In the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...

, a compacton, introduced in , is a

soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...

with

compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...

. 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 In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. Waves can be Periodic function, periodic, in which case those quantities ...

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.

References

* * *{{citation, title=Exact discrete breather compactons in nonlinear Klein-Gordon lattices , last1=Comte , first1=Jean-Christophe , journal=

Physical Review E ''Physical Review E'' is a peer-reviewed, scientific journal, published monthly by the American Physical Society. The main field of interest is collective phenomena of many-body systems. It is currently edited by Uwe C. Täuber. While original r ...

, publisher=American Physical Society , year=2002, volume=65 , issue=6 , pages=067601, doi=10.1103/PhysRevE.65.067601 , pmid=12188877 , bibcode=2002PhRvE..65f7601C Solitons

Example

See also

References