Consider
$$
\frac{\partial^2}{\partial x^2}u+\mu \sin(u) = 0 \\
u(0) = 0 = u(1)
$$
The linearized version is for small $u$
$$
\frac{\partial^2}{\partial x^2}u+\mu u = 0
$$
This gives for the general solution
$$
u(x) = C\sin(\sqrt{\mu}x) \quad n\in N,\mu=4\pi^2n^2 \lor \mu=(\pi+2n\pi)^2
$$
Because of the shape of the linearized version, we can call this general solution of $u$ an eigenfunction.
I plugged the nonlinear PDE in Wolfram Alpha. This showed me that the nonlinear does not have an analytic solution. This made me wonder:
What is the relationship between the solutions of the nonlinear PDE and the eigensolutions?