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Compacton
In the theory of integrable systems, a compacton (introduced in ) 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
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