Related papers: Numerical Solutions to the Sine-Gordon Equation
We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in…
In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon…
We consider an integrable generalization of the sine-Gordon (sG) equation that was earlier derived by one of the authors using bi-Hamiltonian methods. This equation is related to the sG equation in the same way that the Camassa-Holm…
We find one- and two-soliton solutions of shifted nonlocal NLS and MKdV equations. We discuss the singular structures of these soliton solutions and present some of the graphs of them.
We derive the Klein--Gordon equation for a single scalar field coupled to gravity at second order in perturbation theory and leading order in slow-roll. This is done in two ways: we derive the Klein--Gordon equation first using the Einstein…
Two sets of super Riccati equations are presented which result in two linear problems of super sine-Gordon equation. The linear problems are then shown to be related to each other by a super gauge transformation and to the super…
The Cauchy problem for semi-linear Klein-Gordon equations is considered in Friedmann-Lema\^itre-Robertson-Walker spacetimes. The local and global well-posedness of the Cauchy problem is considered in Sobolev spaces. The non-existence of…
A general and systematic regularization is developed for the exact solitonic form factors of exponential operators in the (1+1)-dimensional sine-Gordon model by analytical continuation of their integral representations. The procedure is…
The cubic Klein-Gordon equation is a simple but non-trivial partial differential equation whose numerical solution has the main building blocks required for the solution of many other partial differential equations. In this study, the…
In this paper we describe the integral transform that allows to write solutions of one partial differential equation via solution of another one. This transform was suggested by the author in the case when the last equation is a wave…
We consider the Sine-Gordon model coupled to 2D gravity. We find a nonperturbative expression for the partition function as a function of the cosmological constant, the SG mass and the SG coupling constant. At genus zero, the partition…
In this article, we show how one can restore an unknown nonlinear partial differential equation of a sine-Gordon type from its linearization around an unknown stationary kink. The key idea is to regard the ground state of the linear problem…
We present a set of time quasi-periodic solutions to a nonlinear Klein-Gordon equation with a decaying nonlinear term on the torus in arbitrary dimensions. This paper generalizes the bifurcation method developed in [W2].
In this paper, we investigate the nonlinear Klein-Gordon equation on a metric star graph with three semi-infinite bonds. At the branching point, we impose a weighted continuity condition and a generalized weighted Kirchhoff condition for…
We study the soliton-type solutions of the system introduced by B. Feigin and the author in in [EF]. We show that it reduces to a top-like system, and we study the behaviour of the solutions at the lattice infinity. We compute the…
This article presents several numerical techniques for solving Laplace equation. A numerical FORTRAN solver is developed to solve the 2D laplace equation. The numerical approaches implemented in the solver include Jacobi, Gauss-Siedel,…
We expand a discrete--time lattice sine--Gordon equation on multiple lattices and obtain the partial difference equation which governs its far field behaviour. Such reduction allow us to obtain a new completely discrete nonlinear…
We investigate solutions of the Klein-Gordon equation in a class of five dimensional geometries presenting the same symmetries and asymptotic structure as the Gross-Perry-Sorkin monopole solution. Apart from globally regular metrics, we…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
We study the 2D coupled wave-Klein-Gordon systems with semi-linear null nonlinearities $Q_0$ and $Q_{\alpha\beta}$. The main result states that the solution to the 2D coupled systems exists globally provided that the initial data are small…