$\newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}}\newcommand{\DD}{\mathrm{D}}\newcommand{\dd}{\mathrm{d}}$

**Prerequisites**: Let $\mathrm{T}: C^\infty(\Omega) \rightarrow
C^\infty(\Omega), u(\cdot)\mapsto F(\cdot, \{\partial^\alpha
u(\cdot)\}_{\vert{\alpha}\vert\leq k})$ be a non-linear differential
operator, where $\Omega\subseteq\R^n$ is an open domain, $F:\R^N
\rightarrow \R$ a smooth function and $\alpha \in \N_0^n$ denotes a
multiindex.

**Question**: Is there a generally accepted way of defining the
principal symbol of such a non-linear differential operator? If there
is, how does it extend to systems of PDE's (i.e. $F$ is
vector-valued)?

**Approaches**:

One (rather obvious) idea consists in linearizing $F$ in the $u$-dependent variables and then defining the $m$-th symbol (at $u$!) via $\sigma_m(T,u)(x,\xi) := \sigma_m(\DD_uT)(x,\xi)$ with the Frechet-derivative $\DD$.

Another is in using the Definition $\sigma_m(D)(x,\xi) := i^m \lim_{t\rightarrow\infty} \frac{1}{t^m} e^{-itf}\circ T\circ e^{itf}$ with $f\colon \R^n\rightarrow \R$ such that $\dd f(x)=\xi$. I am having problems with the well-definedness of this generalization. However, if this is a valid approach, am I correct in suspecting that this works out for

*every local operator*?

**Sources on the linear case**: For the definition in the linear case
I consulted the Wikipedia page and this lecture notes on linear analysis on manifolds.