Timeline for Lax pairs in an abstract formalism
Current License: CC BY-SA 3.0
19 events
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| Oct 15 at 6:42 | answer | added | Phil Harmsworth | timeline score: 1 | |
| May 9, 2020 at 10:39 | comment | added | Delio Mugnolo | @AmirSagiv You're probably right. My (naive?) intuition is that things can get interesting in "boundary cases", like, PDEs on bounded domains with periodic BCs. To treat those, a general abstract theory would come in handy. | |
| Aug 26, 2018 at 7:24 | comment | added | mo-user | The two-volume book Soliton equations and their algebro-geometric solutions ( see cambridge.org/0521753074 for volume 1 dealing with the PDE case ) is very much on the rigorous side of things . | |
| Dec 19, 2017 at 19:10 | comment | added | Amir Sagiv | @DelioMugnolo Sure. But, if you want to approach it from a different angle, I would have looked for integrable, nonlinear, bounded-domain system. If no one has ever been able to show in any other way such a system, that's a strong indication (only an indication, not an answer) that the problem is not with Lax' formalism, but an essential problem to integrability. | |
| Dec 19, 2017 at 18:23 | comment | added | Christian Remling | @DelioMugnolo: In some cases, one can do this by hand. For example, the periodic (or finite gap) potentials could be viewed as potentials on $\mathbb R$, by extension, and then one can use the whole line Lax operators. Periodic potentials are invariant because the flows commute with the shift. Similarly, Dirichlet boundary conditions for Jacobi matrices can be implemented by setting the corresponding coefficient $a=0$, and again these sets are invariant under the (whole line) flow, so we can again reduce matters to the original (whole line) Lax operators. | |
| Dec 19, 2017 at 13:03 | comment | added | Delio Mugnolo | @AmirSagiv That's exactly the point. Since Lax does not explain what he means by "commutator" (what domain of the commutator should be taken? the maximal domain? an intersection of two cores?), it is not clear whether one has to make sure that the Lax pair formalism still makes sense. | |
| Dec 19, 2017 at 11:43 | comment | added | Amir Sagiv | @DelioMugnolo you're right. I want to ask - How do we know that there is a nonlinear Lax pairs for any nonlinear PDE with BC? Maybe the introduction of BCs destroys integrability? | |
| Dec 19, 2017 at 8:20 | comment | added | Delio Mugnolo | @AmirSagiv Well, for instance, the Burgers equation with precisely those BCs that arise from the Hopf-Cole-Transformation of a heat equation with, say, Dirichlet or Neumann b.c. But it feels like cheating, you're probably thinking of something different. | |
| Dec 19, 2017 at 6:42 | comment | added | Amir Sagiv | @ChristianRemling and Delio, these are both great examples. Do we have a nonlinear PDE, potential-less equation (such as KG, SG, KdV, NLS), on a bounded domain, which is integrable? | |
| S Dec 18, 2017 at 23:32 | history | suggested | Amir Sagiv |
two important tags that might bring the relevant people
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| Dec 18, 2017 at 22:42 | comment | added | Delio Mugnolo | @AmirSagiv On a much more modest level, transport equations are of course integrable; random evolutions à la Griego-Hersh on bounded domains are sometimes integrable, too; and both heat and wave equations on compact quantum graphs are also integrable, at least in many relevant cases and under most usual boundary conditions. | |
| Dec 18, 2017 at 22:03 | comment | added | Christian Remling | @AmirSagiv: Periodic potentials are of this type. The proper setting here is really finite gap potentials, and there's a huge literature on those (these form finite-dimensional invariant tori, so give you finite-dimensional integrable systems, on which various communities have descended like locusts). | |
| Dec 18, 2017 at 21:24 | comment | added | Amir Sagiv | That's an interesting question. Is there a single example of a PDE on a bounded domain which is known to be integrable, say the NLS with Cauchy BC? | |
| Dec 18, 2017 at 21:22 | review | Suggested edits | |||
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| Dec 18, 2017 at 20:05 | history | edited | Delio Mugnolo | CC BY-SA 3.0 |
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| Dec 18, 2017 at 17:56 | comment | added | Christian Remling | (cont'd) which certainly clarified a few things for me, though it may not do the same for others: arxiv.org/abs/1712.00503 | |
| Dec 18, 2017 at 17:56 | comment | added | Christian Remling | Exactly. The whole literature on these hierarchies is in disastrous shape from a mathematician's point of view. Not only is much of it non-rigorous, but if you actually tried to a give careful version, you'd probably get an outcry from a crowd of people claiming "I did this 50 years ago." If the precise setting doesn't matter much to you, then my recommendation is to focus on the Toda hierarchy, where global existence and uniqueness actually hold (obviously this is totally out of reach in the continuous case), and is it totally immodest if I point out that I recently posted a paper (...) | |
| Dec 18, 2017 at 11:29 | history | edited | Delio Mugnolo | CC BY-SA 3.0 |
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| Dec 18, 2017 at 11:14 | history | asked | Delio Mugnolo | CC BY-SA 3.0 |