How to excite the internal modes of sine-Gordon solitons

Abstract

We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton-antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [J. A. Gonzalez et al., Phys. Rev. E, 65 (2002) 065601(R)].Comment: 14 .eps figures.To appear in Chaos, Solitons and Fractal

Bubblelike structures generated by activation of internal shape modes in two-dimensional sine-Gordon line solitons

“…For the following discussion we consider the case φ(x, 0) = 4 arctan exp (Bx) and F (x) = 2(B 2 − 1)sinh(Bx)sech 2 (Bx). For this force function, the stability analysis for kink internal modes can be solved exactly [11,12]. For B 2 > 1, F (x) has a positive slope around x * , the translational mode is stable and there are no internal shape modes.…”

“…In this work, we investigate the effect on the internal structure, and therefore on the dynamics, of a twodimensional sG line solitons due to space-localized inhomogeneities. The study is based on an analytical model of solitons externally excited by a family of topologically equivalent forces in one dimension [11,12]. We show that the soliton dynamics is highly enriched by the interplay between the activation of internal modes and the kinkantikink interactions.…”

Sine--Gordon Solitons, Kinks and Breathers as Physical Models of Nonlinear Excitations in Living Cellular Structures

“…Considering (for simplicity) purely spatial forcing: F (x, t) = F (x), it has been shown in [47,48] In particular if…”

“…where Λ(Λ + 1) = 2/β 2 . The integer part of Λ, yields the number of eigenvalues in the discrete spectrum, which correspond to the soliton modes (this includes the translational mode Γ 0 , and the internal or shape modes Γ n with n > 0 (see [47,48]). In case of a function F defined in such a way that it possesses many zeroes, maxima and minima, perturbed SGE (18) describes an array of inhomogeneities.…”

“…When the soliton is moving over intervals where dF (x) dx < 0, its internal mode can be excited. The points x i where F (x i ) = 0 and dF (x i ) dx < 0, are 'barriers' which the soliton can overcome due to its kinetic energy (for more details, see [47,48]).…”