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Consider a non-linear operator $\cal H$ which maps a function to a function (e.g., a map from a starting wave function $f(x,y,z)$ to a later wave function according to some non-linear PDE) and an $\epsilon$-perturbation on the input function. All real-world examples that i can think of can then be linearized (to some non-zero linear operator $\cal A$ with some order $m$) for small $\epsilon$: $$ \cal H[f+\epsilon g]-\cal H[f]=\epsilon^m \cal A[g] \qquad +O(\epsilon^{m+1}) $$ Are there any real-world counter-examples which stay non-linear in this limit? I know I haven't defined "real-world"; I'd ideally like a non-linear PDE, but would settle for anything that comes up in some application (e.g., integral equations, or system-control vectors).

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