Sine-Gordon kink-antikink generation on spatially periodic potentials

Scharf, Rainer; Kivshar, Yuri S.; Sánchez, Ángel; Bishop, A. R.

doi:10.1103/physreva.45.r5369

Cited by 29 publications

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“…(6) which shows that 'l/;x is playing a role in the smallamplitude breather evolution. On the contrary, global energetic considerations are more suitable explanations of the nonradiative, f{ -f{ breakup of large-amplitude breathers, as we will report elsewhere [12]. We also want to stress again the fact that we have performed similar simulations on the NLS system [11] and find the same kind of "phase diagram," which is a good indication that NLS-like approaches to the SG breather problem can be helpful: for instance, Eqs.…”

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confidence: 66%

“…We will show that they indeed provide some useful information, without overly involved algebra. With the basic assumption that the effect of the potential on the breather is to allow it to move, without changing its shape, it is possible to derive [further detail is planned to be given elsewhere [12], but the basis is the fact that the evolution under Eq.…”

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confidence: 99%

“…This preliminary work allows usto propose some explanations for this group of phenomena which are qualitatively corred,ancl to seiect further research lines capable to complete and make our picture quantitative. Work along these lines is in progress [11,12]. …”

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Sine-Gordon breathers on spatially periodic potentials

et al. 1992

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We have carried out an extensive simulation program to study the behavior of sine-Gordon breathers initially at rest in the presence of perturbations that are periodic in space and constant in time. We report here a number of different observed phenomena and the range of the relevant parameters (the ratio of the breather width to the perturbation wavelength and the perturbation magnitude) for which each one of them occurs. We also propose some qualitative explanations valid for certain regimes. PACS number(s): 03.40.KfThe study of nonlinear disordered systems has recently become the subject of a great deal of research [1]. A model that has been widely used as an approximate description of quasi-one-dimensional physical phenomena is the sine-Gordoll (SG) equation (see, for instance, [2] and references therein). The effects of a large variety of perturbations on the properties of that equation have been investigated (see [3,4] for recent reviews). However, starting from the seminal work by Fogel et al. [5], only a few papers have been devoted to disordered systems, corresponding to the addition of certain properly chosen inhomogeneous terms to the original equation. In particular, a simple form of inhomogeneity, which to some extent is amenable to analytical work, is a spatially periodic, external potential. The propagation of SG kinks on such a potential has been previously studied by Mkrtchyan and Shmidt [6] and Malomed and Tribelsky [7], and the same.problem has been analyzed as the limiting case of a spatial lattice of impurities when this lattice becomes very dense [8].. In this work we concern ourselves with the influence of these periodic potentials on the SG breathers. Some preliminary work has been done by Pascual [9]. We introduce this issue as an excellent example of the competition between two characteristic length scales, which is an ubiquitous phenomenon in real physical systems: In this case, these length scales are the breather width (AB) and the perturbation period (Ap). The aim of this work is twofold. First, our immediate goal is to learn whether the particle picture of solitons, often very useful, holds or not, and, if not, when and how it breaks down due to this perturbation; second, our ultimate purpose is to apply our results to nonlinear wave propagation in potentials which are stochastic in space (this problem has been considered in [10]), viewing the periodic potential as a particular component or "color" of the noisy one. As is explained below, understanding one color is a fundamental input to treating propagation in a general color distribution. On the other hand, we are also interested in comparing the phenomena we are describing here to those caused by the same kind of potential on the nonlinear Schrodinger (NLS) equation, often viewed as a weak 45 nonlinearity limit of the SG equation. Our investigations on this related system are planned to be reported elsewhere [11].To approach the above issues, we have carried out a number of numerical simulations spanning large intervals of the p...

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Sine-Gordon breathers on spatially periodic potentials

et al. 1992

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“…The binding of a free kink and antikink into a breather has been addressed in [62] in the presence of spatially periodic perturbation. An external dc driving force in the absence of damping for a sufficiently large magnitude of the force causes the breather to split into a kink-antikink pair, while for a small driving force the breather excitations lead to stationary modes [63].…”

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Interaction of sine-Gordon kinks and breathers with a parity-time-symmetric defect

2014

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The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized parity-time-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy, and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated by means of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect depends strongly on the breather parameters (velocity, frequency, and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair, or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions, the energy loss to radiation of small-amplitude extended waves is negligible.

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“…Recently it has been shown that length-scale competition is crucial for understanding dynamics of both one-dimensional nonlinear Schrodinger (NLS) and onedimensional sine-Gordon (SG) systems driven by a static, spatially periodic parametric potential [1][2][3]. The picture that has emerged is that the length-scale competition between the width of a breather and the spatial period of the external potential controls the particlelike coherence of the breather.…”

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confidence: 99%

Length-scale competition in the damped sine-Gordon chain with spatiotemporal periodic driving

1993

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It is shown that there are two different regimes for the damped sine-Gordon chain driven by the spatiotemporal periodic force rsin(wt -knx) with a flat initial condition. For w > kn, the system first bifurcates at a critical re (n) to a translating two-breather excitation from a state locked to the driver. For w < kn, the excitations of the system are the locked states with the phase velocity w/kn in all the regions of r studied. In the first regime, the frequency of the breathers is controlled by w, and the velocity of the breathers, controlled by kn, is shown to be the group velocity determined from the linear dispersion relation for the sine-Gordon equation. A linear stability analysis reveals that, in addition to two competing length scales, namely, the width of the breathers and the spatial period of the driving, there is one more length scale which plays an important role in controlling the dynamics of the system at small driving. In the second regime the length scale k n controls the excitation. The above picture is further corroborated by numerical nonlinear spectral analysis. An energy-balance estimate is also presented and shown to predict the critical value of r in good agreement with the numerical simulations.

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Sine-Gordon kink-antikink generation on spatially periodic potentials

Cited by 29 publications

References 8 publications

Sine-Gordon breathers on spatially periodic potentials

Sine-Gordon breathers on spatially periodic potentials

Interaction of sine-Gordon kinks and breathers with a parity-time-symmetric defect

Length-scale competition in the damped sine-Gordon chain with spatiotemporal periodic driving

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