Soliton equations and the inverse scattering transform

Abstract A universally accepted definition of integrability does not exist for partial differential equations (PDEs). The phase space is infinite dimensional but having ‘infinitely many’ first integrals may not be enough – we could have missed every second one. One instead focuses on properties of solutions and solution-generation techniques. We shall study solitons – solitary non-linear waves which preserve their shape (and other characteristics) in the evolution. These soliton solutions will be constructed by means of an inverse problem: recovering a potential from the scattering data.