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

[math]\displaystyle{ u_t+(u^m)_x+(u^n)_{xxx}=0\, }[/math]

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

[math]\displaystyle{ u_t+(u^2)_x+(u^2)_{xxx}=0 \, }[/math]

has a travelling wave solution given by

[math]\displaystyle{ 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} }[/math]

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