Consider the Klein–Gordon–Zakharov equations in $${\mathbb {R}}^{1+2}$$ , and we are interested establishing small global solution to investigating pointwise asymptotic behavior of solution. The can be regarded as a coupled semilinear wave Klein–Gordon system with quadratic nonlinearities which do not satisfy null conditions, fact that components decay sufficiently fast makes it harder conduct ...

where u = u(t, x) ∈ C(R,R) and V (x) ∈ C(R,R), by using the method of separation of variables (SV). Equations belonging to class (1) are widely used in the modern quantum physics and can be related to other linear and nonlinear equations of mathematical physics (these relations will be discussed below, at the end of the article). In particular, class (1) contains the d’Alembert equation (with V...

This is a comment on a reply to a comment on a paper of Hellaby & Dray, repeating the identification of an important mistake which is still being denied by the authors: their proposed solutions do not satisfy the Einstein-Klein-Gordon equations at a change of signature. Substitution of the proposed solutions into the Einstein-Klein-Gordon equations in unit normal coordinates yields Dirac δ term...

In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a onedimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on so...

The change of signature of a metric is explained using simple examples and methods. The Klein-Gordon field on a signature-changing background is discussed, and it is shown how the approach of Dray et al. can be corrected to ensure that the KleinGordon equation holds. Isotropic cosmologies are discussed, and it is shown how the approach of Ellis et al. can be corrected to ensure that the Einstei...

This paper presents the formulation of time-fractional Klein-Gordon equation using the Euler-Lagrange variational technique in the Riesz derivative sense and derives an approximate solitary wave solution. Our results witness that He’s variational iteration method was very efficient and powerful technique in finding the solution of the proposed equation. The basic idea described in this paper is...

In this paper, the modified simple equation method is used to construct exact periodic and soliton solutions of some nonlinear partial differential equations. Exact solutions of the nonlinear Schrödinger equation, the Hamiltonian amplitude equation, Klein-Gordon equation in 1+2 dimension, the coupled Klein-Gordon equation, the (2 + 1)-dimensional long-wave-short-wave resonance interaction equat...

We study reparametrization-invariant systems, mainly the relativistic particle and its D-dimensional extended object generalization d-brane. The corresponding matter Lagrangians naturally contain background interactions, like electromagnetism and gravity. For a dbrane that doesn’t alter the background fields, we define non-relativistic equations assuming integral sub-manifold embedding of the d...

In this paper the quantum hyperbolic equation formulated in [1] is applied to the study of the propagation of the initial thermal state of the Universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced, VP = h̄/8tP = 1.25 10 GeV where tP is a Planck time. For the barrier height V < VP the master thermal equation is the modified Klein-Gordon ...

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the u5-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.