Why is not Fourier Transform Good for Non-linear Processes
Because complex exponentials $e^{\jmath \omega t}$, which are results of Fourier transform, are the eigenfunctions for linear, time invariant (LTI) systems. See eigenfunction of LTI. Also see this answer on SP.SE.

Thus, Fourier transform is useful for analyzing linear (not suitable for non-linear one) time invariant (can be intepreted as stationnary) system.