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, sound source reconstruction, source reconstruction in ac ...

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 is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...

equations are a class of

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

which can be studied and solved by a method called the

inverse scattering transform In mathematics, the inverse scattering transform is a method that solves the initial value problem for a Nonlinear system, nonlinear partial differential equation using mathematical methods related to scattering, wave scattering. The direct scatte ...

, 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 nonli ...

, the Korteweg–de Vries equation and the KP equation are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem. Inverse scattering has been applied to many problems including

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, particularly radar—as well as detecting buried cables, wate ...

, echolocation,

geophysical Geophysics () is a subject of natural science concerned with the physical processes and properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct investigations acros ...

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 Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to revea ...

, and

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

Citations

References

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