Wave mechanics of sine-Gordon solitons

We continue our study of the acceleration of a single sine-Gordon (SG) soliton kink wave by an external field [cf. G. Reinisch and J. C. Fernandez, Phys. Rev. B 24, 835 (1981), hereafter referred to as paper I]. We exhibit, both qualitatively and quantitatively, the basic physical process which prevents the kink dynamics to be a priori Newtonian: the self-consistent interaction between the small linear oscillations (phonon waves) excited by the external field about the kink profile, and the kink itself. In order to evaluate the importance of this interaction, and therefore build a kink wave mechanics, we define a kink momentum $P=-\left(\frac{\pi }{4}\right)\left(\frac{\partial }{\partial t}\right)\int {-\infty }^{\infty }u-\left(\frac{r}{4}\right)\left(\frac{\partial }{\partial t}\right)\int {-\infty }^{\infty }u\mathrm{dx}$ and check that it measures, in an acceptable sense, the momentum of a particle associated with the kink center, having a mass equal to the energy of the kink. This enables us to recover our previous results (paper I), obtained within the framework of the (rather severe) adiabatic assumption (consisting in retaining only the small-wave-number phonon waves), and correct them by taking into account the whole phonon spectrum. As a matter of fact, we obtain an important correction, of the order of 20%, of the kink dynamics in presence of an external field, and verify that it leads to a better fitting with the numerical results of paper I. We show that the above-mentioned kink wave mechanics is based on the existence of a general force equation of the type $\left(\frac{d}{\mathrm{dt}}\right)P=\Xi \left[{\psi }_{\kappa }\right]$, where $\Xi$ is a linear operator and ${\psi }_k$ is the phonon spectrum. This equation shows that the phonon dressing of SG kinks may lead to first-order (with respect to the perturbation function) dynamical effects concerning the kink, when the phonon-soliton interaction is coherent, i.e., when the characteristic interaction time is comparable with the time of coherence of the excited phonons.