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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.
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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