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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?

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