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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.
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We present a modified version of the one-dimensional sine-Gordon that exhibits a thermodynamic, roughening phase transition, in analogy with the 2D usual sine-Gordon model. The model is suited to study the crystalline growth over an impenetrable substrate and to describe the wetting transition of a liquid that forms layers. We use the transfer integral technique to write down the pseudo-Schrodinger equation for the model, which allows to obtain some analytical insight, and to compute numerically the free energy from the exact transfer operator. We compare the results with Monte Carlo simulations of the model, finding a perfect agreement between both procedures. We thus establish that the model shows a phase transition between a low temperature flat phase and a high temperature rough one. The fact that the model is one dimensional and that it has a true phase transition makes it an ideal framework for further studies of roughening phase transitions.
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.
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.
Nonlinear space-time dynamics, defined in terms of celebrated solitonic equations, brings indispensable tools for understanding, prediction and control of complex behaviors in both physical and life sciences. In this paper, we review sine-Gordon solitons, kinks and breathers as models of nonlinear excitations in complex systems in physics and in living cellular structures, both intra-cellular (DNA, protein folding and microtubules) and inter-cellular (neural impulses and muscular contractions). Key words: Sine-Gordon solitons, kinks and breathers, DNA, Protein folding, Microtubules, Neural conduction, Muscular contraction
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.
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