It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
- The Schrodinger dispersion relation is $\omega(\xi)=|\xi|^2$
- The KdV/Airy dispersion relation is $\omega(\xi)=-|\xi|^3$
- The wave equation dispersion relation is $\omega(\xi)=|\xi|$ (maybe not technically dispersive since particles travel at the same speed)
But the Klein-Gordon equation's dispersion relation $\omega(\xi)=\sqrt{1+|\xi|^2}$ seems to be the only one for which $\omega(0)\neq 0$.
Is there a good physical/mathematical interpretation for dispersion relations which vanish at zero vs. ones that don't?
The best answer I can come up with is that the zeros of $\omega(\xi)$ seems relevant to understanding oscillations in time. More precisely, consider a solution to a linear dispersive equation
$$ u(t,x)=\int e^{ix\cdot\xi+it\omega(\xi)}\hat{u}_0(\xi)d\xi$$
Then to understand time oscillations of $u$, one could consider integrating it in time against some other function and then integrating by parts in time, using $\frac{1}{i\omega(\xi)}\partial_t e^{ix\cdot\xi+it\omega(\xi)}=e^{ix\cdot\xi+it\omega(\xi)}$:
$$\int_0^T u(t,x)f(t) dt = -\int \int_0^T f'(t)\frac{e^{ix\cdot\xi+it\omega(\xi)}}{i\omega(\xi)}\hat{u}_0(\xi)d\xi dt + \text{boundary terms}$$
But if one is after $L^\infty$ estimates, this only works if $\omega(\xi)$ is nonzero on the support of $\hat{u}_0$. Is time oscillation behavior unique to Klein-Gordon?
