Compacton

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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

 u_t+(u^m)_x+(u^n)_{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[edit]

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, so is a compacton.

See also[edit]

References[edit]