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I had occation to link to “formally integrable PDE” from somewhere, and so I created a stub entry just so that the link works. Also created a stub for integrable PDE and cross-linked with integrable system.
So if we have that formally integrable PDEs are coalgebras for the jet comonad, the slice over $\Im \Sigma$ in nice cases, what can be said about integrable PDEs? Is there anything to be done in terms of the cohesive modalities?
what can be said about integrable PDEs?
I doubt that there is something analogously general to be said about these. Global integrability is a tricky business. For classical examples of PDEs the integrability is wide open, such as in one of the “Millenium problems” Navier-Stokes existence
Is there not something between global and formal integrability? Perhaps local integrability?
Looking about, I see that formal integrability often implies local integrability, e.g., in real-analytic and complex situations.
Good point. Maybe one could capture this with making a model for differential cohesion not from infinitesimal neighbourhoods, but from germs.
This is an idea I have been tossing around with Thomas Nikolaus a while back. In the end we seemed to have convinced ourselves that it works, but I didn’t try to write it out in detail.
A comment on terminology: I’m no expert but I suspect that many people would equate “integrable PDE” with “integrable system (given by PDEs)”. See for example this MO question which is marked as “already answered”, linking to What is an integrable system. As far as I understand there is no precise definition of integrable system, but it is somehow related to “sufficiently many conservation laws” or “symmetries”.
Right, I did try to make clear in the Idea-section at integrable PDE that there is much convention involved in what counts as integrable.
In any case, I think “integrating a PDE” is standard terminology, independent of the use of “integrable systems” in physics.
Agreed with your remark on “integrating a PDE”.
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