Harman Patil (Editor)

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 ) x x x = 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 ) x x x = 0

has a travelling wave solution given by

u ( x , t ) = { 4 λ 3 cos 2 ⁡ ( ( x − λ t ) / 4 ) if  | x − λ t | ≤ 2 π , 0 if  | x − λ t | ≥ 2 π .

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

References

Compacton Wikipedia
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