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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$
\includegraphics [scale=0.8]{sol1.eps}

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

$\displaystyle \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$
\includegraphics [scale=0.8]{norm1.eps}


next up previous contents
Next: Second bifurcation Up: First bifurcation Previous: Linear stability theory   Contents
Julien Delbove
2000-11-23