The nverseinverse scattering transform was first introduced by {{harvs|txt|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|year1=1967|year2=1974}} for the [[Korteweg–de Vries equation]], and soon extended to the [[nonlinear Schrödinger equation]], the [[Sine-Gordon equation]], and the [[Toda lattice]] equation. It was later used to solve many other equations, such as the [[Kadomtsev–Petviashvili equation]], the [[Ishimori equation]], the [[Dym equation]], and so on. A further family of examples is provided by the [[Bogomol'nyi–Prasad–Sommerfield bound|Bogomolny equations]] (for a given gauge group and oriented Riemannian 3-fold), the <math>L^2</math> solutions of which are [[magnetic monopoles]].
A characteristic of solutions obtained by the inverse scattering method is the existence of [[solitons]], solutions resembling both particles and waves, which have no analogue for linear partial differential equations. The term "soliton" arises from non-linear optics.
