Theorems (most notably by Hegerfeldt) prove that an initially localized particle whose time evolution is determined by a positive Hamiltonian will violate causality. We argue that this apparent paradox is resolved for a positive energy free particle described by either the Dirac equation or the Klein-Gordon equation because such a particle cannot be localized in the sense required by the theorems.

This paper studies the Klein-Gordon equation with quadratic nonlinearity. The ansatz approach is used to first obtain the singular soliton solution of the equation along with the corresponding domain restriction. The bifurcation analysis is also carried out. By this analysis, a few more traveling wave solutions are retrieved. The bifurcation phase portraits are also given.

In this paper, the auxiliary ordinary differential equation is employed to solve the perturbed Klein–Gordon equation with quadratic nonlinearity in the (1+1)-dimension without local inductance and dissipation effect. By using this method, we obtain abundant new types of exact traveling wave solutions.

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of th...

It is well-known that the solution of the classical linear wave equation with compactly supported initial condition and vanishing initial velocity is also compactly supported in a set depending on time : the support of the solution at time t is causally related to that of the initially given condition. Reed and Simon have shown that for a real-valued Klein-Gordon equation with (nonlinear) right...

In this paper the quantum hydrodynamic approach for the Klein–Gordon equation (KGE) owning a perturbative self-interaction term is developed. The generalized model to non-Euclidean space–time allows for the determination of the quantum energy impulse tensor density of mesons, for the gravitational equation of quantum mechanical systems.

We demonstrate a family of Strichartz estimates for the conformally invariant KleinGordon equation on a class of asymptotically de Sitter spaces with C2 metrics by using well-known local Strichartz estimates and a rescaling argument. This class of metrics includes de Sitter space. We also give an application of the estimates to a semilinear Klein-Gordon equation on these spaces.

We demonstrate for the first time the possibility for explicit construction in a discrete Hamiltonian model of an exact solution of the form exp(−|n|), i.e., a discrete peakon. These discrete analogs of the well-known, continuum peakons of the Camassa-Holm equation [Phys. Rev. Lett. 71, 1661 (1993)] are found in a model different from their continuum siblings. Namely, we observe discrete peakon...

Pointing out the difference between the Discrete Nonlinear Schrödinger equation with the classical power law nonlinearity-for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice-we prove global nonexistence in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (bu...

Mass terms are often introduced into wave equations: for example introducing a mass term for a scalar field gives the Klein-Gordon equation (2 −m2)φ = 0. Proceeding similarly with the metric of general relativity one recovers a vanishing mass term because gab;c = 0. For non-metric theories gab;c = −Qcab, so that the wave equation associated with the metric (2−M)gab = 0 no longer entails vanishi...