Principal symbol for non-linear differential operators

Question

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

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 and this lecture notes on linear analysis on manifolds.

Answer 1

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.

Answer 2

To my knowledge, the principal symbol of a non-linear differential operator is not discussed very often. When I have seen it discussed, the definition basically coincided with your approach 1. For example, this is the definition that you can find in $\S 7$ of

Goldschmidt, H., Integrability criteria for systems of nonlinear partial differential equations, J. Differ. Geom. 1, 269-307 (1967). ZBL0159.14101.

Note that the above reference refers to the kernel of your symbol, interpreted as a linear map $\sigma(D_u\mathrm{T})(x,\xi) \colon \mathbb{R} \otimes S^kT^* \to \mathbb{R}$, as the principal symbol, but the two notions are equivalent up to invertible transformations on the operator $\mathrm{T}$. This definition happens to be the right one in the context of the study of formal properties of a PDE when localized to a point $x$ and the value $u$ of the dependent variable. Perhaps in some other context a different definition would be more appropriate.