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

$u_t+(u^m{)}_x+(u^n{)}_{xxx}=0$

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

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

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

See also

• Peakon
• Vector soliton