Inverse scattering transform: Difference between revisions

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Using a pair of [[differential operator]]s, a 3-step algorithm may solve [[nonlinear system|nonlinear differential equations]]; the initial solution is transformed to scattering data (direct scattering transform), the scattering data evolves forward in time (time evolution), and the scattering data reconstructs the solution forward in time (inverse scattering transform).{{sfn|Drazin|Johnson|1989}}{{rp|66-67}}

This algorithm simplifies solving a nonlinear partial differential equation to solving 2 linear [[ordinary differential equation]]s and an ordinary [[integral equation]], a method ultimately leading to [[Analytic function|analytic solutions]] for many otherwise difficult to solve nonlinear partial differential equations.{{sfn|Drazin|Johnson|1989}}{{rp|72}}

The inverse scattering problem is equivalent to a [[Riemann–Hilbert factorization]] problem, at least in the case of equations of one space dimension.{{sfn|Ablowitz|Fokas|2003|pp=604-620}} This formulation can be generalized to differential operators of order greater than two and also to periodic problems.{{sfn|Osborne|1995}}

In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.

==History==

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The differential equation's solution meets the integrability and Fadeev conditions:{{sfn|Drazin|Johnson|1989}}{{rp|40}}

:Integrability condition:<math>\int^{\infty}_{-\infty} \ |u(x)| \ dx \ < \infty</math>

:Fadeev condition: <math>\int^{\infty}_{-\infty} \ (1+|x|))|u(x)| \ dx \ < \infty</math>

===Differential operator pair===

The [[Lax pair|Lax differential operators]], <math display="inline">L</math> and <math display="inline">M</math>, are linear ordinary differential operators with coefficients that may contain the function <math display="inline">u(x,t)</math> or its derivatives. The [[self-adjoint operator]] <math display="inline">L</math> has a time derivative <math display="inline">L_{t}</math> and generates a <em>eigenvalue (spectral) equation</em> with [[eigenfunction]]s <math display="inline">\psi</math> and time-constant [[eigenvalues and eigenvectors|eigenvalues]] (<em>[[Spectral theory|spectral parameters]]</em>) <math display="inline">\lambda</math>.{{sfn|Aktosun|2009}}{{rp|4963}}{{sfn|Drazin|Johnson|1989}}{{rp|98}}

: <math> L(\psi)=\lambda \psi , \ </math> and <math display="inline"> \ L_{t}(\psi) \overset{def}{=}(L(\psi))_{t}-L(\psi_{t})</math>

The operator <math display="inline">M</math> describes how the eigenfunctions evolve over time, and generates a new eigenfunction <math display="inline">\psi^widetilde{\primepsi} </math> of operator <math display="inline">L</math> from eigenfunction <math display="inline">\psi</math> of <math display="inline">L</math>.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>\psi^widetilde{\primepsi}=\psi_{t}-M(\psi) \ </math>

The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfuctionseigenfunctions <math display="inline">\psi</math>.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>(L_{t}+LM-ML)\psi=0</math>

The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>L_{t}+LM-ML=u_{t}+N(u)=0</math>

The [[AKNS system|AKNS differential operators]], developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.{{sfn|Aktosun|2009}}{{rp|4964}}{{sfn|Ablowitz|Kaup|Newell|Segur|1973}}{{sfn|Ablowitz|Kaup|Newell|Segur|1974}}

===Direct scattering transform===

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==References==

*{{cite journal |last1=Ablowitz |first1=M. J. |last2=Kaup |first2=D. J. |last3=Newell |first3=A. C. |last4=Segur |first4=H. |title=Method for Solving the Sine-Gordon Equation |journal=Physical Review Letters |year=1973 |volume=30 |issue=25 |pages=1262–1264 |doi=10.1103/PhysRevLett.30.1262 |bibcode=1973PhRvL..30.1262A |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1262|url-access=subscription }}

* {{cite journal |last1=Ablowitz |first1=M.J. |last2=Kaup |first2=D.J. |last3=Newell |first3=A.C. |last4=Segur |first4=H. |title=The Inverse Scattering Transform—Fourier Analysis for Nonlinear Problems |journal=Studies in Applied Mathematics |year=1974 |volume=53 |issue=4 |pages=249-315249–315 |doi=10.1002/sapm1974534249 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1974534249|url-access=subscription }}

*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Segur |first2=Harvey |title=Solitons and the Inverse Scattering Transform |year=1981 |publisher=SIAM |isbn=978-0-89871-477-7 |url=https://wwwbooks.google.com/books/edition/Solitons_and_the_Inverse_Scattering_Tran/?id=Bzu4XAUpFZUC?hl |language=en}}

*{{cite journalbook |last1=Ablowitz |first1=Mark J. |titlelast2=NonlinearFokas waves|first2=A. andS. the|title=Complex InverseVariables: ScatteringIntroduction Transformand |journal=OptikApplications |year=20232003 |volumepublisher=278Cambridge University Press |pagesisbn=170710978-0-521-53429-1 |doipages=10.1016/j.ijleo.2023.170710604–620 |url=https://wwwbooks.sciencedirectgoogle.com/science/article/pii/S0030402623002061books?id=SFqbV3i3hO0C |language=en}}

*{{cite journal |last1=Ablowitz |first1=Mark J. |title=Nonlinear waves and the Inverse Scattering Transform |journal=Optik |year=2023 |volume=278 |pages=170710 |doi=10.1016/j.ijleo.2023.170710 |bibcode=2023Optik.27870710A |url=https://www.sciencedirect.com/science/article/pii/S0030402623002061|url-access=subscription }}

*{{cite book |last1=Aktosun |first1=Tuncay |title=Encyclopedia of Complexity and Systems Science |year=2009 |publisher=Springer |isbn=978-0-387-30440-3 |pages=4960–4971 |url=https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 |language=en |chapter=Inverse Scattering Transform and the Theory of Solitons}}

* {{cite book |last1=DrazinAktosun |first1=P. G.Tuncay |last2title=JohnsonEncyclopedia |first2=R.of S.Complexity |title=Solitons:and AnSystems IntroductionScience |year=19892009 |publisher=Cambridge University PressSpringer |isbn=978-0-521387-3365530440-03 |pages=4960–4971 |chapter-url=https://wwwlink.googlespringer.com/booksreferenceworkentry/edition10.1007/Solitons/HPmbIDk2u978-gC?hl=en0-387-30440-3_295 |language=en |chapter=Inverse Scattering Transform and the Theory of Solitons|doi=10.1007/978-0-387-30440-3_295 }}

* {{cite journalbook |last1=GardnerDrazin |first1=CliffordP. SG. |last2=GreeneJohnson |first2=John MR. |last3=Kruskal |first3=Martin D. |last4=Miura |first4=Robert MS. |title=MethodSolitons: forAn Solving the Korteweg-deVries Equation |journal=Physical Review LettersIntroduction |year=19671989 |volumepublisher=19Cambridge |issue=19University |pages=1095–1097Press |doiisbn=10.1103/PhysRevLett.19.1095978-0-521-33655-0 |url=https://journalsbooks.apsgoogle.orgcom/prl/abstract/10.1103/PhysRevLett.19.1095books?id=HPmbIDk2u-gC |language=en}}

*{{cite journal |last1=Gardner |first1=Clifford S. |last2=Greene |first2=John M. |last3=Kruskal |first3=Martin D. |last4=Miura |first4=Robert M. |title=Method for Solving the Korteweg-deVries Equation |journal=Physical Review Letters |year=1967 |volume=19 |issue=19 |pages=1095–1097 |doi=10.1103/PhysRevLett.19.1095 |bibcode=1967PhRvL..19.1095G |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1095|url-access=subscription }}

*{{cite book |last1=Konopelchenko |first1=B.G. |last2=Dubrowsky |first2=V.G. |editor1-last=Sattinger |editor1-first=David H. |editor2-last=Tracy |editor2-first=C.A. |editor3-last=Venakides |editor3-first=Stephanos |title=Inverse Scattering and Applications |year=1991 |publisher=American Mathematical Soc. |isbn=978-0-8218-5129-6 |pages=77-90 |url=https://www.google.com/books/edition/Inverse_Scattering_and_Applications/yTYcCAAAQBAJ?hl=en |language=en |chapter=Localized solitons for the Ishimori equation}}

*{{cite book |last1=OonoKonopelchenko |first1=HB.G. |last2=Dubrowsky |first2=V.G. |editor1-last=AlfinitoSattinger |editor1-first=EDavid H. |editor2-last=BoitiTracy |editor2-first=MC.A. |editor3-last=MartinaVenakides |editor3-first=L.Stephanos |title=Nonlinear Physics:Inverse TheoryScattering and ExperimentApplications |year=19961991 |publisher=WorldAmerican ScientificMathematical Publishing Company Pte LimitedSoc. |isbn=978-9810-028218-25595129-96 |pages=241-24877–90 |url=https://wwwbooks.google.com/books/edition/Nonlinear_Physics/35EfzQEACAAJ?hlid=enyTYcCAAAQBAJ |language=en |chapter=N-SolitonLocalized solutionsolitons offor Harrythe DymIshimori equation by inverse scattering method.}}

*{{cite book |last1=KonopelchenkoOono |first1=B.G. |last2=Dubrowsky |first2=V.GH. |editor1-last=SattingerAlfinito |editor1-first=David HE. |editor2-last=TracyBoiti |editor2-first=C.AM. |editor3-last=VenakidesMartina |editor3-first=StephanosL. |title=InverseNonlinear ScatteringPhysics: Theory and ApplicationsExperiment |year=19911996 |publisher=AmericanWorld MathematicalScientific Soc.Publishing Company Pte Limited |isbn=978-0981-821802-51292559-69 |pages=77-90241–248 |url=https://wwwbooks.google.com/books/edition/Inverse_Scattering_and_Applications/yTYcCAAAQBAJ?hlid=en35EfzQEACAAJ |language=en |chapter=LocalizedN-Soliton solitonssolution forof theHarry IshimoriDym equation by inverse scattering method.}}

*{{cite journal |last1=Osborne |first1=A. R. |title=Soliton physics and the periodic inverse scattering transform |journal=Physica D: Nonlinear Phenomena |year=1995 |volume=86 |issue=1 |pages=81–89 |doi=10.1016/0167-2789(95)00089-M |url=https://www.sciencedirect.com/science/article/abs/pii/016727899500089M |issn=0167-2789|url-access=subscription }}

== Further reading ==

*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Clarkson |first2=P. A. |title=Solitons, Nonlinear Evolution Equations and Inverse Scattering |date=12 December 1991 |publisher=Cambridge University Press |isbn=978-0-521-38730-9 |url=https://wwwbooks.google.com/books/edition/Solitons_Nonlinear_Evolution_Equations_a/?id=RH44_EsqjGkC?hl |language=en}}

*{{cite book |last1=Bullough |first1=R. K. |last2=Caudrey |first2=P. J. |title=Solitons |date=11 November 2013 |publisher=Springer Science & Business Media |isbn=978-3-642-81448-8 |url=https://wwwbooks.google.com/books/edition/Solitons/?id=kFT-CAAAQBAJ?hl=en&gbpv=0 |language=en}}

*{{citation|mr=0336122|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|title=Korteweg-deVries equation and generalization. VI. Methods for exact solution.

|journal=Comm. Pure Appl. Math.|volume= 27 |year=1974|pages= 97–133|doi=10.1002/cpa.3160270108}}

*{{cite book |last1=Gelʹfand |first1=Izrailʹ Moiseevich |title=On the Determination of a Differential Equation from Its Spectral Function |date=1955 |publisher=American Mathematical Society |page=253-304|url=https://wwwbooks.google.com/books/edition/On_the_Determination_of_a_Differential_E/tSusYgEACAAJ?hlid=entSusYgEACAAJ |language=en}}

*{{cite book |last1=Marchenko |first1=Vladimir A. |title=Sturm-Liouville Operators and Applications |series=Operator Theory: Advances and Applications |date=1986 |volume=22 |___location=Basel|publisher=Birkhäuser|doi=10.1007/978-3-0348-5485-6 |isbn=978-3-0348-5486-3 |url=https://link.springer.com/book/10.1007/978-3-0348-5485-6 |language=en}}

*{{cite book |last1=Shaw |first1=J. K. |title=Mathematical Principles of Optical Fiber Communication |date=1 May 2004 |publisher=SIAM |isbn=978-0-89871-556-9 |url=https://wwwbooks.google.com/books/edition/Mathematical_Principles_of_Optical_Fiber/?id=GxpTjLWWAfcC?hl |language=en}}

==External links==