Compacton

---

title: "Compacton" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["solitons"] topic_path: "general/solitons" source: "https://en.wikipedia.org/wiki/Compacton" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In the theory of integrable systems, a compacton (introduced in ) is a soliton with compact support.

An example of an equation with compacton solutions is the generalization

: u_t+(u^m)x+(u^n){xxx}=0,

of the Korteweg–de Vries equation (KdV equation) with m, n 1. The case with m = n is the Rosenau–Hyman equation as used in their 1993 study; the case m = 2, n = 1 is essentially the KdV equation.

Example

The equation

: u_t+(u^2)x+(u^2){xxx}=0 ,

has a travelling wave solution given by

: u(x,t) = \begin{cases} \dfrac{4\lambda}{3}\cos^2((x-\lambda t)/4) & \text{if }|x - \lambda t| \le 2\pi, \ \ 0 & \text{if }|x - \lambda t| \ge 2\pi. \end{cases}

This has compact support in x, and so is a compacton.

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::