Compacton

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|\leq 2\pi ,\\\\0&{\text{if }}|x-\lambda t|\geq 2\pi .\end{cases}}

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

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, American Physical Society, 70 (5): 564–567, Bibcode:1993PhRvL..70..564R, doi:10.1103/PhysRevLett.70.564

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