Given a 1+1 dimensional wave equation ($c$ constant) plus a small ($\left|k\right|\ll 1$, $k$ imaginary (?)) third order derivative in $x$ term,
$$
f\_{tt}=c^2\ f\_{xx}+k\ f\_{xxx}
$$
 is there a sensible answer to what is the propagation speed?


OK, IF there is such an answer, it seems it would be dependent on frequency, so let us be more specific: Given an almost compactly supported initial condition with a bound on frequency, what is the propagation speed? If that sounds a mess, yes making sense of it is part of the problem. I am hoping that this problem is simpler than I think it is, and I apologise for my lack of knowledge of the problem.


I suppose I need to say where the problem comes from to convince people that it is meaningful: The equation above is an approximation to various models giving a potentially `varying speed of light', including some non commutative geometry models. Let us suppose that there was such a modification, and take a scenario that several people have outlined to measure such effects (see \*): A supernova explodes in a distant galaxy, and the effect of varying $k$ can be measured in different arrival times of electromagnetic radiation of different wavelengths, from gamma to visible, potentially measuring even very small values of $k$. 


This can be modelled numerically. It can be guessed by Fourier transform. What is missing is some sort of formal justification of what the propagation speed means in this context. Apologies if this is simple for people in differential equations, it has puzzled me and several other people!


* Amelino-Camelia G. & Majid S., Waves on noncommutative space-time and gamma ray bursts, Int. J. Mod. Phys. A 15, 4301 (2000).



propagation speed for a modified wave equation
One can certainly prove something along these lines; here's a straightforward perturbative statement: Let's impose the initial conditions $f(0,x)=u$, $f\_t(x,0)=0$ (for simplicity), with (as you propose) $\widehat{u}$ compactly supported, by $|\xi|\le C$. Let's also take $u\in\mathcal S$, $\|u\|\_2=1$. Finally, let me do this for $c=1$.


If we now take $L$ so large that $u$ is almost supported by $(-L,L)$ in the sense that $\int\_{|x|>L}|u|^2<\epsilon$ and $v\in L^2$, $\|v\|=1$ is supported off $(-L-|t|, L+|t|)$, then
$$
|\langle v, f(t) \rangle | \lesssim Ck+\epsilon .
$$
This is of course true for the solution $\widehat{f\_0}=\widehat{u}\cos t\xi$ of the unperturbed equation ($k=0$) by finite propagation speed, and then we just compare $\widehat{f\_0}$ with $\widehat{f}=\widehat{u}\cos t\sqrt{\xi^2-ik\xi^3}$ by a Taylor expansion in $k$ and look at $\langle v, f\rangle = \langle \widehat{v}, \widehat{f}\rangle$.


