Inhomogeneous Klein-Gordon Equation Research Articles

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity.

Semiclassical theories as initial value problems

Motivated by the initial value problem in semiclassical gravity, we study the initial value problem of a system consisting of a quantum scalar field weakly interacting with a classical one. The quantum field obeys a Klein–Gordon equation with a potential proportional to the classical field. The classical field obeys an inhomogeneous Klein–Gordon equation sourced by the renormalized expectation value of the squared quantum field in a Hadamard state, ⟨Ψ|Φ2Ψ⟩. Thus, the system of equations for the scalar field is reminiscent of the semi-classical Einstein field equations with a Klein–Gordon field, where classical geometry is sourced by the renormalized stress-energy tensor of the quantum field, and the Klein–Gordon equation depends on the metric explicitly. We show that a unique asymptotic solution for the system can be obtained perturbatively at any fixed finite order in the weak coupling from initial data, provided that the interaction is switched on and off smoothly in a spacetime region to the future of the initial data surface. This allows one to provide “free” initial data for the decoupled system that guarantees that the Wightman function of the quantum field be of Hadamard form and hence that the renormalized ⟨Ψ|Φ2Ψ⟩ exist (in a perturbative sense) and be smooth. We comment on how to relax the switching of the interaction, which might be relevant for the corresponding problem in semiclassical gravity.

Scattering problem for Klein-Gordon equation with cubic convolution nonlinearity

The scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation, we prove the existence of the scattering operator, which improves the known results in some sense.

Alternative solution of the inhomogeneous linear differential equation of order two

In this paper, we present an alternative method for solving the general inhomogeneous linear ordinary differential equation (ODE) of order two. The solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand. The main advantage of the method exposed herein is that the particular solution is computable from two different integrals. This allows the problem solver to choose the simplest integral with which to work with in order to get the final solution. For illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous Klein–Gordon equation.

Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity

We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation.

Light-cone expansion of the Dirac sea to first order in the external potential.

The perturbation of the Dirac sea to first order in the external potential is calculated in an expansion around the light cone. It is shown that the perturbation consists of a causal contribution, which describes the singular behavior of the Dirac sea on the light cone and contains bounded line integrals over the potential and its partial derivatives, and a non-causal contribution, which is a smooth function. As a preparatory step, we construct a formal solution of the inhomogeneous Klein-Gordon equation in terms of an infinite series of line integrals. More generally, the method presented can be used for an explicit analysis of Feynman diagrams of the Dirac, Klein-Gordon, and wave equations in position space.

Decay estimates for the three-dimensional inhomogeneous klein-gordon equation and applications

We introduce a new pointwise decay estimate for solutions to the Cauchy problem for the three-dimensional inhomogeneous Klein-Gordon equation: (∂ t 2 -Δ+1)u(t,x)=g(t,x), (t,x)∈R + ×R 3 ; u(0,x)=0, ∂ t u(0,x)=0

The role of the shear mach number in earthquake source dynamics

abstract The effect of the rupture velocity upon the spatial and temporal dependence of earthquake source functions is investigated. To this end a model is suggested in which the fault zone is realized as a flexible membrane under the action of a moving force with additional stiffness forces provided by the surrounding medium. The motion of each particle of the membrane is impeded by a displacement-dependent friction and radiation damping. The particle motion along the fault is found to obey an inhomogeneous Klein-Gordon equation whose solutions are derived in closed form. In the time domain, the solutions yield a particle-motion function that has frequently been derived by analysis of earthquake seismograms. The physical parameters in the theoretical source function are found to depend strongly on the Mach number, as already predicted by the theoretical directivity function. The theory excludes the possibility of supersonic rupture and asserts a transonic rupture for major shallow earthquakes and subsonic rupture for seismic events with low and intermediate magnitudes. It predicts new functional relations between the initial particle velocity at the fault's tip, D˙0, the Mach number, M = ν/β, the rise time τ, the stress drop σ∞ and the fault length L.

Singular Functions of Tachyon Field Theories

We construct the singular functions of tachyon field theories in two dimensions and in four dimensions. Specifically, we give Poincar\'e-invariant solutions of the inhomogeneous Klein-Gordon equation with unit point source (the Green's functions), and the corresponding solutions of the homogeneous Klein-Gordon equation.

Scalar radiation from a point source

We consider a classical scalar field, obeying the inhomogeneous Klein-Gordon equation, in the case of a single point source. We propose a definition for the radiated energy-momentum and give an expression for it in terms of the prescribed world line of the source where we assume that the acceleration vanishes outside a finite interval. We find that only the part of the world line with non-vanishing acceleration contributes to the radiation, which travels at all speeds less than but not equal to the speed of light. We briefly discuss the case with more than one point source.

Radiation in a Plasma. I. Čerenkov Effect

We start with the linearized plasma equations containing an isotropic pressure term, plus extra source terms ${\mathrm{J}}^{s}$ and ${\ensuremath{\rho}}^{s}$ in the Maxwell equations. The fields of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) can be decomposed into two modes. The electromagnetic (EM) mode has all the magnetic field and no charge accumulation; it is the ordinary EM field of (${\mathrm{J}}^{s}, {\ensuremath{\rho}}^{s}$) in a dispersive medium of relative dielectric constant ${\ensuremath{\epsilon}}_{r}=1\ensuremath{-}{(\frac{{\ensuremath{\omega}}_{p}}{\ensuremath{\omega}})}^{2}$. The plasma (P) mode has all the charge accumulation and no magnetic field; at great distances from the source, it becomes a longitudinal (radial) plasma wave with the usual dispersion relation for plane plasma waves. Various potentials for the EM and P modes are given by the inhomogeneous Klein-Gordon equation. The fields of a uniformly moving charged particle are found by a Lorentz transformation. When $(\frac{u}{{v}_{0}})l1$ ($u=\mathrm{particle}\mathrm{velocity}$, ${v}_{0}=\mathrm{rms}\mathrm{thermal}\mathrm{velocity}$), the EM and P fields are exponentially screened outside oblate spheroids foreshortened in the direction of motion. When $(\frac{u}{{v}_{0}})g1$, the $P$ field exists only within the Mach (\ifmmode \check{C}\else \v{C}\fi{}erenkov) cone trailing the particle. The frequency and angular spectra of the \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation are found, and the total radiated energy is found by assuming an arbitrary high-frequency cutoff due to Landau damping. The expression for total radiated energy agrees with that given by Pines and Bohm, except for the logarithmic terms.