Inverse scattering transform: Difference between revisions

Revision as of 05:17, 11 June 2024

Latest revision as of 23:27, 19 June 2025

(9 intermediate revisions by 4 users not shown)

Line 6:

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.

Line 21:

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===

Line 112:

==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 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=609-613604–620 |url=https://wwwbooks.google.com/books/edition/Complex_Variables/SFqbV3i3hO0C?hlid=en&gbpv=0SFqbV3i3hO0C |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 |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 book |last1=Drazin |first1=P. G. |last2=Johnson |first2=R. S. |title=Solitons: An Introduction |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-33655-0 |url=https://wwwbooks.google.com/books/edition/Solitons/?id=HPmbIDk2u-gC?hl=en |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=77-9077–90 |url=https://wwwbooks.google.com/books/edition/Inverse_Scattering_and_Applications/yTYcCAAAQBAJ?hlid=enyTYcCAAAQBAJ |language=en |chapter=Localized solitons for the Ishimori equation}}

*{{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=241-248241–248 |url=https://wwwbooks.google.com/books/edition/Nonlinear_Physics/35EfzQEACAAJ?hlid=en35EfzQEACAAJ |language=en |chapter=N-Soliton solution of Harry Dym 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==