Relation eigenfunction of linearized PDE and solution of the original PDE

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?