Let's say I start with the linear differential equation
$$ y''=-y, $$
which has (for example) the two solutions $y=e^{\pm i x}$, therefore following the superposition principle the general solution is $y=a e^{ i x}+b e^{- i x}$, with $a$ and $b$ determined by the initial conditions.
Now let's say I add a perturbative term to my equation:
$$ y''=-y + \alpha y |y|^2, $$
with $\alpha$ a constant such that the new term is small compared to $y$ ($||^2$ is the absolute square). With this new equation I can still find two solutions, by assuming a solution of the form $ae^{i w x}$, I find $w\approx \pm (1 -\alpha |a|^2/2)$, defining two solutions $a e^{ i(1-\alpha |a|^2/2) x}$ and $b e^{ -i(1-\alpha |b|^2/2) x}$.
Now I would like to find the general solution to the new equation, however as it is nonlinear, I can no longer use the superposition principle. Still, as my perturbative term is small, I would expect the new general solution to be very close to the first one I obtained with the unperturbed equation (at least it should tend to it when $\alpha$ tends to zero). Is there any procedure to obtain this? I suspect there is no general procedure for any given nonlinear equation, but I thought there might be one when the nonlinearity comes from a perturbative term like here.
Numerically, I find that a linear superposition of the two special solutions is a very good approximation, at the expense of changing $w$ from $\pm (1 -\alpha |a|^2/2)$ to $\pm (1 + 0.5\alpha |a|^2/2+\epsilon)$, with $\epsilon$ to be adjusted empirically.