In the context of orbital stability, probably one of the most used theorem to show the orbital stability of traveling waves is the one from Grillakis-Shatah-Strauss "Stability theory of solitary waves in the presence of symmetry". This theorem roughly reduces the problem (under the right setting) to study the linearized operator around the traveling wave. For the sake of notation let's denote the traveling wave $\phi_c$, where $c\in\mathbb{R}$ is the speed of the wave. Then, following the same notation of GSS, we have the linearized operator is given by $$ L_c:=E''(\phi_c)-cQ''(\phi_c), $$ where $E$ is the energy and $Q$ the "charge" (depending in the physical context, of course both conserved quantities related to the equation that the traveling wave solves).
During the last couple of months I've seen at least three articles where the authors cannot get the "right properties" on the linearized operator, say at most one negative eigenvalue and one-dimensional kernel, so they consider and additional space (say $X$) where they can get rid of the extra negative directions or the extra dimensions in the kernel. Let say, if we denote by $\mathcal{H}$ the "energy space" associated to our PDE, now we restrict the analysis to $$ \mathcal{H}\cap X, $$ and we try to prove orbital stability in $\mathcal{H}\cap X$. Now let me explain my question:
For me this is ok (in my naive intuition) as soon as these extra conditions are preserved by the flow (so the auxiliary space is somehow "compatible" with the evolution of the PDE, or in other words, the solution remains always in the auxiliary space). What is completely weird to me is to consider an additional space that is not preserved by the flow.
For example, in this paper, the author consider the following equation:
$$
u_{tt}-u_{xx}-\sinh(u)=0.
$$
posed on $\mathbb{T}$. Then, after working a while he got "bad properties" on the spectra of the linearized operator. So he said (bottom page 15), by restricting the analysis with the following additional space,
$$
X:=\{f\in H^1(\mathbb{T}):\ \int_\mathbb{T}f=0\},
$$
he can get rid of all extra negative direction and extra dimensions in the kernel, and obtain that the spectra of the linearized operator has only one negative direction and a simple kernel in this restricted space. And hence, the orbital stability follows for initial data in this restricted space. However, notice that this condition $\int f=0$ is not preserved by the PDE.
My question is, doesn't this kind of argument have any problem with Grillakis-Shatah-Strauss' result? I mean, is it true that we can just restrict the domain of the linearized operator (and hence the space of initial data) to any space we want in order to get rid of the bad directions, even when these additional spaces have no relation with the PDE? I've been a lot of time trying to understand how you match both things. I am asking here because is not the first time I see something like this (with conditions that are not preserved).
My naive answer would be that this additional space must be assumed to holds for all times (unless you prove it). Something like "suppose you have solutions such that for all times $t\in\mathbb{R}$ satisfies that $\int_{\mathbb{T}} \phi(t,x)dx=0$, then.....(statement of orbital stability). In other words, doesn't GSS result see the evolution in time of the equation or is it enough to have the conservation laws (only of $E$ and $Q$) and the initial data in the right space ($\mathcal{H}\cap X$)?