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

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

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

See also

• peakon
• vector soliton

References

• Rosenau, Philip (2005), "What is a compacton?" (PDF), Notices of the American Mathematical Society: 738–739
• Rosenau, Philip; Hyman, James M. (1993), "Compactons: Solitons with finite wavelength", Physical Review Letters (American Physical Society) 70 (5): 564–567, Bibcode:1993PhRvL..70..564R, doi:10.1103/PhysRevLett.70.564