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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 matters 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.

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.

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, 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 matters 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.

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 matters 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.