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This is one of those "what is an $X$?" questions so let me apologize in advance.

By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so called "Kdv hierarchy" which is apparently related to enumerative geometry of curves in some ways which are a total mystery to me) enough times to care about the meaning of these words.

However when i type these words in google most of the results are links to physics papers and I haven't been able to find anything clarifying about the meaning of this phrase, let alone a precise mathematical definition. Hence the question:

What is an integrable hierarchy? Where do they come from? What are their applications?

EDIT: Let me emphasize that this question is mainly about the "hierarchy" part. I understand (at least on a basic level) what is an integrable system and I know of several good mathematical references for this topic. The emphasis here is on what constitutes an integrable hierarchy as apposed to a plain old "integrable system".

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    $\begingroup$ Is "hierarchy" the word you're getting tripped up on here? I don't think it means anything much more than "family" in this context. $\endgroup$ – Sam Hopkins Nov 11 '17 at 15:08
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    $\begingroup$ @SamHopkins Yes. I have a feeling that for some people this word means more than just an arbitrary family of integrable systems. $\endgroup$ – Saal Hardali Nov 11 '17 at 16:15
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    $\begingroup$ The answer to this question could make a good "What is..." article for the Notices. (Both the "integrable" and the "hierarchy" parts.) $\endgroup$ – Willie Wong Nov 11 '17 at 22:35
  • $\begingroup$ Another point of confusion might be that, especially in these infinite-dimensional settings, usually things are understood in a rather formal/algebraic way as opposed to analytic: convergence issues are not considered, flows are not actual flows but just "evolutionary operators". Some applications to enumerative geometry, if I'm not mistaken, consider "integrable hyerarchies" over a space that is not a manifold but the formal completion at $0$ of the cohomology ring of a manifold. $\endgroup$ – Qfwfq Aug 13 '18 at 20:24
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An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.
[For the definition of a commuting flow, see for example the first part of this MO question.]

They arise from integrals of motion which are in involution (meaning that the Poisson bracket of any pair vanishes).

Commuting flows are useful, because they can be solved by the inverse scattering transform technique.

For an introduction from a mathematical perspective, see for example Introduction to integrable systems: open Toda lattice, KP, and KdV-hierarchies.

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  • $\begingroup$ Sounds like you're describing an integrable system. $\endgroup$ – Saal Hardali Nov 11 '17 at 16:16
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    $\begingroup$ This would be an integrable system for a finite number of degrees of freedom, but the concept of an integrable hierarchy is typically used in the infinite-dimensional case. $\endgroup$ – Carlo Beenakker Nov 11 '17 at 16:25
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    $\begingroup$ The definition of the elementary Schur polynomials using the o.g.f. on pg. 32 in Shapiro's introductory paper and the specific polynomial in Example 3.14 don't agree. Correct interpretation? $\endgroup$ – Tom Copeland Nov 12 '17 at 18:18
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    $\begingroup$ @TomCopeland --- This is a mistake, arising from two different definitions of Schur polynomials; they are denoted $S_n$ and $s_n$ in this paper, page 12. Shapiro defines $S_n$ on page 32, but then in example 3.14 he should have used $s_n$. If you start from the polynomial $S_n$ you should replace $p_k$ by $p_k/k$ to get $s_n$. $\endgroup$ – Carlo Beenakker Nov 12 '17 at 20:49
  • $\begingroup$ As I thought, so the first o.g.f. is related to the refined Lah polynomials, oeis.org/A130561, and the Example 3.14, to the refined Stirling polynomials of the first kind, a.k.a., the cycle index polynomials for the symmetric groups, oeis.org/A036039. $\endgroup$ – Tom Copeland Nov 12 '17 at 21:48
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One more remark perhaps as a supplement to the existing answers, to further motivate the term hierarchy.

The standard way to generate the common hierarchies (Toda, KdV are the most standard examples) is via the Lax equation $$ \dot{J} = [P(J),J] . $$ Let's focus on the Toda hierarchy specifically (but it works the same way for any of the other hierarchies that have been mentioned). $J:\ell^2\to\ell^2$ is the operator that is evolving, $(Ju)_n = a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$, and $P(J)$ is the anti-symmetric part of $p(J)$, thought of as an infinite matrix, where $p(J)$ is defined in the obvious way, given a polynomial $p$.

Each choice of polynomial $p$ gives a flow, and these together form the "hierarchy" (it's even better to think of this as the abelian group of polynomials acting on Jacobi matrices). So it's not just a random collection of flows that happen to commute (and act by unitary conjugation), the individuals flows of a hierarchy all come from the same construction.

Of course, this is only one point of view, and quite a few different approaches are possible too. I'm about to finish a paper that will give center stage to another property of Toda flows, namely the existence of an associated cocycle for the action of $G=\mathbb R\times \mathbb Z$, where the action of $\mathbb R$ implements the flow and $\mathbb Z$ acts by shifting the coefficients. This property too can be made the starting point of the construction of the whole hierarchy.

That didn't really address the original question, but the (disappointing) answer to that is simply, I believe, that "integrable hierarchy" is a term like "trigonometric function:" given some background in the area, you know what it refers to without ever having defined it rigorously.

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A great starting point for learning integrable hierarchies is Jimbo-Miwa-Date:

enter image description here

They start off with the KdV equation, and observe it has a very large space of symmetries. So we can take a solution and boostrap it to generate other solutions.

$$ \partial_t \phi + \partial^3_x \phi - 6 \phi \, \partial_x \phi = 0 $$

The Korteweg de Vries equation is a nonlinear cubic differential equation, describing water waves. (See Boussinesq approximation).

In this case, I believe there is a traveling wave solution: $$ \phi(x,t) = - \frac{1}{2} c \, \text{sech} \left[ \frac{\sqrt{c}}{2} (x - ct - a) \right] $$

However, I suspect the more attractive features of these equations are the integrals of motion:

  • $\int \phi \, dx$ the "mass"
  • $\int \phi^2 \, dx$ the "momentum"
  • $\int \big[2\phi^3 - (\partial_x \phi)^3 \big]\, dx$ the "energy"

and we can check these are integrals of motion using the rules of ordinary calculus, showing that $\frac{\partial}{\partial t} \int [\dots ] \,dx = 0$.

I was never 100% sure about these two styles of studying PDE. One generates solutions using algebraic manipulations, another studying the well-posedness conditions and things like that.

Huygens' principle leads to the KdV hierarchy, Schrödinger equation and many other famous equations. In that case, we are studying the scattering of light: rainbows.

enter image description here

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  • $\begingroup$ when I have a moment, I'd like to explain why this is a "hierarchy" and exhibit other herarchies than KdV (such as Benjamin Ono). However, I think the link to Huygens' principle (which becomes the action principle) offers a variational way to look at this eq. Also... I have not mentioned the Lax pairs, which is what people really think of with integrable systems. $\endgroup$ – john mangual Nov 11 '17 at 20:51
  • $\begingroup$ at the danger of lumping all physicists into the same bucket, here's string theory paper linking topological string theory to integrable hierarchies. [1] $\endgroup$ – john mangual Nov 11 '17 at 20:53
  • $\begingroup$ +1. Will you please explain why this is a "hiearchy" and exhibit other hierarchies and describe the Lax pair as you said you would like to do? Thanks. $\endgroup$ – Hans Mar 24 '18 at 5:11
  • $\begingroup$ In addition to the MJD book mentioned by John, it could be a good idea to look at Alan Newell's book Solitons in mathematics and physics ( epubs.siam.org/doi/book/10.1137/1.9781611970227 ) which deals with the integrable hierarchies quite a bit. $\endgroup$ – mathphysicist Sep 9 '18 at 15:22
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In brief, an integrable hierarchy is an infinite (usually countable) set of integrable partial differential systems such that any two systems in this set are compatible. Such hierarchies are usually generated by recursion operators or master symmetries, cf. e.g. the introduction of this recent paper and the references given there regarding the recursion operators and this book and references therein regarding master symmetries.

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  • $\begingroup$ Off-topic, but could you please slow down with your barrage of recent edits to questions (mainly adding tags) since this bumps questions to the front page and pushes other questions off the front page $\endgroup$ – Yemon Choi Sep 9 '18 at 15:11

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