Inverse scattering problem: Difference between revisions
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{{Short description|Problem in Applied Mathematics}} | |||
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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. | 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}} | [[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== | ==Citations== | ||
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*{{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=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. | *{{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}} | * {{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}} | ||
Latest revision as of 02:15, 23 September 2025
Template:Short description Template:Refimprove In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles.Template:Sfn 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.Template:Sfn Inverse scattering has been applied to many problems including radiolocation, echolocation, geophysical survey, nondestructive testing, medical imaging, and quantum field theory.Template:SfnTemplate:Sfn
Citations
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