We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε ...

The Coulomb potential is derived in " one space-one time " dimension, and introduced into Dirac and Klein-Gordon equations. The equations are solved, and somewhat surprising result-nonexistence of bound state solutions in the lower dimension-discussed and identified as another fine example of the " Klein paradox " .

We give the governing equations for multiple scalar fields in a flat Friedmann-Robertson-Walker (FRW) background spacetime on all scales, allowing for metric and field perturbations up to second order. We then derive the Klein-Gordon equation at second order in closed form in terms of gaugeinvariant perturbations of the fields in the uniform curvature gauge. We also give a simplified form of th...

We derive the long-time decay in weighted norms for solutions of the discrete 3D Schrödinger and Klein-Gordon equations.

We prove that the Maxwell-Klein-Gordon equations on R relative to the Coulomb gauge are locally well-posed for initial data in H for all ε > 0. This builds on previous work by Klainerman and Machedon [3] who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as well as cubic terms.

We consider positive, radial and exponentially decaying steady state solutions of the general reaction-diffusion and Klein-Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction-diffusion equation and a precise center-stable manifold theorem for th...

The long-time asymptotics is derived for solutions of the discrete 3dimensional Schrödinger and Klein–Gordon equations. §

We consider a collisionless ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, the relativistic Vlasov-Klein-Gordon system, we prove localin-time existence of classical solutions and a continuation criterion which says that a solution can blow up only if the particle momenta become large. We also show that class...