Inverse scattering transform: Difference between revisions

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{{Short description|Method for solving certain nonlinear partial differential equations}}

[[File:Inverse scattering transform.png|thumb|The 3-step algorithm: transform the initial solution to initial scattering data, evolve initial scattering data, transform evolved scattering data to evolved solution]]In [[mathematics]], the '''inverse scattering transform''' is a method that solves the [[initial value problem]] for a [[Nonlinear system|nonlinear]] [[partial differential equation]] using mathematical methods related to [[scatterscattering|wave scattering]].{{sfn|Aktosun|2009}}{{rp|4960}} The <em>direct scattering transform</em> describes how a [[Function (mathematics)|function]] scatters waves or generates [[Bound state|bound-states]].{{sfn|Drazin|Johnson|1989}}{{rp|39-43}} The <em>inverse scattering transform</em> uses wave scattering data to construct the function responsible for wave scattering.{{sfn|Drazin|Johnson|1989}}{{rp|66-67}} The direct and inverse scattering transforms are analogous to the direct and inverse [[Fourier transform]]s which are used to solve [[Linear differential equation|linear]] partial differential equations.{{sfn|Drazin|Johnson|1989}}{{rp|66-67}}

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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===Scattering data time evolution===

The equations describing how scattering data evolves over time occur as solutions to a 1st order linear ordinary differential equation with respect to time. Using varying approaches, this first order linear differential equation may arise from the thelinear differential operators (Lax pair, operatorsAKNS pair), a combination of Laxthe linear differential operators and the nonlinear differential equation, or through additional substitution, integration or differentiation operations. Spatially asymptotic equations (<math display="inline">x \to \pm \infty</math>) simplify solving these differential equations.{{sfn|Aktosun|2009}}{{rp|4967-4968}}{{sfn|Drazin|Johnson|1989}}{{rp|68-72}}{{sfn|Gardner|Greene|Kruskal|Miura|1967}}

===Inverse scattering transform===

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The nonlinear differential Korteweg–De Vries equation is

{{sfn|Ablowitz|Segur|1981}}{{rp|4}}

: <math>u_{xt}-6uu_{x}+u_{xxx}=0</math>

===Lax operators===

The Lax operators are:{{sfn|Drazin|Johnson|1989}}{{rp|97-102}}

: <math>L= -\partial^{2}_{x}+u(x,t) \ </math> and <math display="inline"> \ M= -4\partial^{3}_{x}+6u\partial_{x}+3u_{x} </math>

The multiplicative operator is:

: <math>L_{t}+LM-ML=u_{xt}-6uu_{x}+u_{xxx}=0</math>

===Direct scattering transform===

The solutions to this differential equation

: <math display="inline">L(\psi)=-\psi_{xx}+u(x,0)\psi= \lambda \psi</math>

may include <em>scattering solutions</em> with a continuous range of eigenvalues (<em>continuous spectrum</em>) and <em>[[bound state|bound-state]]</em> solutions with discrete eigenvalues (<em>discrete spectrum</em>). The scattering data includes transmission coefficients <math display="inline">T(k,0)</math>, left reflection coefficient <math display="inline">R_{L}(k,0)</math>, right reflection coefficient <math display="inline">R_{R}(k,0)</math>, discrete eigenvalues <math display="inline">-\kappa^{2}_{1}, \ldots,-\kappa^{2}_{N}</math>, and left and right bound-state <em>normalization (norming) constants</em>.{{sfn|Aktosun|2009}}{{rp|4960}}

: <math>c(0)_{Lj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{L}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>

: <math> c(0)_{Rj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{R}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>

===Scattering data time evolution===

The spatially asymptotic left <math display="inline">\psi_{L}(k,x,t)</math> and right <math display="inline">\psi_{R}(k,x,t)</math> [[Jost function]]s simplify this step.{{sfn|Aktosun|2009}}{{rp|4965-4966}}

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\end{align}</math>

The <em>dependency constants</em> <math display="inline"> \gamma_{j}(t)</math> relate the right and left Jost functions and right and left normalization constants.{{sfn|Aktosun|2009}}{{rp|4965-4966}}

:<math>\gamma_{j}(t)=\frac{f_\psi_{L}(x,ki\kappa_{j},t)}{f_\psi_{R}(x,ki\kappa_{j},t)}=(-1)^{N-j} \frac{c_{Rj}(t)}{c_{Lj}(t)}</math>

The Lax <math display="inline">M</math> differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions.{{sfn|Aktosun|2009}}{{rp|4967}}

:<math>\partial_{t}\psi_{L}(k,x,t)-M\psi_{L}(x,k,t)=

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c_{Lj}(t)&=c_{Lj}(0)e^{+4\kappa^{3}_{j}t}, \ j=1, \ldots, N \\

c_{Rj}(t)&=c_{Rj}(0)e^{-4\kappa^{3}_{j}t}, \ j=1, \ldots, N \end{align}</math>

===Inverse scattering transform===

The <em>Marchenko kernel</em> is <math display="inline">F(x,t)</math>.{{sfn|Aktosun|2009}}{{rp|4968-4969}}

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The [[Gelfand–Levitan–Marchenko integral equation|Marchenko integral equation]] is a linear integral equation solved for <math display="inline">K(x,y,t)</math>.{{sfn|Aktosun|2009}}{{rp|4968-4969}}

: <math> K(x,z,t)+F(x+z,t)+ \int^{\infty}_{x} K(x,y,t)F(y+z,t) \ dy=0 </math>

The solution to the Marchenko equation, <math display="inline">K(x,y,t) </math>, generates the solution <math display="inline">u(x,t)</math> to the nonlinear partial differential equation.{{sfn|Aktosun|2009}}{{rp|4969}}

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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=249–315 |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://books.google.com/books?id=Bzu4XAUpFZUC |language=en}}

* {{cite journalbook |last1=Ablowitz |first1=M.Mark J. |last2=KaupFokas |first2=D.J. |last3=Newell |first3=A.C. |last4=Segur |first4=HS. |title=TheComplex InverseVariables: ScatteringIntroduction Transform—Fourierand AnalysisApplications for Nonlinear Problems|year=2003 |journalpublisher=StudiesCambridge inUniversity Applied MathematicsPress |yearisbn=1974 |volume=53978-0-521-53429-1 |pages=249-315604–620 |url=https://onlinelibrarybooks.wileygoogle.com/doi/abs/10.1002/sapm1974534249books?id=SFqbV3i3hO0C |language=en}}

*{{cite bookjournal |last1=Ablowitz |first1=Mark J. |last2=Segur |first2=Harvey |title=SolitonsNonlinear waves and the Inverse Scattering Transform |journal=Optik |year=19812023 |publishervolume=SIAM278 |isbnpages=978-0-89871-477-7170710 |doi=10.1016/j.ijleo.2023.170710 |bibcode=2023Optik.27870710A |url=https://www.googlesciencedirect.com/booksscience/editionarticle/Solitons_and_the_Inverse_Scattering_Tranpii/Bzu4XAUpFZUC?hl S0030402623002061|languageurl-access=ensubscription }}

*{{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 |chapter-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|doi=10.1007/978-0-387-30440-3_295 }}

*{{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 |url=https://www.sciencedirect.com/science/article/pii/S0030402623002061}}

* {{cite book |last1=AktosunDrazin |first1=TuncayP. G. |titlelast2=EncyclopediaJohnson of|first2=R. ComplexityS. and|title=Solitons: SystemsAn ScienceIntroduction |year=20091989 |publisher=SpringerCambridge University Press |isbn=978-0-387521-3044033655-3 |pages=4960–49710 |url=https://linkbooks.springergoogle.com/referenceworkentry/10.1007/978books?id=HPmbIDk2u-0-387-30440-3_295gC |language=en |chapter=Inverse Scattering Transform and the Theory of Solitons}}

* {{cite bookjournal |last1=DrazinGardner |first1=P.Clifford GS. |last2=JohnsonGreene |first2=RJohn M. S|last3=Kruskal |first3=Martin D. |last4=Miura |first4=Robert M. |title=Solitons:Method Anfor IntroductionSolving the Korteweg-deVries Equation |journal=Physical Review Letters |year=19891967 |publishervolume=Cambridge19 University|issue=19 Press|pages=1095–1097 |isbndoi=978-0-521-33655-010.1103/PhysRevLett.19.1095 |bibcode=1967PhRvL..19.1095G |url=https://wwwjournals.googleaps.comorg/booksprl/editionabstract/Solitons10.1103/HPmbIDk2uPhysRevLett.19.1095|url-gC?hlaccess=ensubscription |language=en}}

*{{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 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 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1095}}

*{{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=AblowitzOsborne |first1=MarkA. JR. |title=NonlinearSoliton wavesphysics and the Inverseperiodic Scatteringinverse Transformscattering transform |journal=OptikPhysica D: Nonlinear Phenomena |year=20231995 |volume=27886 |issue=1 |pages=17071081–89 |doi=10.1016/j.ijleo.2023.1707100167-2789(95)00089-M |url=https://www.sciencedirect.com/science/article/abs/pii/S0030402623002061016727899500089M |issn=0167-2789|url-access=subscription }}

*{{cite book |last1=Oono |first1=H. |editor1-last=Alfinito |editor1-first=E. |editor2-last=Boiti |editor2-first=M. |editor3-last=Martina |editor3-first=L. |title=Nonlinear Physics: Theory and Experiment |year=1996 |publisher=World Scientific Publishing Company Pte Limited |isbn=978-981-02-2559-9 |pages=241-248 |url=https://www.google.com/books/edition/Nonlinear_Physics/35EfzQEACAAJ?hl=en |language=en |chapter=N-Soliton solution of Harry Dym equation by inverse scattering method.}}

== 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==