Inverse scattering transform

In mathematics, the inverse scattering transform is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical methods related to wave scattering.^[1]: 4960 The direct scattering transform describes how a function scatters waves or generates bound-states.^{[2]: 39–43} The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering.^{[2]: 66–67} The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations.^{[2]: 66–67}

Using a pair of differential operators, a 3-step algorithm may solve 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).^{[2]: 66–67}

This algorithm simplifies solving a nonlinear partial differential equation to solving 2 linear ordinary differential equations and an ordinary integral equation, a method ultimately leading to analytic solutions for many otherwise difficult to solve nonlinear partial differential equations.^[2]: 72

The inverse scattering problem is equivalent to a Riemann–Hilbert factorization problem, at least in the case of equations of one space dimension.^[3] This formulation can be generalized to differential operators of order greater than two and also to periodic problems.^[4] In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.