Compacton Explained

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 ^{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 ^{2 )} _{xxx} =0

has a travelling wave solution given by

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

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

See also

• Peakon
• Vector soliton