In the physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 field of of scattering theory

Scattering theory

In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

, 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

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, inverse scattering refers to the determination of the solutions of a set of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s based on known asymptotic solutions, that is, on solving the S-matrix. Examples of equations that have been solved by inverse scattering are the Schroedinger equation, the Korteweg–de Vries equation

Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

 and the KP equation.

The inverse scattering problem is the inverse problem

Inverse problem

An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...

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

Scattering

Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

.

Since its early statement for radiolocation

Radar

Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

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

Acoustic location

Acoustic location is the science of using sound to determine the distance and direction of something. Location can be done actively or passively, and can take place in gases , liquids , and in solids .* Active acoustic location involves the creation of sound in order to produce an echo, which is...

, geophysical survey, nondestructive testing

Nondestructive testing

Nondestructive testing or Non-destructive testing is a wide group of analysis techniques used in science and industry to evaluate the properties of a material, component or system without causing damage....

, medical imaging

Medical imaging

Medical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...

, quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

.

See also: Inverse scattering transform

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...