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Register a new userOn the existence of internal modes of sine-Gordon kinks
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We study whether or not sine-Gordon kinks exhibit internal modes or ``quasimodes. By considering the response of the kinks to ac forces and initial distortions, we show that neither intrinsic internal modes nor ``quasimodes exist in contrast to previous reports. However, we do identify a different kind of internal mode bifurcating from the bottom edge of the phonon band which arises from the discretization of the system in the numerical simulations, thus confirming recent predictions.
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In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch [Phys. Rev. B 42, 2290 (1990)] were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his comment. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our reply. As a consequence, our result that quasimodes do not exist in the sG equation [Phys. Rev. E 62, R60 (2000)] remains true.
We study the dynamics of $phi^4$ kinks perturbed by an ac force, both with and without damping. We address this issue by using a collective coordinate theory, which allows us to reduce the problem to the dynamics of the kink center and width. We carry out a careful analysis of the corresponding ordinary differential equations, of Mathieu type in the undamped case, finding and characterizing the resonant frequencies and the regions of existence of resonant solutions. We verify the accuracy of our predictions by numerical simulation of the full partial differential equation, showing that the collective coordinate prediction is very accurate. Numerical simulations for the damped case establish that the strongest resonance is the one at half the frequency of the internal mode of the kink. In the conclusion we discuss on the possible relevance of our results for other systems, especially the sine-Gordon equation. We also obtain additional results regarding the equivalence between different collective coordinate methods applied to this problem.
We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We analytically calculate the correlation function of the position of the kink center as well as the diffusion coefficient, both up to second-order in temperature. We find that the kink behavior is very similar to that obtained in the overdamped limit: There is a quadratic dependence on temperature in the diffusion coefficient that comes from the interaction among the kink and phonons, and the average value of the wave function increases with $sqrt{t}$ due to the variance of the centers of individual realizations and not due to kink distortions. These analytical results are fully confirmed by numerical simulations.
Motivated by the recently developed duality between elasticity of a crystal and a symmetric tensor gauge theory by Pretko and Radzihovsky, we explore its classical analog, that is a dual theory of the dislocation-mediated melting of a two-dimensional crystal, formulated in terms of a higher derivative vector sine-Gordon model. It provides a transparent description of the continuous two-stage melting in terms of the renormalization-group relevance of two cosine operators that control the sequential unbinding of dislocations and disclinations, respectively corresponding to the crystal-to-hexatic and hexatic-to-isotropic fluid transitions. This renormalization-group analysis compactly reproduces seminal results of the Coulomb gas description, such as the flows of the elastic couplings and of the dislocation and disclination fugacities, as well the temperature dependence of the associated correlation lengths.
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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