Internal modes of sine-Gordon solitons in the presence of spatiotemporal perturbations

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.

Internal oscillations of spatial optical solitons in a cubic-quintic nonlinear medium

“…These oscillations introduce qualitatively new features into the system dynamics being responsible for long-lived of the solitary wave shape and resonant soliton interaction [12,13]. Internal oscillations have been analyzed for the so-called kink solitons, topological solitary waves of the Klein-Gordon type models [13], sine-Gordon solitons [14], FS in saturable nonlinear medium (2+1)D [15] and (3+1)D [16], quadratic medium [17]. Besides, internal oscillations also exist in periodic layered medium [18].…”

Interaction of sine‐Gordon solitons in the model with attracting impurities

“…Soliton internal modes can be responsible for the non-trivial effects of their interactions [13,14]. The internal modes can include the translational and pulsation modes describing long-lived oscillations of soliton width [15]. The effect of various perturbations on the excitation of SGE soliton's internal modes attracts a lot of attention of researchers.…”

“…Local inhomogeneities are ubiquitous in many physical systems, including those described by SGE, and it is very important to study the soliton scattering on such impurities [5]. For instance, there exist many works devoted to the analysis of external force that varies in time and space [5,9,[15][16][17]. Weak perturbations on the SGE solutions can be studied in frame of the perturbation theory well-developed for solitons [5,7] but the effect of strong perturbations is more often addressed numerically [18][19][20][21].…”

“…The dependence of the stabilized soliton amplitude on the impurity width W at fixed K D 2.8. Curve 1 gives numerical result, while curve 2 corresponds to(15). [Colour figure can be viewed at wileyonlinelibrary.com] Space of impurity parameters showing the range where passing kink excites a breather (above the lines 1 and 2), or a soliton (below the lines 1 and 2).…”

Resonant interaction of ϕ4 kink with PT-symmetric perturbation with spatially periodic gain/loss coefficient

“…In the context of this model, in the damped case, the strongest resonance has been found at half the frequency of the kink's IM [24]. The sine-Gordon model with spatiotemporal perturbations, φ tt − φ xx + sin φ = −γφ t + F (x, t) either with some spatially inhomogeneous forces F (x) or with spatiotemporal inhomogeneous forces F (x, t), also manifests interesting kink dynamics [25]. The influence of the IM on the soliton ratchet mobility has been investigated for the driven and damped double sine-Gordon equation φ tt − φ xx = −dU/dφ − γφ t + f (t), with U = (1−cos φ)+(λ/2)(1−cos 2φ), λ is a parameter, and f (t) = ǫ 1 sin(ωt + δ 1 ) + ǫ 2 sin(2ωt + δ 2 ) [26].…”