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

ut+(um)x+(un)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

ut+(u2)x+(u2)xxx=0

has a travelling wave solution given by

u(x,t)={4λ3cos2((x−λt)/4)if |x−λt|≤2π,0if |x−λt|≥2π.

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

## See also

* Peakon
* Vector soliton

## References

* Rosenau, Philip (2005), "What is a compacton?", Notices of the American Mathematical Society: 738–739, https://www.ams.org/notices/200507/what-is.pdf
* Rosenau, Philip; Hyman, James M. (1993), "Compactons: Solitons with finite wavelength", Physical Review Letters (American Physical Society) 70 (5): 564–567, doi:10.1103/PhysRevLett.70.564, PMID 10054146, Bibcode: 1993PhRvL..70..564R
* Comte, Jean-Christophe (2002), "Exact discrete breather compactons in nonlinear Klein-Gordon lattices", Physical Review E (American Physical Society) 65 (6): 067601, doi:10.1103/PhysRevE.65.067601, PMID 12188877, Bibcode: 2002PhRvE..65f7601C

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