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is a solution of the KSE. In order to determine if this solution is always
stable, we supperpose on it a perturbation. The evolution of the perturbation is imposed
by the KSE.
We have the choise of the perturbation. The solution of the KSE is invariable in and in
, and we have considered periodic limit conditions, so we can decide that we are looking
for perturbation under a normal form. It's to say that :
The amplitude of the perturbation is supposed to be small, so we will just retain
the order one terms in amplitude.
The KSE become :
With the form of the perturbation we find the dispersion relation :
k is the wave number and is equal to
. Now there is two possibilities.