scholarly journals Stationary multi-kinks in the discrete sine-Gordon equation

Nonlinearity ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 1036-1060
Author(s):  
Ross Parker ◽  
P G Kevrekidis ◽  
Alejandro Aceves
Keyword(s):  
Spectral Problem ◽  
Matrix Equation ◽  
Gordon Equation ◽  
Point Spectrum ◽  
Sine Gordon Equation ◽  
Time Stepping ◽  
The Family ◽  
Analytic Expressions ◽  
Klein Gordon ◽  
Good Agreement

Abstract We consider the existence and spectral stability of static multi-kink structures in the discrete sine-Gordon equation, as a representative example of the family of discrete Klein–Gordon models. The multi-kinks are constructed using Lin’s method from an alternating sequence of well-separated kink and antikink solutions. We then locate the point spectrum associated with these multi-kink solutions by reducing the spectral problem to a matrix equation. For an m-structure multi-kink, there will be m eigenvalues in the point spectrum near each eigenvalue of the primary kink, and, as long as the spectrum of the primary kink is imaginary, the spectrum of the multi-kink will be as well. We obtain analytic expressions for the eigenvalues of a multi-kink in terms of the eigenvalues and corresponding eigenfunctions of the primary kink, and these are in very good agreement with numerical results. We also perform numerical time-stepping experiments on perturbations of multi-kinks, and the outcomes of these simulations are interpreted using the spectral results.

NON-UNITARY SUPERSYMMETRIC CONFORMAL FIELD THEORIES AND SUPERSYMMETRIC SINE-GORDON EQUATION

1990 ◽  
Vol 05 (25) ◽  
pp. 2071-2077 ◽  
Author(s):  
SOONKEON NAM
Keyword(s):  
Gordon Equation ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Sine Gordon Equation ◽  
Moody Algebra ◽  
Superconformal Field Theories ◽  
Conformal Field ◽  
The Family ◽  
Coset Construction

We study coset construction of superconformal minimal models using admissible representations of Kac-Moody algebra. In particular, we study supersymmetric minimal models of Wn algebra, and in particular we argue that c = −5/2 cannot be considered as a minimal model of superconformal or super-W3 algebra. In the second part of the paper, we consider superconformal field theories whose perturbations correspond to breather-breather scattering in supersymmetric sine-Gordon equations, and find a family of theories with c = −3N(4N + 3)/2(N + 1), N = 1, 2, 3, …, which is the counterpart of the family of non-unitary theories with c = −2N(6N + 5)/(2N + 3), N = 1, 2, 3, …, among which N = 1 (c = −22/5) is the Yang-Lee edge singularity.

Symmetries of generalized Klein-Gordon (including sine-Gordon) equations in three or more dimensions

1987 ◽  
Vol 101 (2) ◽  
pp. 343-348 ◽  
Author(s):  
T. J. Gordon
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Higher Dimensions ◽  
Higher Order ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Klein Gordon ◽  
Image Position ◽  
Recent Attention

Much recent attention has been devoted to those nonlinear partial differential equations admitting higher-order conservation laws (e.g. [2] and references therein) or equivalently admitting higher-order symmetries. In particular the sine-Gordon equation possesses such symmetries [5, 7] where is the two-dimensional d'Alembertian operator. The question posed and solved here is whether such behaviour is possible in higher dimensions. We therefore consider the ‘Generalized Klein–Gordon’ (GKG) equationin N dimensions where and N ≥ 3.

A method for generating exact solutions of the nonlinear Klein–Gordon equation

1992 ◽  
Vol 70 (6) ◽  
pp. 467-469 ◽  
Author(s):  
A. Grigorov ◽  
N. Martinov ◽  
D. Ouroushev ◽  
Vl. Georgiev
Keyword(s):  
Electromagnetic Field ◽  
Phase Velocity ◽  
Exact Solutions ◽  
Gordon Equation ◽  
External Electromagnetic Field ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Simple Method ◽  
Klein Gordon ◽  
Klein Gordon Equation

A simple method for generating the exact solutions of the nonlinear Klein–Gordon equation is proposed. The solutions obtained depend on two arbitrary functions and are in the form of running waves. An application of one of the solutions for the (2 + 1) – dimensional sine-Gordon equation is proposed. It concerns the selective properties of a two-dimensional semi-infinite Josephson junction with regard to an external electromagnetic field in the form of running waves with a phase velocity equal to the Swihart velocity. A method for measuring the Swihart velocity is presented.

scholarly journals SIMPLICITY OF EXTREMAL EIGENVALUES OF THE KLEIN–GORDON EQUATION

2011 ◽  
Vol 23 (06) ◽  
pp. 643-667 ◽  
Author(s):  
MARIO KOPPEN ◽  
CHRISTIANE TRETTER ◽  
MONIKA WINKLMEIER
Keyword(s):  
Spectral Problem ◽  
Gordon Equation ◽  
Boundary Points ◽  
Klein Gordon ◽  
Electric Potentials ◽  
Klein Gordon Equation

We consider the spectral problem associated with the Klein–Gordon equation for unbounded electric potentials such that the spectrum is contained in two disjoint real intervals related to positive and negative energies, respectively. If the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.

Symmetries of generalized Klein–Gordon (including sine-Gordon) equations in two dimensions

1987 ◽  
Vol 102 (3) ◽  
pp. 573-586
Author(s):  
T. J. Gordon
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Two Dimensions ◽  
Sine Gordon Equation ◽  
Fundamental Significance ◽  
Klein Gordon ◽  
Countably Infinite ◽  
Infinite Set

Much attention has been devoted over the years to the sine-Gordon equation φuv = sin φ (e.g. [2] and references therein). Of fundamental significance is the existence of a countably infinite set of conservation laws, which arises from a corresponding set of symmetries (e.g. [6, 7]).

BÄCKLUND TRANSFORMATIONS AND LAX PAIRS FOR THE NONLINEAR KLEIN–GORDON EQUATION WITH SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (11) ◽  
pp. 2511-2521
Author(s):  
XIAO-GE XU ◽  
YI-TIAN GAO ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Gordon Equation ◽  
Bäcklund Transformations ◽  
Sine Gordon Equation ◽  
Lax Pairs ◽  
Special Cases ◽  
Backlund Transformations ◽  
Klein Gordon ◽  
Pulse Waves ◽  
Klein Gordon Equation

In this paper, the nonlinear Klein–Gordon equation describing the propagation of pulse waves in plasma or waveguide is investigated. With symbolic computation, the generalized Bäcklund Transformations (BTs) for this equation are constructed under different conditions. It is shown that the BTs published in the previous literature for the Sine–Gordon equation, Sinh–Gordon equation, and Liouville equation all turn out to be special cases of the results in the present paper. Moreover, the corresponding Lax pairs are explicitly derived from the obtained BTs through some transformations.

scholarly journals Solutions of Klein-Gordon Equation by the Laplace Decomposition Method and Modified Laplace Decomposition Method

2021 ◽  
pp. 11-19
Author(s):  
R. M. Wayal
Keyword(s):  
Decomposition Method ◽  
Gordon Equation ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Initial Profile ◽  
Computational Work ◽  
Klein Gordon ◽  
Laplace Decomposition Method ◽  
Klein Gordon Equation ◽  
Good Agreement

In this article, the Laplace decomposition method and Modified Laplace decomposition method have been employed to obtain the exact and approximate solutions of the Klein-Gordon equation with the initial profile. An approximate solution obtained by these methods is in good agreement with the exact solution and shows that these approaches can solve linear and nonlinear problems very effectively and are capable to reduce the size of computational work.

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