Inverse scattering transform: Difference between revisions

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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, at least in the case of equations of one space dimension. This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.

In higher space dimensions on has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.

==Example: the Korteweg–de Vries equation==