In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the

inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating th ...

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

Soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...

equations are a class of

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

which can be studied and solved by a method called the

inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...

, which reduces the nonlinear PDEs to a linear inverse scattering problem. The

nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...

, the Korteweg–de Vries equation and the

KP equation KP may refer to: Businesses and organizations * ''Komsomolskaya Pravda'', a daily Russian newspaper * ''KP'' (newspaper), a Ukrainian newspaper * KP Snacks, a United Kingdom food manufacturer * Kaiser Permanente, a U.S. health maintenance organi ...

are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem. Since its early statement for

radiolocation Radiolocation, also known as radiolocating or radiopositioning, is the process of finding the location of something through the use of radio waves. It generally refers to passive uses, particularly radar—as well as detecting buried cables, ...

, many applications have been found for inverse scattering techniques, including echolocation,

geophysical Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...

survey,

nondestructive testing Nondestructive testing (NDT) is any of a wide group of analysis techniques used in science and technology industry to evaluate the properties of a material, component or system without causing damage. The terms nondestructive examination (NDE), n ...

, medical imaging,

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...

References

* . * ''Inverse Acoustic and Electromagnetic Scattering Theory''; Colton, David and Kress, Rainer Scattering theory Scattering, absorption and radiative transfer (optics) Inverse problems {{math-physics-stub