---
tag: RN8Y1823FX
kind: definition
name: Soliton
msc: [35C08, 35Q51, 37K40]
defines: 63D4AXYCT4
uses: [M37M411KP7, KTT75QC72Z, 01PPDSDP2N]
sources:
  - url: https://en.wikipedia.org/wiki/Soliton
---

## Definition

Let $E$ be an evolution equation for $u(x, t)$ with $x, t \in \R$.
A *solitary wave* of $E$ is a [[traveling wave|M37M411KP7]] solution $u(x, t) = \varphi(x - ct)$ with $c \in \R$ and a profile $\varphi$ that tends to constant limits as $x \to \pm \infty$.
A *soliton* is a solitary wave of $E$ that survives interaction with other solitary waves of $E$.
That is, after each such interaction it emerges with the same shape and speed, shifted only in position.

## Remarks

The second clause is often checked by the existence of exact multi-soliton solutions, which split for $t \to \pm \infty$ into separated solitary waves with the same speeds.
There is no single formal definition used by all authors.

Example: for the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t + 6 u u_x + u_{xxx} = 0$ and $c > 0$, put $\varphi(\xi) = \frac{c}{2} \operatorname{sech}^2\big(\frac{\sqrt{c}}{2} \xi\big)$, using the [[hyperbolic secant|01PPDSDP2N]].
For $u = \varphi(x - ct)$ the equation becomes $-c\varphi' + 6\varphi\varphi' + \varphi''' = 0$, which integrates to $\varphi'' = c\varphi - 3\varphi^2$.
With $\varphi = A \operatorname{sech}^2(k\xi)$ one has $\varphi'' = 4k^2 A \operatorname{sech}^2(k\xi) - 6k^2 A \operatorname{sech}^4(k\xi)$, so the equation holds with $4k^2 = c$ and $A = 2k^2 = c/2$.
The KdV equation also has explicit two-soliton solutions, so this solitary wave is a soliton.

Non-example: the linear equation $u_t + u_{xxx} = 0$ has no solitary wave with $c \ne 0$ that decays at infinity.
Indeed, $\varphi''' = c\varphi'$ gives $\varphi'' = c\varphi$, whose decaying solutions on all of $\R$ are only $\varphi = 0$.

## Explanations

### L1 human

A solitary wave is a single hump that moves without changing shape.
It is a soliton when it also survives collisions with other such humps, keeping its size and speed.