# Compacton

In the theory of integrable systems, a compacton, introduced in (Philip Rosenau & James M. Hyman 1993), is a soliton with compact support.

An example of an equation with compacton solutions is the generalization

${\displaystyle u_{t}+(u^{m})_{x}+(u^{n})_{xxx}=0\,}$

of the Korteweg–de Vries equation (KdV equation) with mn > 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

${\displaystyle u_{t}+(u^{2})_{x}+(u^{2})_{xxx}=0\,}$

has a travelling wave solution given by

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

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