Numerical Schemes for KdVs?

qspeechc

Hi

I am working with the DE
[tex] u_t + u_{xxx} + u^su_x = 0 [/tex]
where s>0 is any real number.

I am trying to simulate the collision of two travelling solitons numerically (something like this: http://www.tu-chemnitz.de/physik/KSND/abb/anim.gif), using a finite-difference scheme, but my numerical solutions either blow up or do something crazy.
I used the initial conditions
[tex]f(x) = e^{-x^2}[/tex]
and then 10*f(x-10)+f(x+10) or something like that to get two solitons with different velocities. Any help?