Interpretation of dispersion relation I'm going through Terry Tao's book on nonlinear dispersive PDE, and am a little confused by some of the things going on. The setup is that we are working with a PDE given by 
$$
\partial_t u = Lu.
$$
In particular, on page 56, he defines the "frequency operator" $D = \frac{1}{i}\nabla$, and says that we can write a general partial differential operator $L$ as $L = ih(D)$ where $h$ is some polynomial. While I do not doubt this, I just don't understand the usefulness of doing any of it. I am completely new to dispersive PDE, so this might be the issue. 
He also calls $h$ the "dispersion relation" of the dispersive PDE $\partial_t u = Lu$, and I don't understand a) what a dispersion relation is, or b) why this polynomial should be called it. 
If anyone could point me in the right direction, it would be much appreciated!
 A: In order to give a complete notion of what a dispersive partial differential equation is, consider the one-dimensional frame. We look for plane wave solutions of the form $$
u(x,t)=Ae^{i(kx-\omega t)},
$$ 
where $A$, $k$ and $\omega$ are constants representing the amplitude, the wavenumber, and the frequency, respectively. Hence $u$ will be a solution of $$\partial_t u=Lu, \qquad Lu:=\sum_{\vert \alpha \vert\leq k}c_\alpha \partial^\alpha_xu(x),$$ if and only if $$
\omega+\sum_{\alpha\leq k}c_\alpha i^{\alpha-1}k^\alpha=0.
$$
This equation is called the dispersion relation. A commonly used defining criteria for dispersive equations is that $\omega(k)$ is a real valued function of $k$ such that $\tfrac{d^2\omega}{dk^2}\neq 0$. In the physical context this means that different frequencies in this equation will tend to propagate at different velocities, thus when time evolves, the different waves disperse in the medium, with the result that a single hump breaks into wave-trains. 
The importance of these definitions comes from the fact that the relation between $\omega$ and $k$ characterizes the plane wave motion. Consider for instance the linear Schrödinger equation $$
iu_t+u_{xx}=0,
$$
and the plane wave $u(t,x)=e^{i(k x-\omega t)}$. Then, $u(t,x)$ is a solution of the equation if and only if the dispersion relation $\omega=k^2$ holds. Note that in this case $\omega$ is a real valued function of the frequency. Now, consider the quantity $$
\nu_p(k):=\dfrac{\omega}{k},
$$
which is commonly call the phase velocity of the wave. With this definition one can re-write the solutions of the linear Schrödinger equation as: $$
u(t,x)=e^{ik(x-\nu_p(k)t)}=u(0,x-\nu_p(k)t),
$$
and thus conclude that the wave travels with velocity $\nu_p(k)$. In particular, large frequency data travel faster than smaller ones.
