Etude des équations stationnaire de Schrödinger, intégrale de Gelfand-Levitan et de Korteweg-de-Vries. Solitons et méthode de la diffusion inverse

The Korteweg-de Vries (KdV) equation is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1 + 1 dimensions whose solutions can be exactly specified; it has a soliton like solution or solitary wave of sech 2 form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. This article will cover in detail: the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly.