Related papers: The FD-method for solving nonlinear Klein-Gordon e…
We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical…
We consider finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation and find the condition on the nonlinearity so that the standard, one-frequency solitary waves are the only solutions with…
The Klein-Gordon equation is solved approximately for the Hulth\'{e}n potential for any angular momentum quantum number $\ell$ with the position-dependent mass. Solutions are obtained reducing the Klein-Gordon equation into a…
We report on the well-posedness of the Feynman problem for the Klein-Gordon equation on asymptotically Minkowski spacetimes. The main result is the invertibility of the Klein-Gordon operator with Feynman conditions at infinite times.…
This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely on…
We provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the…
In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-\Delta_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \…
In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine-Gordon equation and the non-integrable $\phi^4$ model. We focus, in particular, on two of their prototypical solutions, namely the kink-like…
We study the Dirac--Klein-Gordon system in $1+2$ spacetime dimensions. We show global existence of the solutions, as well as sharp time decay and linear scattering. One key advance is that we provide the first asymptotic stability result…
We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schr\"odinger equation. Here, we show…
A q-Gauss-Newton algorithm is an iterative procedure that solves nonlinear unconstrained optimization problems based on minimization of the sum squared errors of the objective function residuals. Main advantage of the algorithm is that it…
In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is…
In this paper we provide a new technique to find solutions to the Klein-Gordon-Maxwell system. The method, based on an iterative argument, permits to improve previous results where the reduction method was used. We also show how this device…
In this work, two novel classes of structure-preserving spectral Galerkin methods are proposed which based on the Crank-Nicolson scheme and the exponential scalar auxiliary variable method respectively, for solving the coupled fractional…
We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit the discrete energy conservation for the undamped model theoretically. Due to the semi-implicit treatment for the…
In this article we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not…
We extend our previous result on the focusing cubic Klein-Gordon equation in three dimensions to the non-radial case, giving a complete classification of global dynamics of all solutions with energy at most slightly above that of the ground…
Highly localized explicit solutions to multidimensional wave and Klein--Gordon--Fock equations are presented. Their Fourier transform is also found explicitly. Solutions depend on a set of parameters, and demonstrate astigmatic properties.…
For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global…
We are concerned with the existence of global in time solutions to the Cauchy problem for semi-linear Klein-Gordon equations with memory-type dissipation in $\mathbb{R}^n$. In the first place, we consider the linearized equation: applying…