TITLE: What does the flow of the principal symbol of the differential operator tell us about the PDE?
QUESTION [8 upvotes]: Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there's the obvious linear PDE:
$$P(f)=0$$
Naturally associated with $P$ we also have a hamiltonian system  $$\Phi= (T^*M, \omega,H=\sigma(P))$$ where the symplectic form $\omega$ is the standard one and principal symbol $\sigma(P)$ of $P$ is taken as the hamiltonian.

Question: What does the dynamics of $\Phi$ tell us about the original differential equation?

Obviously since we are only considering the principal symbol we won't get a terrible amount of information. On the other hand we are not taking only the zero locus $\{ \sigma(P)=0\}$ (AKA the characteristic variety / set ) so one might hope that we can at least find some information in $\Phi$ which is not already present in the geometry of $\{\sigma(P)=0\}$.

REPLY [5 votes]: It goes something like this (I can't promise that what I've written below is completely correct. It is only to help you read the rigorous details in more definitive reference):
The initial observation is that the smoothness of a distribution $f$ on $\mathbb{R}^n$ can be measured using the decay rate of its Fourier transform $\hat{f}(\xi)$ as a function of $|\xi|$. Microlocal analysis means to localize this idea in the cotangent bundle.
The wavefront set $WF(u) \subset T^*M$ of a distribution $u$ on $M$ has the property that
$$ WF_x(U) = WF(u)\cap T^*_xM $$
is a closed conic subset of $T^*_xM$.
First, it suffices to restrict to a compactly supported distribution $u$ on a coordinate chart $O \subset M$. We can therefore assume everything is on $\mathbb{R}^n$. Then $WF_x(u)$ at $x \in \mathbb{R}^n$ is a closed cone in $\mathbb{R}^n\backslash\{0\}$, which should be viewed as $T^*_x\mathbb{R}^n\backslash\{0\}$. In particular, $\xi_0 \notin WF_x(u)$ if and only if for any neighborhood $N$ of $x$ there exists $\chi \in C^\infty_0(N)$ such that the Fourier transform $\widehat{\chi u}$ decays rapidly (faster than polynomial decay) in a conical neighborhood of $\xi_0$.
Let $\sigma$ be the symbol of a real linear differential operator $P$ and $\Sigma = \sigma^{-1}(0)$ its characteristic variety. We say that $P$ is of real principal type if the Hamiltonian vector field of $\sigma$ restricted to $\Sigma$ is everywherer nonzero.
Hormander's original propagation of singularities theorem (which has many antecedents and by now many generalizations) says that if $P$ is a real differential operator of real principal type and $u$ is a distribution on $M$ such that $Pu \in C^\infty(M)$, then $WF(u) \subset \Sigma$ and is invariant under the null bicharacteristic flow, which is the Hamiltonian flow of $\sigma$ restricted to $\Sigma$.