The inverse scattering transform, nonlinear waves, singular perturbations and synchronized solitons

It is not the intention of this article to cover, in detail, all of the material presented in my lecture notes at the Conference.Rather, it is the plan to attempt to point out (a) the universal nature and origin of the partial differential equations which the inverse scattering transform enables one to solve and (b) the rich potential for examining partial differential equations which are close, in some perturbation sense, to integrable ones. Several examples are investigated. We discuss the synchronous response of a nonlinear Schrôdinger soliton to an applied field and also examine the effects of density gradients, damping and diffusion. We also consider the behavior of a kink of the sine-Gordon equation in the presence of an impurity and finally develop 4π pulse solutions of the double sine- Gordon equation which are formed by a pair of synchronized 2π pulses.