## Equations for Integrable Systems

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So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further assume that the fibers of the map $f:M\to\mathbb{C}^n$ determined by the $f_i$ is an open subset of an abelian variety and the vector fields $X_{f_i}$ are linear. We call such a thing an algebraically completely integrable Hamiltonian system.

Now, I'm told that there's a definition of integrable system in PDEs that acts as some sort of stability condition, though I don't understand it. I've also been told that there's a way to, from an integrable system of PDEs, construct an integrable Hamiltonian system (just drop the condition that fibers be in abelian varieties), and that these two types of objects should be equivalent.

My questions:

1) What's the correct formulation for PDEs to make something like this work out? (I know virtually nothing about PDEs, and would be quite grateful just to be pointed at a good reference, if there's too much I need to read to get a quick, understandable answer)

2) Is there a general method of going from an algebraically completely integrable Hamiltonian system, which is algebro-geometric in nature to working out the PDEs explicitly? Does it help if the symplectic variety is known to be the cotangent bundle of something? How about if the base is unirational? rational?

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## 3 Answers

I'm no expert on this, but since nobody answers:

one book which should definitely help is this big introductory one by Babelon, Bernard and Talon. There's also a very technical older paper by Ben-Zvi and Frenkel where apparently some sort of general construction is made, and there's a more readable paper by Inoue, Vanhaecke and Yamazaki on algebraic complete integrability and integrable hierarchies of PDEs intended for your second question (especially section 6 for the relationship).

Just glancing at all this I'm not so sure a general explicit method to obtain the PDEs exists (I could be wrong) but some cases seem to be understood. Hope this helps...

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Many definitions for "integrable" equations exist, your current definition is quite limiting. I would say what you are describing are a subset of Hamiltonian systems. When most people say "integrable" in PDE's and/or dynamical systems, they usually mean equations for which the Inverse Scattering Transform can be used to construct an analytic solution, which is a much larger class of problems than Hamiltonian systems.

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The question is intentionally restrictive.  I'm only actually interested in a very special class of systems (in fact, algebraic ones, which are cotangent bundles of unirational varieties) and was hoping that restricting would be more likely to get me the result I'm looking for. – Charles Siegel Nov 23 2009 at 20:01

Regarding 2), I'm not an expert in algebraic integrable systems, but shouldn't it be the other way around? That is, given an integrable evolutionary system of PDEs which is further assumed to be Hamiltonian, you can consider the fixed points of the (linear combinations of) associated higher flows, and then the equations describing these fixed points (these equations are also referred to as the (higher) stationary equations) are Hamiltonian and integrable themselves. As for a reference on the things just described, you can try this book and references therein.

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