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Register a new userKink stability, propagation, and length scale competition in the periodically modulated sine-Gordon equation
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We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on this and related nonlinear problems in four directions. First, we present the results of a numerical simulation program which are not compatible with the existence of a radiative threshold, predicted by earlier calculations. Second, we carry out a perturbative calculation which helps interpret those previous predictions, enabling us to understand in depth our numerical results. Third, we apply the collective coordinate formalism to this system and demonstrate numerically that it accurately reproduces the observed kink dynamics. Fourth, we report on a novel occurrence of length scale competition in this system and show how it can be understood by means of linear stability analysis. Finally, we conclude by summarizing the general physical framework that arises from our study.
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Our principal focus in the present work is on one-dimensional kink-antikink and two-dimensional kink-antikink stripe interactions in the sine-Gordon equation. Using variational techniques, we reduce the interaction dynamics between a kink and an antikink on their respective time, and space (the latter in the case of the two-dimensional stripes) dependent widths and locations. The resulting reduced system of coupled equations is found to accurately describe the width and undulation dynamics of a single kink stripe as well as that of interacting ones. As an aside, we also discuss two related topics: the computational identification of the kink center and its numerical implications and alternative perturbative and multiple scales approaches to the transverse direction induced dynamics for a single kink stripe in the two-dimensional realm.
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.
We consider $lambdaphi^{4}$ kink and sine-Gordon soliton in the presence of a minimal length uncertainty proportional to the Planck length. The modified Hamiltonian contains an extra term proportional to $p^4$ and the generalized Schrodinger equation is expressed as a forth-order differential equation in quasiposition space. We obtain the modified energy spectrum for the discrete states and compare our results with 1-loop resummed and Hartree approximations for the quantum fluctuations. We finally find some lower bounds for the deformations parameter so that the effects of the minimal length have the dominant role.
We consider the reflectionless transport of sine-Gordon solitons on a line. Transparent boundary conditions for the sine-Gordon equation on a line are derived using the so-called potential approach. Our numerical implementation of these novel boundary conditions proves the absence of the backscattering in transmission of sine-Gordon solitons through the boundary of the considered finite domains.
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 Lins 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.
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