Timeline for Stability and symmetries
Current License: CC BY-SA 4.0
7 events
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| Apr 14, 2020 at 14:09 | comment | added | Willie Wong | 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. | |
| Apr 14, 2020 at 14:00 | comment | added | Willie Wong | ... 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. | |
| Apr 14, 2020 at 13:56 | comment | added | Willie Wong | @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. | |
| Apr 14, 2020 at 13:47 | comment | added | Sharik | @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. | |
| Apr 14, 2020 at 13:46 | comment | added | Sharik | 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). | |
| Apr 13, 2020 at 7:13 | vote | accept | Sharik | ||
| Apr 10, 2020 at 21:52 | history | answered | Willie Wong | CC BY-SA 4.0 |