Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory

Abstract

In this article, we develop a singular perturbation theory for describing the long time cumulative effects of weak perturbations on solitons. In all cases, the solitons behave in a similar fashion to either relativistic or Newtonian particles or nonlinear oscillators under the influence of external forces. We show how the ubiquitous nonlinear Schrödinger soliton can become synchronized to a periodic external field and how it moves in gradual field gradients. We examine how the kink of the sine-Gordon equation acts both as a relativistic particle and a Newtonian particle in the presence of a general impurity and demonstrate the relaxation of a kink-antikink pair to a breather under the influence of damping. Finally, we discuss the motion of a soliton of the Korteweg de Vries equation under various perturbations and discover that while the soliton remains dominant, the continuous spectrum is excited and plays a crucial rôle in balancing the ‘mass’ and ‘energy’ depletion rates. In each case, we briefly discuss the result in the context of a physical situation.