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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. This formulation can be generalized to differential operators of order greater than two and also to periodic problems.
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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