View source for Inverse scattering problem

{{Short description|Problem in Applied Mathematics}}
{{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}}

==Citations==
{{Reflist}}

==References==
*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Fokas |first2=A. S. |title=Complex Variables: Introduction and Applications |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-53429-1 |pages=609–613 |url=https://books.google.com/books?id=SFqbV3i3hO0C |language=en}}

*{{cite book |last1=Bao |first1=Gang |editor1-last=Beliaev |editor1-first=D. |editor2-last=Smirnov |editor2-first=S. |title=ICM International Congress of Mathematics 2022 July 6-14 |year=2023 |publisher=EMS Press |isbn=9783985470587 |pages=5034–5055 |url=https://ems.press/books/standalone/279/5589 |access-date=19 March 2024 |language=en |chapter=Mathematical analysis and numerical methods for inverse scattering problems}}[https://ems.press/content/book-chapter-files/33328 Reprint]

*{{cite book |last1=Colton |first1=David |last2=Kress |first2=Rainer |title=Inverse Acoustic and Electromagnetic Scattering Theory |year=2013 |publisher=Springer Science & Business Media |isbn=978-3-662-02835-3 |url=https://books.google.com/books?id=gmrmCAAAQBAJ |language=en}}

*{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, Instantons, and Twistors |year=2010 |publisher=OUP Oxford |isbn=978-0-19-857062-2 |url=https://books.google.com/books?id=AJUUDAAAQBAJ |language=en}}

*{{cite book |last1=Grinev |first1=A. Y. |last2=Chebakov |first2=I. A. |last3=Gigolo |first3=A. I. |chapter=Solution of the inverse problems of subsurface radiolocation |title=4th International Conference on Antenna Theory and Techniques (Cat. No.03EX699) |year=2003 |volume=2 |pages=523–526 |doi=10.1109/ICATT.2003.1238792 |location=Sevastopol, Ukraine|isbn=0-7803-7881-4 }}

* {{citation|mr=2798059|last1=Marchenko|first1=V. A.|title=Sturm-Liouville operators and applications|edition=revised|publisher=[[American Mathematical Society]]|location=Providence|year=2011|isbn=978-0-8218-5316-0}}

{{DEFAULTSORT:Inverse Scattering Problem}}
[[Category:Scattering theory]]
[[Category:Scattering, absorption and radiative transfer (optics)]]
[[Category:Inverse problems]]

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