Inverse scattering transform: Difference between revisions

Revision as of 13:16, 28 March 2024

Revision as of 17:41, 28 March 2024

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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}}