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


**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**:


1. 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 $\DD$.
2. 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](https://en.wikipedia.org/wiki/Symbol_of_a_differential_operator) and [this lecture notes on linear analysis on manifolds](https://faculty.math.illinois.edu/~palbin/AnalysisOnMfds/LectureNotes.pdf).



Principal symbol for non-linear differential operators
I've seen only the first. It is indeed used mostly for identifying whether a nonlinear PDE is elliptic, hyperbolic, or parabolic. If so, one can use the respective linear theory, along with the appropriate implicit function theorem to prove existence theorems. Look up fully nonlinear elliptic PDEs for one well studied area. 


