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

A singular perturbation theory for describing the long time cumulative effects of weak perturbations on solitons is developed. 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. It is shown how the ubiquitous nonlinear Schroedinger soliton can become synchronized to a periodic external field and how it moves in gradual field gradients. 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 is examined and the relaxation of a kink-antikink pair to a breather under the influence of damping is demonstrated. Finally, the motion of a soliton of the Korteweg de Vries equation under various perturbations is discussed and it is discovered that while the soliton remains dominant, the continuous spectrum is excited and plays a crucial role in balancing the 'mass' and 'energy' depletion rates. In each case, the result is briefly discussed in the context of a physical situation. (author)