Unlinear stability

As soon as $\nu> 39,5$, the first mode become unstable. The theory of the linear stability had us predicted this behaviour. But in this domain the solution $u(x,t)$ of the KSE don't keep a weak magnitude. So the there is a time when the linear stability can not be amplied. To conclude, the prediction of the final behaviour is not predicted by the linear stability. A study of unlinear stability is really hard to develop. That's why, I propose a not exhaustive list of the bifurcations that I have meet in the experimental study.

I have let simulation turn to see if the solution converged or not. I made this experiment for different values of $\nu$. For $\nu [39.5,130]$, the solution converged toward a function similar to this one which is represented on figure 3.1.

Figure 3.1: Solution for $\nu = 100$

To prouve that the solution is effectively converged, I have plot the norm of the solution. The norm that I have used, is :

$$\vert\vert f\vert\vert=\sqrt{\int_0^1 \vert f(x)\vert^2 dx}$$

The graph obtained is on the figure 3.2

Figure 3.2: Norm versus time for $\nu = 100$