Editing Inverse scattering problem (section)

{{Refimprove|date=June 2015}}
In mathematics and physics, the '''inverse scattering problem''' is the problem of determining characteristics of an object, based on data of how it [[scattering|scatters]] incoming radiation or particles.{{sfn|Ablowitz|Fokas|2003}} It is the [[inverse problem]] to the '''direct scattering problem''', which is to determine how radiation or particles are scattered based on the properties of the scatterer.

[[Soliton]] equations are a class of [[partial differential equations]] which can be studied and solved by a method called the [[inverse scattering transform]], which reduces the nonlinear PDEs to a linear inverse scattering problem. The [[nonlinear Schrödinger equation]], the [[Korteweg–de Vries equation]] and the [[KP equation]] are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a [[Riemann-Hilbert problem]].{{sfn|Dunajski|2010}}  Inverse scattering has been applied to many problems including [[radar|radiolocation]], [[Acoustic location|echolocation]], [[geophysical]] survey, [[nondestructive testing]], [[medical imaging]], and [[quantum field theory]].{{sfn|Bao|2023}}{{sfn|Grinev|Chebakov|Gigolo|2003}}