Inverse Scattering Transform

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 inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. These include the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equations, the Sine-Gordon equation, the Kadomtsev–Petviashvili equation, the Toda lattice equation, the Ishimori equation, the Dym equation etc. A further, particularly interesting, family of examples is provided by the Bogomolny equations (for a given gauge group and oriented Riemannian 3-fold), the 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.

The inverse scattering problem can be written as a Riemann–Hilbert factorization problem. This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.