---
tag: 6GY3X4Y8X5
kind: concept
name: Inverse scattering transform
msc: [37K15, 35Q53, 35Q51]
specializes: []
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_scattering_transform
same_as:
  wikidata: Q17098188
  wikipedia: Inverse scattering transform
---

## Summary

The inverse scattering transform is a method to solve the initial value problem of some nonlinear evolution equations, such as the [[Korteweg–De Vries equation|KTT75QC72Z]].
The solution at time $t$ is the potential of a linear spectral problem, such as a [[Schrödinger operator|89AA6XJY5K]] in one variable.
Its scattering data evolve in time by simple explicit formulas, and the potential is then recovered from the data.
It plays for these nonlinear equations the role that the Fourier transform plays for linear equations with constant coefficients.

## Explanations

### L1 human

The method works in three steps.
First, one turns the initial profile of the wave into spectral data: its bound states and how it reflects incoming waves.
Second, one moves these data forward in time, which is easy because each piece evolves on its own.
Third, one rebuilds the profile from the new data by solving a linear integral equation.
The bound states give the [[solitons|63D4AXYCT4]], which is why a localised initial wave splits into a train of solitons and a small spreading tail.