Compacton

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

:$u_t+(u^m)_x+(u^n)_=0\,$

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

:$u_t+(u^2)_x+(u^2)_=0 \,$

has a travelling wave solution given by

:$u(x,t) = \begin \dfrac\cos^2((x-\lambda t)/4) & \text, x - \lambda t, \le 2\pi, \\ \\ 0 & \text, x - \lambda t, \ge 2\pi. \end$

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

See also

*Peakon
*Vector soliton

References

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*{{citation, title=Exact discrete breather compactons in nonlinear Klein-Gordon lattices , last1=Comte , first1=Jean-Christophe , journal=Physical Review E, publisher=American Physical Society , year=2002, volume=65 , issue=6 , pages=067601, doi=10.1103/PhysRevE.65.067601 , pmid=12188877 , bibcode=2002PhRvE..65f7601C

Solitons