Sine-Gordon breathers on spatially periodic potentials

Abstract

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...

Length scale competition in nonlinear Klein—Gordon models: A collective coordinate approach

“…Indeed, even behaving as particles, kinks do have a characteristic width; however, for most perturbations, that is not a relevant parameter and one can consider kinks as point-like particles. This is not the case when the perturbation itself gives rise to certain length scale of its own, a situation that leads to the phenomenon of length-scale competition, first reported in [8] (see [9] for a review). This phenomenon is nothing but an instability that occurs when the length of a coherent structure approximately matches that of the perturbation: Then, small values of the perturbation amplitude are enough to cause large modifications or even destroy the structure.…”

“…This phenomenon is nothing but an instability that occurs when the length of a coherent structure approximately matches that of the perturbation: Then, small values of the perturbation amplitude are enough to cause large modifications or even destroy the structure. Thus, in [8], the perturbation considered was sinusoidal, of the form…”

Sine-Gordon kink-antikink generation on spatially periodic potentials

“…The question arises whether kinks and anti kinks also interact radiationlessly under these circumstances. Finally it is interesting to see under which conditions a breather can break up into a kinkantikink (f{-f{) pair under the perturbation [2].In this Rapid Communication we address the question of how large-amplitude SG breathers behave under the influence of spatially periodic potentials. These potentials include the effects of discreteness as a particular case; they also can be relevant to understand the interaction of domain wall excitations in discrete solid-state and materials science problems governed by SG-like equations of motion, for example, dislocations in a crystal, walls in ferroelectrics or ferromagnets, or discommensurations in superlattices.…”

“…Kf, 61.70.Ga Adding spatial disorder to completely integrable nonlinear dynamics like the one governed by the nonlinear Schrodinger (NLS) equation or the sine-Gordon (SG) equation leads to a variety of novel effects having practical relevance [1]. Competition of the length scales introduced by the perturbation and by the nonlinearity is one example we have studied recently [2]. If these length scales are very different from each other the perturbed dynamics can support solitonlike or breatherlike excitations.…”

Kink propagation through disordered media

“…The last type of solitary waves is that of topological kinks, which exist only in nonlinear systems with two or more equivalent ground states, i.e., when nonlinearity is not small. In many 8867 cases, kink dynamics can be described in the framework of a collective-coordinate approach, when the evolution of the kink parameters is similar to the evolution of an effective particle (see (12)(13)(14)(15)(16) and references therein), the particle coordinate being the kink cent er position. Such a simplification holds because of an exponentially small radiative loss of the kink in the spatially inhomogeneous medium (see, e.g., [14]); under this condition, one can use the collective-coordinate formalism with high accuracy (see [17] for an example).…”