Comment on “Existence of internal modes of sine-Gordon kinks”

Abstract

In Ref. [1] [Phys. Rev. B. 42, 2290Rev. B. 42, (1990] we used a rigorous projection operator collective variable formalism for nonlinear Klein-Gordon equations to prove the continuum Sine-Gordon (SG) equation has a long lived quasimode whose frequency ωs= 1.004 Γ0 is in the continuum just above the lower phonon band edge with a lifetime (1/τs ) = 0.0017 Γ0. We confirmed the analytic calculations by simulations which agreed very closely with the analytic results. In Ref.[3] [Phys. Rev. E. 62, R60 (2000)] the authors performed two numerical investigations which they asserted "show that neither intrinsic internal modes nor quasimodes exist in contrast to previous results." In this paper we prove their first numerical investigation could not possibly observe the quasimode in principle and their second numerical investigation actually demonstrates the existence of the SG quasimode. Our analytic calculations and verifying simulations were performed for a stationary Sine-Gordon soliton fixed at the origin. Yet the authors in Ref.[3] state the explanation of our analytic simulations and confirming simulations are due to the Doppler shift of the phonons emitted by our stationary Sine-Gordon soliton which thus has a zero Doppler shift.

Wobbles and other kink-breather solutions of the sine-Gordon model

“…Recently there has been some controversy as to whether a kink possesses an internal mode [7] - [8]. Such a mode was claimed to exist by Boesch and Willis [9] and then disclaimed by Quintero et al [10].…”

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

“…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].…”

Collective influence of impurities on the dynamics of kinks of modified sine-Gordon equation

“…При этом, возможно возбуждение внутренних степеней свободы солитонов, которые могут играть определяющую роль в физических процессах. Внутренние моды включают в себя трансляционную и, связанную с долгоживущими осцилляциями ширины солитона, пульсационную моду [Willis, 2006].…”