitude is analytical for time-like (but not for space-like) photons and relations similar to the Gell-Mann-Goldberger-Thirring sum rule exist.     

Universal relations for the strongly-interacting Fermi gas

Shina Tan has derived some universal relations that hold for any state of a system consisting of fermions with two spin states that have a large scattering length. These relations involve an intensive quantity called the contact that measures the number of pairs of atoms that are very close together. We show how these relations can be derived in the framework of quantum field theory using standard renormalization methods and the operator product expansion. They allow the contact density to be identified as the expectation value of a local operator constructed out of quantum fields.     

Anomalies of the external and internal gravitational fields of the Earth’s isostatically balanced crust in the quadratic approximation

Formulas that include the contribution from dipole-distributed anomalous masses represented as layers distributed in height relative to the reference ellipsoid to the gravitational field in the quadratic approximation have been derived. The relationships between the expansion coefficients of some function and its square in terms of spherical functions have been established. The contribution from the relief masses and the density jump at the Mohorovicic discontinuity is used to illustrate the results.     

Continuous transitions between discontinuous magnetohydrodynamic flows of plasma and its heating

The possibility that the type of discontinuous flow changes as the conditions gradually (continuously) change is investigated in connection with the problems arising when the results of numerical simulations of magnetic reconnection in plasma are interpreted. The conservation laws at a discontinuity surface in magnetohydrodynamics admit such transitions, but the so-called transition solutions for the boundary conditions that simultaneously satisfy two types of discontinuities should exist in this case. The specific form of such solutions has been found, and a generalized scheme of permitted transitions has been constructed on their basis. An expression for the jump in internal energy at discontinuity is derived. The dependence of the plasma heating efficiency on the type of discontinuity is considered.     

Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries

An approach to express an incident shaped beam with respect to an arbitrarily oriented spheroidal particle having layers with non-confocal boundaries is presented. To overcome the difficulty of non-confocal boundary conditions connected with different spheroidal coordinate systems, a theoretical procedure is developed to deal with the non-confocal boundary conditions by virtue of a transformation for vector wave functions. The unknown coefficients of scattered and internal electromagnetic fields are determined by solving a system of linear equations derived from the boundary conditions and relations between the spheroidal vector wave functions and spherical ones. Numerical results of the normalized scattering cross section for a two-layered non-confocal prolate spheroid are evaluated. PACS 42.25.Fx; 42.25.Bs     

Calculation of the green functions by the coupling constant dispersion relations

The discontinuities of the Green functions on the cut in the complex plane of the coupling constant are calculated by the steepest descent method. The saddle points are given by the solutions of the classical field equations at those values of the coupling constant for which the classical theory has no ground state. The Green functions at the physical values of the coupling constant are determined by dispersion relations.     

Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media

We study the coherence properties of the field generated by beam radiated from quasi-homogeneous (QH) electromagnetic source scattering on QH media. Formulas for the spectral density and spectral degree of coherence of the three dimensional scattered field are derived. The results show under assumption that the diagonal correlation coefficients of the source are proportional to each other, the far field of the scattered light satisfy two reciprocity relations analogous to that in the scalar case, that, the spectral density is proportional to the convolution of the spectral density of the source and the spatial Fourier transform of the correlation coefficient of the scattering potential; the spectral degree of coherence is proportional to the convolution of the diagonal correlation coefficients and the strength of the scattering potential.