Inverse scattering problem

In the physics field of scattering theory, the inverse scattering problem is that of determining characteristics of an object (its shape, internal constitution, etc.) based on data of how it scatters incoming radiation or particles.

In mathematics, inverse scattering refers to the determination of the solutions of a set of differential equations based on known asymptotic solutions, that is, on solving the S-matrix.

Soliton equations can be studied and solved by inverse scattering. Examples are the Nonlinear Schrödinger equation, the Korteweg–de Vries equation and the KP equation. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem.

The inverse scattering problem is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the characteristics of the scatterer.

Since its early statement for radiolocation, many applications have been found for inverse scattering techniques, including echolocation, geophysical survey, nondestructive testing, medical imaging, quantum field theory.

References

• Marchenko, V. A. (2011), Sturm-Liouville operators and applications (2 ed.), Providence: American Mathematical Society, ISBN 978-0-8218-5316-0, MR 2798059 .

Inverse Acoustic and Electromagnetic Scattering Theory, Colton, David, Kress, Rainer http://www.springer.com/mathematics/dynamical+systems/book/978-1-4614-4941-6