Consider the scalar phase equation $$ \theta_t=\theta_{xx}+f(\theta),\qquad f(\theta+2\pi).\qquad (1) $$ Traveling waves profiles $\theta(x-ct)$ can be found using phase-plane analysis for $$ \theta_{\xi\xi}=-c\theta_{\xi}-f(\theta).\qquad (2) $$ Writing (1) in moving coordinates $t$ and $\xi$, we have $$ \theta_t=\theta_{\xi\xi}+c\theta_{\xi}+f(\theta).\qquad (3) $$ Traveling waves solutions are then time-independent solutions of (3).
Now, it is said:
One can obtain stability information fairly easy in this situation: all monotone profiles (now with \theta considered as a function in $\mathbb{R}$) are stable and all-non-monotone profiles are unstable. This can be readily seen by inspecting the linearization, which is a Sturm-Liouville problem with eigenvalue zero and eigenfunction given by the derivative of the wave profile. Since the first eigenfunction (when it exists, that is, when the most unstable eigenvalue is not contained in the essential spectrum) of scalar eigenvalue problems has a sign. This shows that the monotone pulses and fronts, with derivative belonging to the point spectrum, are stable [...]"
I would like to understand this!
As far as I see, what is meant is linearization of (3) in a traveling wave profile $\theta^*=\theta^*(\xi), \xi=x-ct$. This should be given by the ODE
\begin{equation} \theta_t=\theta_{\xi\xi}+c\theta_{\xi}+\partial_{\theta}f(\theta^*)\theta,\qquad (1) \end{equation} hence the linearizing operator should be given by $$ L_{*}\theta:=\theta_{\xi\xi}+c\theta_{\xi}+\partial_{\theta}f(\theta^*)\theta=u_t $$
Putting this into the Sturm-Liouville form, I get $$ \frac{d}{d\xi}\left(e^{c\xi}\frac{d\theta}{d\xi}\right)+e^{c\xi}\partial_{\theta}f(\theta^*)\theta=e^{c\xi}u_t. $$ As boundary conditions, I think that $$ \theta(0)=\theta(2\pi), \theta'(0)=\theta'(2\pi) $$ could be reasonable here.
Next, one can show that $$ \mathcal{L}P(\xi)=0,\text{ with }P(\xi)=\frac{d}{dk}\theta^*(\xi+k)_{|k=0}=\frac{d}{d\xi}\theta^*(\xi). $$
which shows that $\theta^*_{\xi}$ is indeed an eigenfunction to eigenvalue $\lambda=0$.
Now, I only can guess what is meant with the stability claims given above.
Open questions for me are:
(1) What is meant with the "first eigenfunction"?
I guess this means the eigenfunction belonging to the largest eigenvalue.
(2) Why does the "first eigenfunction" not exist if the most unstable eigenvalue is contained in the essential spectrum? And why does it exist if it is not?
(3) Why does the "first eigenfunction" have a sign and why does this imply that monotone profiles (monotone pulses and monotone fronts) with derivative belonging to the point spectrum are stable?
