Misplaced Pages

Compacton

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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 {\displaystyle 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 ) x x x = 0 {\displaystyle u_{t}+(u^{2})_{x}+(u^{2})_{xxx}=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 . {\displaystyle 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, and so is a compacton.

See also

References

Category: