Existence of an internal quasimode for a sine-Gordon soliton

We apply our recently derived Hamiltonian theory of constrained nonlinear Klein-Gordon systems to the problem of a single sine-Gordon (SG) kink and show that there exists a quasi-internal degree of freedom which we describe by a collective variable. We show that the collective variable used by Rice to describe the regular oscillations of a ${\mathrm{\phi }}^4$ and SG kink internal mode is actually, in the exact theory, coupled to the phonon field. In the ${\mathrm{\phi }}^4$ case, the internal mode is an exact eigenstate of the linearized ${\mathrm{\phi }}^4$ equation (when linearized about the single-kink solution) whose eigenfrequency lies in the gap below the phonon band edge. In the SG case there is no exact bound eigenstate (other than the zero-frequency Goldstone mode) and so the frequency calculated by Rice corresponding to the quasi-internal mode for the Sg system is in the phonon continuum. Therefore, any bound oscillation at the Rice frequency in the SG system decays via spontaneous emission of phonons. However, rather surprisingly, we find by numerical solution of the SG equation of motion that the internal mode is extremely long-lived with a lifetime of well over 300 oscillations at a frequency ${\mathrm{\omega }}_{\mathit{s}}$=(1.004±0.001)${\mathrm{\Gamma }}_0$, where ${\mathrm{\Gamma }}_0$ is the frequency at the phonon band edge. We calculate the phonon dressing of the bare kink ansatz using the collective variable theory in lowest order and show the renormalized ‘‘dressed’’ frequency ${\mathrm{\Omega }}_{\mathit{d}}$ of the internal mode agrees with the frequency observed from simulation ${\mathrm{\omega }}_{\mathit{s}}$ to within 5%. We calculate the linewidth of the radiation from simulation and obtain 1/${\mathrm{\tau }}_{\mathit{s}}$=(0.003±0.001)${\mathrm{\Gamma }}_0$. Using collective variable theory and the simple model of radiation reaction we obtain for the lifetime the value 1/τ=0.002${\mathrm{\Gamma }}_0$. The physical observability and relationship to other investigations of collective variable treatments of internal modes are analyzed and discussed.