I am reading Integrals of Nonlinear Equations of Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide precise explanations: he sticks to the PDE Weltanschauung and simply differentiates and integrates all the way through. This is fully ok, as long his results are groundbreaking and he's interested in a very specific PDE (KdV on $\mathbb R$, in this case) where his ideas work.

However, things break down soon if one tries to apply Lax' ideas to other examples; this is especially the case if one thinks of PDEs on bounded domains, where boundary conditions matter and make it tricky to figure out what's the precise meaning of a commutator.

So I was wondering whether there is some good reference about an abstract approach to the Lax pair idea. I'm ideally thinking of something along the lines of:

Let $H$ be a complex Hilbert space, $D_1,D_2$ subspaces of $H$ that are densely and compactly embedded in $H$, $\mathbb R_+\ni t\mapsto L(t)\in {\mathcal L}(D_1,H)$ be a $C^1$-family such that each $L(t)$ is self-adjoint as an operator on $H$ with domain $D_1$, $\mathbb R_+\ni t\mapsto P(t)\in {\mathcal L}(D_2,H)$ be a $C^1$-family of operators that generate an invertible evolution family on $H$ ...

and so on. Basically, what I'm looking for is a precise translation of the idea of Lax pairs to Hilbert space theory.

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    $\begingroup$ 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 (...) $\endgroup$ – Christian Remling Dec 18 '17 at 17:56
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    $\begingroup$ (cont'd) which certainly clarified a few things for me, though it may not do the same for others: arxiv.org/abs/1712.00503 $\endgroup$ – Christian Remling Dec 18 '17 at 17:56
  • $\begingroup$ 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? $\endgroup$ – Amir Sagiv Dec 18 '17 at 21:24
  • $\begingroup$ @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). $\endgroup$ – Christian Remling Dec 18 '17 at 22:03
  • $\begingroup$ @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. $\endgroup$ – Delio Mugnolo Dec 18 '17 at 22:42

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