Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory

Book excerpt: - of nonlinear the of solitons the the last 30 theory partial theory During years - has into solutions of a kind a differential special equations (PDEs) possessing grown and in view the attention of both mathematicians field that attracts physicists large and of the of the problems of its novelty problems. Physical important applications for in the under consideration are mo- to the observed, example, equations leading mathematical discoveries is the Makhankov One of the related V.G. by [60]. graph from this field methods that of certain nonlinear by equations possibility studying inverse these to the problem; equations were analyze quantum scattering developed this method of the inverse called solvable the scattering problem (on subject, are by known nonlinear At the the class of for same time, currently example [89,94]). see, the other there is solvable this method is narrow on hand, PDEs sufficiently and, by of differential The latter called the another qualitative theory equations. approach, the of various in includes on pr- investigations well-posedness approach particular solutions such or lems for these the behavior of as stability blowing-up, equations, these and this of approach dynamical systems generated by equations, etc., properties in wider class of a makes it to an problems (maybe possible investigate essentially more general study).