Content deleted Content added
No edit summary |
m Journal cites: format journal names, using AWB (11880) |
||
Line 1:
In [[mathematics]], the '''inverse scattering transform''' is a method for solving some non-linear [[partial differential equations]]. It is one of the most important developments in mathematical physics in the past 40 years. The method is a non-linear analogue, and in some sense generalization, of the [[Fourier transform]], which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential.
The inverse scattering transform may be applied to many of the so-called [[exactly solvable model]]s, that is to say completely integrable infinite dimensional systems. It was first introduced by {{harvs|txt|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|year1=1967|year2=1974}} for the [[Korteweg–de Vries equation]], and soon extended to the [[nonlinear Schrödinger equation]], the [[Sine-Gordon equation]], and the [[Toda lattice]] equation. It was later used to solve many other equations, such as the [[Kadomtsev–Petviashvili equation]], the [[Ishimori equation]], the [[Dym equation]], and so on. A further family of examples is provided by the [[Bogomol'nyi–Prasad–Sommerfield bound|Bogomolny equations]] (for a given gauge group and oriented Riemannian 3-fold), the <math>L^2</math> solutions of which are [[magnetic monopoles]].
A characteristic of solutions obtained by the inverse scattering method is the existence of [[solitons]], solutions resembling both particles and waves, which have no analogue for linear partial differential equations. The term "soliton" arises from non-linear optics.
The inverse scattering problem can be written as a [[Riemann–Hilbert factorization]] problem, at least in the case of equations of one space dimension. This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.
Line 37:
'''Step 1.''' Determine the nonlinear partial differential equation. This is usually accomplished by analyzing the [[physics]] of the situation being studied.
'''Step 2.''' Employ ''forward scattering''. This consists in finding the [[Lax pair]]. The Lax pair consists of two linear [[Operator (mathematics)|operator]]s, <math>L</math> and <math>M</math>, such that <math>Lv=\lambda v</math> and <math>\frac{dv}{dt}=Mv</math>. It is extremely important that the [[eigenvalue]] <math>\lambda</math> be independent of time; i.e. <math>\frac{d\lambda}{dt}=0.</math> Necessary and sufficient conditions for this to occur are determined as follows: take the time [[derivative]] of <math>Lv=\lambda v</math> to obtain
:<math>\frac{dL}{dt}v+L\frac{dv}{dt}=\frac{d\lambda}{dt}v+\lambda \frac{dv}{dt}.</math>
Plugging in <math>Mv</math> for <math>\frac{dv}{dt}</math> yields
:<math>\frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+\lambda Mv.</math>
Line 49:
:<math>\frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+MLv.</math>
Thus,
:<math>\frac{dL}{dt}v+LMv-MLv=\frac{d\lambda}{dt}v.</math>
Line 65:
After concocting the appropriate Lax pair it should be the case that Lax's equation recovers the original nonlinear PDE.
'''Step 3.''' Determine the time evolution of the eigenfunctions associated to each eigenvalue <math>\lambda</math>, the norming constants, and the reflection coefficient, all three comprising the so-called scattering data. This time evolution is given by a system of linear [[ordinary differential equations]] which can be solved.
'''Step 4.''' Perform the ''inverse scattering'' procedure by solving the [[Gelfand–Levitan–Marchenko integral equation]] ([[Israel Moiseevich Gelfand]] and [[Boris Moiseevich Levitan]];<ref>Gel’fand, I. M. & Levitan, B. M., "On the determination of a differential equation from its spectral function". American Mathematical Society Translations, (2)1:253–304, 1955.</ref> [[Vladimir Aleksandrovich Marchenko]]<ref>V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.</ref>), a linear [[integral equation]], to obtain the final solution of the original nonlinear PDE. All the scattering data is required in order to do this. Note that if the reflection coefficient is zero, the process becomes much easier. Note also that this step works if <math>L</math> is a differential or difference operator of order two, but not necessarily for higher orders. In all cases however, the ''inverse scattering'' problem is reducible to a [[Riemann–Hilbert factorization]] problem.
Line 79:
* [[Dym equation]]
Further examples of integrable equations may be found on the article [[Integrable system#
==References==
Line 86:
*N. Asano, Y. Kato, ''Algebraic and Spectral Methods for Nonlinear Wave Equations'', Longman Scientific & Technical, Essex, England, 1990.
*M. Ablowitz, P. Clarkson, ''Solitons, Nonlinear Evolution Equations and Inverse Scattering'', Cambridge University Press, Cambridge, 1991.
*{{citation|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
*{{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}}
| |||