---
tag: 63D4AXYCT4
kind: concept
name: Soliton
msc: [35C08, 35Q51, 37K40]
specializes: []
sources:
  - url: https://en.wikipedia.org/wiki/Soliton
same_as:
  wikidata: Q464949
  wikipedia: Soliton
---

## Summary

A soliton is a localised wave that moves at constant speed without changing its shape.
It keeps its shape and speed after it collides with other solitons; the only trace of the collision is a shift in position.
Solitons arise when nonlinear steepening and dispersion balance, in equations such as the [[Korteweg–De Vries equation|KTT75QC72Z]], the [[nonlinear Schrödinger equation|GEJHRXC176]] and the [[sine-Gordon equation|802QSF9NM3]].
They are central in the theory of integrable equations and in models of water waves and of light pulses in optical fibres.

## Explanations

### L1 human

A soliton is a single hump of water, or of light, that travels far without spreading out or breaking.
Usually a wave packet spreads because its parts travel at different speeds, and a tall wave steepens and breaks because its top moves faster.
In a soliton these two effects cancel.
For the KdV equation $u_t + 6 u u_x + u_{xxx} = 0$ the soliton is $u = \frac{c}{2} \operatorname{sech}^2\big(\frac{\sqrt{c}}{2}(x - ct)\big)$: taller solitons are narrower and faster.
When a tall fast soliton overtakes a small slow one, both come out of the collision unchanged.