Comment on "Existence of Internal Modes of Sine-Gordon Kinks"

TL;DR

This paper confirms the existence of a long-lived quasimode in the continuum Sine-Gordon equation through analytic and simulation methods, addressing previous conflicting claims about internal modes.

Contribution

It provides a rigorous proof and simulation validation for the existence of a quasimode in the continuum Sine-Gordon model, countering prior numerical assertions to the contrary.

Findings

Confirmed the existence of a long-lived quasimode in the Sine-Gordon equation.

Demonstrated previous numerical studies could not detect the quasimode due to methodological limitations.

Validated analytic predictions with high-precision simulations.

Abstract

In Ref.[1] [Phys. Rev. B. {\bf 42}, 2290 (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 $\omega_s$= 1.004 $\Gamma_0$ is in the continuum just above the lower phonon band edge with a lifetime ($1/\tau_s$) = 0.0017 $\Gamma_0$. We confirmed the analytic calculations by simulations which agreed very closely with the analytic results. In Ref.[3] [Phys. Rev. E. {\bf 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.