The Continuing Story of the Wobbling Kink

Abstract

The wobbling kink is the soliton of the φ 4 model with an excited internal mode. We outline an asymptotic construction of this particle-like solution that takes into account the coexistence of several space and time scales. The breakdown of the asymptotic expansion at large distances is prevented by introducing the long-range variables "untied" from the short-range oscillations. We formulate a quantitative theory for the fading of the kink's wobbling due to the second-harmonic radiation, explain the wobbling mode's longevity and discuss ways to compensate the radiation losses. The compensation is achieved by the spatially uniform driving of the kink, external or parametric, at a variety of resonant frequencies. For the given value of the driving strength, the largest amplitude of the kink's oscillations is sustained by the parametric pumping -at its natural wobbling frequency. This type of forcing also produces the widest Arnold tongue in the "driving strength versus driving frequency" parameter plane. As for the external driver with the same frequency, it brings about an interesting rack and pinion mechanism that converts the energy of external oscillation to the translational motion of the kink.

Stability of solitary waves in nonlinear Klein–Gordon equations

“…The kinks are stable, and are a linear superposition of two terms: the first is either the sine-Gordon kink (for l odd numbers) or the φ 4 kink (for even l), while the second is a localized function. These kinks resemble the φ 4 wobbling kinks studied in [11]. The corresponding spectra of the Sturm-Liouville problem associated to the stability of these kinks have several internal modes, some of which have a localized odd eigenfunction, while others have a localized even eigenfunction.…”

“…Equation ( 2) with potential (10) is known in the literature as the sine-Gordon equation. Equation (11) represents its static kink solution. In the study of the linear stability of this topological wave, it is necessary to solve equation (19) with the potential (22) with l = 1 and ω ph = 1.…”

“…Nonlinear Klein-Gordon equations model a plethora of phenomena such as the existence of bound oscillatory states and resonance windows in the kink-antikink interaction [1][2][3][4][5] in the presence and in the absence of internal modes [6][7][8][9], the fading of the kink's wobbling due to the second-harmonic radiation [10,11], the phase transitions in the Ginzburg-Landau theory [12][13][14], the motion of domain walls [15], and the existence of kinks with power-law tail asymptotics that give rise to long-range interactions in the even-higher-order field theories [16,17].…”

“…The pulses also satisfy equation ( 3) setting Q = 0. As a matter of fact, the equations that govern physical systems, such as the propagation of magnetic flux along the Josephson junctions [26], the dynamics of the azimuthal angle of the unit vector of magnetization of ferromagnetic materials [27], the resonant soliton-impurity interactions [28], the unidirectional motion of kinks due to zero-average forces [29][30][31][32], the stabilization of wobbling kinks [11], and the traveling of dislocations along the colloids [33], are modeled by nonlinear Klein-Gordon equations with external and parametric forces and damping. Therefore, the observation of these waves in nature and in experiments depends on their stability [34].…”