Non-linear Perturbation Operator Examples - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:42:22Z http://mathoverflow.net/feeds/question/103424 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103424/non-linear-perturbation-operator-examples Non-linear Perturbation Operator Examples bobuhito 2012-07-29T04:46:01Z 2012-07-29T04:46:01Z <p>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).</p>