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In the theory of stability of solitary waves I have seen many times that people mention some of the symmetries of the equation in order to introduce the "right" notion of stability. For instance, if we consider the KdV equation $$ u_t+u_{xxx}+uu_x=0, \qquad (t,x)\in\mathbb{R}^2, $$ this equation has solitary wave solutions $u(t,x)=\phi_c(x-ct)$ and is invariant under space translations, that is, if $u(t,x)$ is a solution of the equation, then so is $u(t,x-\gamma)$ for any $\gamma\in\mathbb{R}$. I have seen plenty of times authors saying that: since the equation is invariant under space translations, then the right notion of stability is "stability modulo this symmetry". I understand that the right notion of stability is "orbital stability" (otherwise we could just perturb the speed of the traveling wave and then it will be impossible to "stay close" to the evolution of the initial wave if we don't consider its orbit). However, I don't understand this specific sentence because, for instance, I could ask, what about the other symmetries? Why we never consider the "right notion" of stability modulo all the other symmetries of the equation? I am aware that the KdV has infinitely many of them, so it seems a little bit strange for me to state this supposed relation between orbital stability and the symmetries of the equation and to forget about all the other symmetries at the same time.

Of course, this is not a peculiarity of the KdV equation, we could also consider for instance the cubic NLS equation, which also has traveling wave solutions and infinitely many symmetries, however, the notion of orbital stability only consider some of them. So I am wondering if there is something that I am not understanding properly. Does anyone has any explanation for this, I would really appreciate it.

Edit: Sorry I just noticed that I never state the notion of stability of the KdV equation. This notion corresponds exactly to stay always close to the "orbit" of the solitary wave, that is, to stay close to the set $$ \Omega_c:=\{\phi_c(x-y): \ y\in\mathbb{R}\}. $$

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  • $\begingroup$ It depends on the class of perturbations one maybe interested in. For example, in study of navier stokes, it is common to consider symmetric subspace stability, i.e. only consider initial perturbations that respect a certain symmetry possessed by the solution under consideration , and then study the growth of those perturbations. So this process does infact consider the 'other' types of symmeteries than just orbital $\endgroup$ Apr 10, 2020 at 14:24

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First, you seem to be misunderstanding what orbital stability means.

The notion of orbital stability should be considered as in contrast against asymptotic stability. The latter notion is the stability that one usually expects around, say, hyperbolic fixed points of a finite dimensional dynamical system, where a stable fixed point is one where perturbations eventually decay away and the state of the system returns to the fixed point. In many physical settings, however, one cannot expect asymptotic stability, because the fixed point may be such that there are small perturbations in orbit around it. The simplest example is considering the two dimensional dynamical system generated by the vector field $x\partial_y - y\partial_x$. The origin is a fixed point, but small perturbations of the origin do not return to the initial data.

One can then generalize this to non-equilibrium points (such as the traveling KdV solitons), by asking whether initial data which are small perturbations of a particular solution (such as the solitons), will remain for all time small perturbations of the reference solution. Notice that for linear systems orbital stability is trivial for any solution. For nonlinear systems generically there can be positive feedback which would cause perturbations to diverge from the reference solution at large times; and hence orbital stability is not trivial for general nonlinear systems.

For the KdV system, around a soliton, that orbital stability is the right notion (as opposed to asymptotic stability) can be justified via the symmetry of the equation. (The symmetry of the equation itself is not used to "derive orbital stability".) The reason is that given a soliton solution $u(t,x) = \phi_c(x - ct)$, if you consider the initial data given by $v(0,x) = \phi_c(x + \epsilon)$, the corresponding solution will be $v(t,x) = \phi_c(x + \epsilon - ct)$. For small $\epsilon$ you have that $v(0,x) = u(0,x) + \epsilon \partial_x u(0,x) + O(\epsilon^2)$ and hence $v(0,x)$ is a small perturbation of $u(0,x)$. Asymptotic stability must fail for the KdV system because $\| v(t,x) - u(t,x)\|$ is constant in time for any reasonable norm, and so the perturbed solution never returns to the original soliton. And therefore the best one can hope for is orbital stability.

Second, one can also alternatively try to quotient out the influence of the symmetries if one wants better control of the solution at large times. For example, given the discussion above, one may ask whether the translation symmetry is the only mechanism preventing asymptotic stability from holding. That is, is it in fact possible that if we factor in this effect of the symmetry, the perturbed solutions can always be regarded as approaching some other soliton? One way to formulate this is, for example, to say: for a solution $v(t,x)$ evolving from an initial data that is a perturbation of my reference soliton solution $u(t,x)$, I am not going to ask whether $v(t,x)$ eventually approaches $u(t,x)$. Instead, I am going to look at the set of all solutions of the form $u(t,x + \xi)$ for some $\xi\in \mathbb{R}$, and ask whether $v$ eventually approaches one of these solutions. This is the setting where one now considers instead of a singular solution, a manifold of solutions (the soliton manifold). One can then ask whether a perturbed solutions $v(t,x)$ can be optimally decomposed as a (time-dependent) element of the soliton manifold plus a small (perhaps decaying) perturbation. And this gets us to modulation theory.

To conclude, however, the point of view of the soliton manifold has sometimes some additional (coincidental) advantages. Returning to the original KdV equation, one notes that $v(x,t) = \phi_{c+\epsilon}(x - (c+\epsilon) t)$ is also a solution. For sufficiently small $\epsilon$ this $v$ is again a small perturbation (in terms of initial data) from that of $u(x,t) = \phi_c(x - ct)$. However, evidently at large times, the differing speeds will make $v$ and $u$ very different solutions. If we consider the point of view of the soliton manifold relative to space translations this happen to be neatly resolved: while $v$ is not going to be close to $u$ at late times, one can easily check that $v(x,t)$ will be in fact close to $u(x - \epsilon t, t)$, a time-dependent translation of the original soliton solution. This means that if we relax ourselves to study orbital stability about the soliton manifold, then we can also account for initial data perturbations that changes the soliton velocity. (This is not to say that we shouldn't consider the manifold of solutions corresponding to all translations of all different velocity solitons! Maybe this will give us better information!)

To give a more pedestrian example: time translation is another symmetry of the KdV equation. However a time-translated soliton can be also identified with a spatially translated soliton. So trivially the soliton manifold (defined with respect to spatial translations) also captures non-decay due to time-translation.

In this business the general idea is that if you have some obstruction to decay, you can start by trying to see whether you can embed your background solution to some parametrized family of solutions, and see if your obstruction can be characterized as arising from a change in the parameter. One easy way to obtain a parametrized family of solutions is to exploit the symmetries inherent in the equations.

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  • $\begingroup$ Thank you very much for your extended answer. However, there is something that still bothers me. I don't know if you have heard about this other kind of "soliton" solutions called "breathers", which are solitary waves that also oscilates in time. For this kind of solution the "profiles" are two dimensional, in the sense that the solution cannot be written just as translations of a fixed profile $Q(x-ct)$ (of course because they oscilate). $\endgroup$
    – Sharik
    Apr 14, 2020 at 13:46
  • $\begingroup$ @Willie_Wong (Sorry I had to split the message because it was too long for MO) Thus, it is not clear for me what would be the right notion of orbital stability for such kind of solutions for example (for some fixed PDE having some previously studied soliton, and hence having the same symmetries). I mean, in this case it is not enough (I think) to consider the distance to space-translations of a breather for a given time. I guess that maybe would be to stay close to translations in space and time (separately) of breathers (so a 2-dimensional manifold), but I am not sure how to justify this. $\endgroup$
    – Sharik
    Apr 14, 2020 at 13:47
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    $\begingroup$ @Sharik: the examples I gave in my answers are just that, examples. In each case you study you have to contemplate what the correct manifold of solutions is. The examples are meant to convey the fact that there is not a one-to-one mapping of symmetries with the dimension of the correct manifold of solutions. Indeed, part of the goal was to emphasize that while the manifold of solutions can be built out of (and necessitated by) considerations of symmetries, the correct manifold of solutions maybe bigger/smaller depending on the situation considered. $\endgroup$ Apr 14, 2020 at 13:56
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    $\begingroup$ ... in the case of breather solutions: if you start with already a two-dimensional family of solutions, then that's a good indication that you may want to use that family as your manifold. But generally one does not start by diving too deeply into what should be ultimately the correct notion of stability: one starts by trying to prove stability of the system and seeing what obstructions there are, and then trying to see if those obstructions can be explained in terms of known properties of the solution. $\endgroup$ Apr 14, 2020 at 14:00
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    $\begingroup$ Returning to the two-dimensional family: if you have a family of non-decaying solutions into which your solution embeds, then the tangent space to this family at your solution gives rise to zero eigenvalues of the corresponding linearized evolution. This means that even on the linear level you cannot expect asymptotic stability at your solution. Whether you have orbital stability depends on the nonlinearities, and in particular whether the non-decaying linearized solutions contribute to instability. If so, you are forced to consider the manifold of solutions etc. $\endgroup$ Apr 14, 2020 at 14:09

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