8 Solitons wave animation (non-linear wave pde)

             3
∂u+ 6u ∂u+  ∂-u3 = 0
∂t     ∂x   ∂x

Assuming special solution u = f (ξ)  where ξ = x− c ⁄ t  , this PDE is transformed to non-linear first order ODE

  f2    3  1 (df)2
− c 2 + f + 2 dξ   = 0

The above is solved analytically (Krvskal, Zabrsky 1965) and the solution is

      (   )     ( √-       )
f (ξ) = 1c sech2  -c(x − ct)
        2         2

Tall waves move fast but have smaller period, short wave move slow. Tall wave pass through short wave and leave as they enter. Here are two animations and the above solution. This first animation has one tall wave passing though short wave

  

This animation shows three waves

  

Code used for the above is

PDF

Mathematica notebook