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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
    Commented Dec 18, 2017 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
    Commented Dec 18, 2017 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
    Commented Dec 18, 2017 at 21:24
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    $\begingroup$ @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. $\endgroup$
    – Amir Sagiv
    Commented Dec 19, 2017 at 19:10
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    $\begingroup$ 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 . $\endgroup$
    – mo-user
    Commented Aug 26, 2018 at 7:24

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The paper by R. Beals and R. R. Coifman, "Inverse Scattering and Evolution Equations" Communications on Pure and Applied Mathematics 38(1) (1985) pp. 29-42 may be helpful.

They do go beyond the KdV equation. From the abstract:

We prove a global existence result for some ... evolution equations, including the cubic nonlinear Schrodinger equation, the modified KdV equation, the three-wave interaction equation, and the light-cone sine-Gordon equation.

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