The Physics

Start with a relatively small value for $M\quad (\ll 1)$ and show that the wave maintains its shape and moves at the correct velocity. Then increase $M$ to find out what happens. The stability analysis for a non-linear equation like this is difficult. Try halving the distance between grid points. How does the behaviour of the system change? Do the effects you observe describe real physics or are they numerical artefacts? One common numerical problem only manifests itself for large $M$. Try running for a few steps at $M=1$, look at the values for $x$ and try to work out what has gone wrong. Think about how to prevent this problem (Hint: you should find something very similar to the Courant-Friedrichs-Lewy condition).