Gravitational radiation from a spatially compact source

A formula is obtained for the retarded solution of the Einstein field equations for a spatially compact source to second order in a weak field expansion. The energy radiated away from the source is evaluated and is found to be equal to the linearized expression despite the inclusion of back-scatter terms. It is also shown that, to the order considered, back-scatter cannot destroy the asymptotic flatness of a space-time.     

A note on reflection of spherical waves

In 1909 Sommerfeld gave an exact solution for the reflection of a spherical wave from a plane surface in terms of an oscillatory integral and also presented an asymptotic solution for the case where both source and receiver are at the boundary. Weyl (1919) presented an alternative solution and also an asymptotic solution for the case where the source is at the boundary. It is known that the general case is solved if a general solution for the case where the source is at the boundary is known. Here it is demonstrated that it is sufficient to have the general solution for the case where both source and receiver are at the boundary. This is mainly of theoretical interest, but may have practical applications. As an example it is demonstrated that Sommerfeld's approximate solution gives Ingard's (1951) approximate solution which is valid for arbitrary source and receiver heights.     

A rigorous formulation of LSZ field theory

An exact formulation of LSZ field theory is given. It is based on the Wightman axioms, asymptotic completeness, and a technical assumption stating the existence of retarded products of field operators. Reduction formulae are derived directly from the strong asymptotic condition. The GLZ-theorem, which states the conditions under which a given set of retarded functions defines a field theory, is formulated and proved in a rigorous way.     

An asymptotic outer solution applied to the Keller box method

The Keller box method (“Numerical Solutions of Partial Differential Equations, Vol. 2” (B. Hubbard Ed.), pp. 327–350, Academic Press, New York, 1970) was applied to incompressible flow past a flat plate to demonstrate that the basic computation region must extend outward from the wall until the outer boundary conditions are effectively obtained. The Keller box method was modified to include an asymptotic outer solution for the case of the self-similar solution for compressible flow in a boundary layer. Initial application of the basic and modified Keller box methods to incompressible flow past a flat plate showed similar rates of convergence but smaller RMS error for the same basic range of the independent variable when the asymptotic outer solution is applied. Furthermore, extension of the solution beyond the range of the independent variable for the numerical solution using the resulting asymptotic solution produced RMS error at least as small as the RMS error on the range of the numerical solution. Also, when the asymptotic solution was applied, a smaller range of independent variables could be used in the numerical solution to obtain the same RMS error. Numerical results for compressible flow were qualitatively the same as for the case with the incompressible velocity profile except the rate of iterative convergence was slightly slower. Application of asymptotic outer solution for incompressible flow at a two dimensional stagnation point produced similar results with smaller relative improvements. For compressible flow with smaller favorable pressure gradients than the stagnation point and with adverse pressure gradients, significant improvements were again obtained. Examination of the errors associated with the asymptotic solution reveals that greatest success is obtained for flows with thicker boundary layers and shows that the boundary layer at a two dimensional stagnation point is too thin for small error in the asymptotic solution. Despite relatively large errors in the asymptotic solutions for boundary layer in strong favorable pressure gradients where the boundary layer is thin, the boundary layer solutions generally showed improvement in error and reduction in computation times.     

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.     

On the covariance of retarded commutators

The Lorentz structure of the retarded commutators relevant to the electroproduction of pions and the weak radiative decay of hyperons is investigated within the framework of the Jost-Lehmann-Dyson representation, assuming a given asymptotic behavior for the spectral functions. In particular we study the connection between the covariance of these retarded commutators and the absence of Schwinger terms in the corresponding equal time commutators.     

Gravitational radiation in asymptotic de Sitter space

A solution of the gravitational field equations is found by using an axially symmetric metric which is asymptotically a de Sitter space metric. We use the general approach of Bondi, van der Burg, and Metzner as applied to the asymptotic flat-space case and search for the necessary conditions for gravitational radiation in asymptotic de Sitter space. We find that the character of the gravitational radiation, if it exists at all, is considerably different from that obtained in the case of asymptotic flat space.     

Exact and asymptotic solutions of the Cauchy–Poisson problem with localized initial conditions and a constant function of the bottom

In the paper, the asymptotic solutions for a problem of Cauchy–Poisson type with localized initial conditions are constructed. The bottom of the basin under consideration which was constant before the perturbation, is instantly perturbed at the initial time moment by a spatially localized function. Simplifications of the corresponding formulas are presented inside and outside the vicinity of the leading front, as well as in the case of a special choice of the initial condition. It is shown that, in the vicinity of the leading front, the asymptotic solution coincides with the asymptotic solution of the linear Boussinesq equation.     

Validity of linear acoustics for prediction of waveforms caused by sonically moving laser beams

The question is raised as to whether the analysis of the generation of sound by a laser beam moving over a water surface at the sound speed c for an interminable time period requires consideration of nonlinear effects. A principal consideration in this regard is whether the linear acoustics theory predicts a pressure waveform that is bounded in the asymptotic limit when the laser irradiation time is arbitrarily large. It is shown that a bounded asymptotic limit exists when the upper boundary condition corresponds (as is more nearly appropriate) to that of a pressure release surface, but not when it corresponds to that of a rigid surface. The asymptotic solution to the appropriate inhomogeneous wave equation is given exactly for the former case, and it is shown that the highest asymptotic amplitudes, given specified laser power and beam radius a, occur in the limit of a very small light absorption coefficient mu. In this limit, the peak amplitude is independent of mu and occurs at a depth of 0.88/mu. An approximate solution for the pressure waveform at intermediate times establishes that the characteristic time for buildup to the asymptotic limit is of the order of 2.5/(c mu 2a). If this time is substantially shorter than the time that a plane-wave pulse with the asymptotic waveform would take to develop a shock wave, then accumulative nonlinear effects are of minor importance.     

Asymptotic solutions of the Cauchy problem with localized initial conditions for linearized two-dimensional Boussinesq-type equations with variable coefficients

In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.     

Kinematic characterization of valvular opening

The evaluation of the valvular opening due to a pulse flow, as is the case of cardiac valves, requires the knowledge of the leaflets material properties and the coupled solution of the fluid and solid equations. This approach is not commonly feasible. A different approach is introduced here to describe the opening behavior of valvular leaflets by a functional kinematic relationship. The asymptotic analysis, in the limit of leaflet opening without vortex shedding, is presented for a two-dimensional rigid leaflet model under the irrotational scheme. The approach is then verified by numerical solution of the Navier-Stokes equation in asymptotic and nonasymptotic conditions.     

On Global Stability for Lifschitz–Slyozov–Wagner Like Equations

This paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz–Slyozov–Wagner (LSW) system in the case when the initial data has compact support. The main result of the paper is a proof of weak global asymptotic stability for LSW like systems. Previously strong local asymptotic stability results were obtained by Niethammer and Velázquez for the LSW system with initial data of compact support. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose, and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self-similar solution of the LSW model.     

Kinetic theory of the interaction in an electron current-electromagnetic wave system

The interaction between an electron beam and a retarded electromagnetic field with an accelerating electrostatic field (traveling wave tube with bunching) is considered. An exact steady-state solution of the kinetic equation is found for the case of a zero electrostatic field and an approximate solution is found for the case of a slowly varying electrostatic potential. A theory is constructed for the amplification of the electromagnetic wave; a critical value is indicated for the power of the amplified wave, above which stable amplification is possible. The dependence of the differential efficiency on the power of the amplified wave is calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 17–21, April, 1982.     

The Dynamics of the Autowave Front in a Model of Urban Ecosystems

A nonlinear singularly perturbed system of parabolic equations in a two-dimensional domain is considered. The system can be used to simulate the motion of an autowave front in a model of the evolution of urban ecosystems in the case of an inhomogeneous medium whose parameters vary with time. An asymptotic analysis of the problem is performed using methods of the theory of contrast structures. An asymptotic approximation of a front-type solution of the zero and first orders is obtained.     

Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering

In this paper, an exact explicit solution for the complex cubic-quintic Ginzburg-Landau equation is obtained, by using Lambert W function or omega function. More pertinently, we term them as Lambert W-kink-type solitons, begotten under the influence of intrapulse Raman scattering. Parameter domains are delineated in which these optical solitons exit in the ensuing model. We report the effect of model coefficients on the amplitude of Lambert W-kink solitons, which enables us to control efficiently the pulse intensity and hence their subsequent evolution. Also, moving fronts or optical shock-type solitons are obtained as a byproduct of this model. We explicate the mechanism to control the intensity of these fronts, by fine tuning the spectral filtering or gain parameter. It is exhibited that the frequency chirp associated with these optical solitons depends on the intensity of the wave and saturates to a constant value as the retarded time approaches its asymptotic value.     

Dissipation in the Klein-Gordon Field

The formalism of (±)-frequency parts , previously applied to solution of the D'Alembert equation in the case of the electromagnetic field, is applied to solution of the Klein-Gordon equation for the K-G field in the presence of sources. Retarded and advanced field operators are obtained as solutions, whose frequency parts satisfy a complex inhomogeneous K-G equation. Fourier transforms of these frequency parts are solutions of the central equation, which determines the time dependence of the destruction/creation operators of the field. The retarded field operator is resolved into kinetic and dissipative components. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two—a kinetic and a dissipative component. As in the analogous electromagnetic case the dissipation theorem is derived according to which work done by the dissipative power/force is negative: energy/momentum is dissipated from the sources to the K-G field. Boson quantization conditions are satisfied by the kinetic component but not by the dissipative component of the retarded K-G field.     

Form-factor representation and multipole expansion of the retarded interatomic dispersion energy

Dispersion-relation methods are used to derive a form-factor representation for the retarded dispersion energy of two hydrogen atoms that are described by relativistic electron theory. By expressing the electromagnetic form factors in terms of atomic transition matrix elements the complete multipole expansion of the interatomic dispersion energy is obtained. The long-range asymptotic limit of the successive multipole interactions is given explicitly.     

HIGH-FREQUENCY ASYMPTOTICS FOR THE HELMHOLTZ EQUATION IN A HALF-PLANE

Base on the integral representations of the solution being derived via Fokas' transform method, the high-frequency asymptotics for the solution of the Helmholtz equation, in a half-plane and subject to the Neumann condition is discussed. For the case of piecewise constant boundary data, full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals.