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
)
x
x
x
=
0
{\displaystyle 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
)
x
x
x
=
0
{\displaystyle u_{t}+(u^{2})_{x}+(u^{2})_{xxx}=0\,}
has a travelling wave solution given by
u
(
x
,
t
)
=
{
4
λ
3
cos
2
(
(
x
−
λ
t
)
/
4
)
if
|
x
−
λ
t
|
≤
2
π
,
0
if
|
x
−
λ
t
|
≥
2
π
.
{\displaystyle u(x,t)={\begin{cases}{\dfrac {4\lambda }{3}}\cos ^{2}((x-\lambda t)/4)&{\text{if }}|x-\lambda t|\leq 2\pi ,\\\\0&{\text{if }}|x-\lambda t|\geq 2\pi .\end{cases}}}
This has compact support in x , and so is a compacton.
See also
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 70 (5), American Physical Society: 564– 567, Bibcode :1993PhRvL..70..564R , doi :10.1103/PhysRevLett.70.564 , PMID 10054146 Comte, Jean-Christophe (2002), "Exact discrete breather compactons in nonlinear Klein-Gordon lattices", Physical Review E 65 (6), American Physical Society: 067601, Bibcode :2002PhRvE..65f7601C , doi :10.1103/PhysRevE.65.067601 , PMID 12188877 
