Compacton

In the theory of integrable systems, a compacton, introduced in {{harvs| last1=Rosenau | first1=Philip | authorlink1=Philip Rosenau | last2=Hyman | first2=James M. | authorlink2=James (Mac) Hyman | title=Compactons: Solitons with finite wavelength | publisher=American Physical Society | year=1993 | journal=Physical Review Letters | volume=70 | issue=5 | pages=564–567}}, 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, and so is a compacton.

References

{{citation|url=https://www.ams.org/notices/200507/what-is.pdf|title=What is a compacton? |last1=Rosenau | first1=Philip |journal=Notices of the American Mathematical Society |year=2005|pages=738–739}}
{{Citation | last1=Rosenau | first1=Philip | last2=Hyman | first2=James M. | title=Compactons: Solitons with finite wavelength | publisher=American Physical Society | year=1993 |journal= Physical Review Letters |volume= 70| issue=5 | pages=564–567 |doi= 10.1103/PhysRevLett.70.564 | bibcode=1993PhRvL..70..564R | pmid=10054146}}
{{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 }}

Category:Solitons

Compacton

Example

See also

References