### 2.5 Solitons wave animation (non-linear wave PDE)

\[ \frac{\partial u}{\partial t}+6u\frac{\partial u}{\partial x}+\frac{\partial ^{3}u}{\partial x^{3}}=0 \]

Assuming special solution \(u=f\left ( \xi \right ) \) where \(\xi =x-ct\), this PDE is transformed to non-linear ﬁrst order ODE

\[ -c\frac{f^{2}}{2}+f^{3}+\frac{1}{2}\left ( \frac{df}{d\xi }\right ) ^{2}=0 \]

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

\[ f\left ( \xi \right ) =\left ( \frac{1}{2}c\right ) \operatorname{sech}^{2}\left ( \frac{\sqrt{c}}{2}\left ( x-ct\right ) \right ) \] 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

Source code for the above 3 cases is