Inverse scattering transform: Difference between revisions

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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|mr=0336122|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|yearyear1=19741967|pagesyear2= 97–1331974}} 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.