Relativistic dynamics of sine-Gordon solitons trapped in confining potentials

Abstract

A collective-coordinate method is used to study theoretically and numerically the stability and the dynamics of a sine-Gordon soliton trapped in a confining potential. The example of a harmonic well is emphasized. A remarkably simple approximated solution is found and checked by numerical simulations. The perturbed soliton is stable up to high (relativistic) energies and its profile has the following kinklike dependence on space and time: U(x,t)=4 ${\tan }^{\mathrm{-}1}$exp[( 1+(1/4)V(y) / [1-(y’ ${)}^2$] ${)}^{1/2}$[x-y(t)]] , where V(y) is the potential energy of the particlelike kink at x=y(t). When an external driving force is present, resonances are pointed out and their nonlinear nature is stressed.