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Compacton

From HandWiki - Reading time: 1 min

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

ut+(um)x+(un)xxx=0

of the Korteweg–de Vries equation (KdV equation) with mn > 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

ut+(u2)x+(u2)xxx=0

has a travelling wave solution given by

u(x,t)={4λ3cos2((xλt)/4)if |xλt|2π,0if |xλt|2π.

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

See also

References




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