Towards soliton solutions of a perturbed sine-Gordon equation (original) (raw)
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On kinks and other travelling-wave solutions of a modified sine-Gordon equation
Meccanica, 2015
We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. On the real line candidate orbitally stable solutions with bounded energy density are either the constant one, or of kink (i.e. soliton) type, or of array-of-kinks type, or of "half-array-of-kinks" type. While the first three have unperturbed analogs, the last type is essentially new. We also propose a convergent method of successive approximations of the (anti)kink solution based on a careful application of the fixed point theorem.
New Travelling Wave Solutions for Sine-Gordon Equation
Journal of Applied Mathematics, 2014
We propose a method to deal with the general sine-Gordon equation. Several new exact travelling wave solutions with the form ofJacobiAmplitudefunction are derived for the general sine-Gordon equation by using some reasonable transformation. Compared with previous solutions, our solutions are more general than some of the previous.
Nonlinear stability of 2-solitons of the sine-Gordon equation in the energy space
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018
In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem. The solutions that we study are the 2-kink, kink-antikink and breather of SG. In order to prove this result, we will use Bäcklund transformations implemented by the Implicit Function Theorem. These transformations will allow us to reduce the stability of the three solutions to the case of the vacuum solution, in the spirit of previous results by Alejo and the first author [3], which was done for the case of the scalar modified Korteweg-de Vries equation. However, we will see that SG presents several difficulties because of its vector valued character. Our results improve those in [5], and give a first rigorous proof of the stability in the energy space of SG 2-solitons.
Soliton evolution and radiation loss for the sine-Gordon equation
Physical Review E, 1999
An approximate method for describing the evolution of solitonlike initial conditions to solitons for the sine-Gordon equation is developed. This method is based on using a solitonlike pulse with variable parameters in an averaged Lagrangian for the sine-Gordon equation. This averaged Lagrangian is then used to determine ordinary differential equations governing the evolution of the pulse parameters. The pulse evolves to a steady soliton by shedding dispersive radiation. The effect of this radiation is determined by examining the linearized sine-Gordon equation and loss terms are added to the variational equations derived from the averaged Lagrangian by using the momentum and energy conservation equations for the sine-Gordon equation. Solutions of the resulting approximate equations, which include loss, are found to be in good agreement with full numerical solutions of the sine-Gordon equation.
Stability of Travelling Wave Solutions to the Sine-Gordon Equation
2010
We give a geometric proof of spectral stability of travelling kink wave solutions to the sine-Gordon equation. For a travelling kink wave solution of speed cneqpm1c \neq \pm 1cneqpm1, the wave is spectrally stable. The proof uses the Maslov index as a means for determining the lack of real eigenvalues. Ricatti equations and further geometric considerations are also used in
The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its kink solutions (one-dimensional fronts) are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. Although it has been derived over the years for quite a few physical systems that have nothing to do with Special Relativity, the Sine-Gordon equation emerges as a non-linear relativistic wave equation. This opens the way for exploiting the tools of the Theory of Special Relativity. Using no more than the relativistic kinematics of tachyonic momentum vectors, from which the solutions are constructed through the Hirota algorithm, the existence and classification of N-moving-front solutions of the (1+2)-and (1+3)-dimensional equations for all N ! 1 are presented. In (1+2) dimensions, each multi-front solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than a limiting speed, c = 1, or are greater than or equal to c. To connect with concepts of the Theory of Special Relativity, c will be called "the speed of light." In (1+3)-dimensions, multifront solutions are characterized by spatial structure and by velocity composition. The spatial structure is either planar (rotated (1+2)-dimensional solutions), or genuinely three-dimensionalbranes. Planar solutions, propagate rigidly at one velocity, which is lower than, equal to, or higher than c. Branes must contain clusters of fronts whose speed exceeds c = 1. Some branes are "hybrids": different clusters of fronts propagate at different velocities. Some velocities may be lower than c but some must be equal to, or exceed, c. Finally, the speed of light cannot be approached from within the subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.
Dynamics of sine-Gordon solitons
After reviewing a few physical examples in which the sine-Gordon equation arises as the governing dynamical equation, we discuss various solutions exhibiting multisoliton dynamics. Interaction of solitons and the corresponding velocitydependent interaction potentials are derived and discussed. Numerical experiments are carried out in order to study kink dynamics in an inhomogeneous medium. Finally, we introduce two kinds of generalized sine-Gordon equations and discuss their properties.
Soliton Solutions in a Modified Double and Triple Sine-Gordon Models
2016
We modify both the double sine-Gordon (DSG) and triple sine-Gordon (TSG) model in (1,1) dimensions by the addition of an extra kinetic term and a potential term to their Lagrangian density and present a modified DSG (MDSG) and a modified TSG (MTSG) models. We obtain soliton solutions of the presented modified models and find that both of them possesses the same solutions of the unmodified model with some extra conditions imposed on the parameters of the models. We study some properties of the modified models, in particular, we show that the corresponding governing equation has two solutions, a special ones, which are the exact solutions of the unmodified models and a general ones, and these two types of solutions are coincides in our presented models. We end the paper with conclusions and some features and comments.
Solitons for the nonlinear Klein-Gordon equation
2007
In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the
Perturbation theory for the double sine-Gordon equation
Wave Motion, 2005
This paper presents the perturbation theory for the double-sine-Gordon equation. We received the system of differential equations that shows the soliton parameters modification under perturbation's influence. In particular case λ = 0 the results of the research transform into well-known perturbation theory for the sine-Gordon equation.