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Register a new userTransparent boundary conditions for the sine-Gordon equation:Modeling the reflectionless propagation of kink solitons on a line
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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.
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We consider the problem of absence of backscattering in the transport of Manakov solitons on a line. The concept of transparent boundary conditions is used for modeling the reflectionless propagation of Manakov vector solitons in a one-dimensional domain. Artificial boundary conditions that ensure the absence of backscattering are derived and their numerical implementation is demonstrated.
We explore boundary scattering in the sine-Gordon model with a non-integrable family of Robin boundary conditions. The soliton content of the field after collision is analysed using a numerical implementation of the direct scattering problem associated with the inverse scattering method. We find that an antikink may be reflected into various combinations of an antikink, a kink, and one or more breathers, depending on the values of the initial antikink velocity and a parameter associated with the boundary condition. In addition we observe regions with an intricate resonance structure arising from the creation of an intermediate breather whose recollision with the boundary is highly dependent on the breather phase.
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 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.
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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