Questions tagged [integrable-systems]

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What is an integrable system?

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "...
Gil Kalai's user avatar
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25 votes
1 answer
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What's up with Wick's theorem?

Sorry about the dumb title. I'd like to understand Wick's theorem. More specifically, I have seen it pop up in several different contexts and I am really puzzled by the different statements of it ...
Dan Petersen's user avatar
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How to use these higher symmetries and conservation laws?

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (...
W. mu's user avatar
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1 vote
1 answer
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Construct super Poisson brackets on the coordinate rings of Lie super groups

On line 7 of page 61 of the book a guide to quantum groups, a Poisson bracket is defined on $\mathbb{C}[GL_n]$ for every classical $r$-matrix as follows. Let $V$ be a vector space with a basis $v_1, \...
Jianrong Li's user avatar
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74 votes
4 answers
6k views

What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
Gil Kalai's user avatar
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10 votes
3 answers
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Is the 'massive' Calogero-Moser system still integrable?

Background The (rational) Calogero-Moser system is the dynamical system which describes the evolution of $n$ particles on the line $\mathbb{C}$ which repel each other with force proportional to the ...
Greg Muller's user avatar
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10 votes
2 answers
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What does it mean for a differential equation "to be integrable"? [duplicate]

What does it mean for a differential equation "to be integrable"? Are "integrable" and "solvable" synonyms? The first thing that comes to my mind is to say: it's integrable if we can find the ...
Simone Gaiarin's user avatar
10 votes
1 answer
2k views

basic questions on quantum integrable systems

I have been learning about (classical) integrable systems lately, e.g. in the examples of a Lax pair etc. I frequently run into the term 'quantum integrable system'. May I ask a few questions: What ...
Qiao's user avatar
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8 votes
1 answer
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What is exactly the (singularity) confinement property ?

This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata. It seems to be related in certain case to the Painlevé ...
ogerard's user avatar
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7 votes
3 answers
460 views

Kernel of a non-integrable connection

The Riemann-Hilbert correspondence states that the kernel of an integrable (zero curvature) connection is a local system. Here, a connexion on a vector bundle $E$ over a manifold $X$ is a morphism of ...
B. Pillet's user avatar
7 votes
1 answer
355 views

How to solve the system of PDEs defining Killing vectors

Recently I came across the following problem. Here's the setting: Let $(M^n,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection, and $U$ a coordinate neighbourhood with coordinates $\{x^...
Stefan Vasilev's user avatar
6 votes
1 answer
264 views

Bialgebraic structure of Sklyanin algebra

Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
Kevin Ye's user avatar
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1 answer
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From Sato grassmannian to spectral curve

Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi_k(z)=\sum_{m>0}a_{km}z^{-k+m}$). Can ...
Sasha's user avatar
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4 votes
0 answers
348 views

Constructing Markov traces simply

Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...
Thirsty's user avatar
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1 answer
318 views

What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]

. Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link. . . . . . Below is ...
Diego Santos's user avatar
2 votes
0 answers
152 views

Pulled back foliation is completely integrable

There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm". Assume $M$ is a symplectic ...
Olorin's user avatar
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2 votes
2 answers
588 views

Explanation of definition of George Wilson's adelic Grassmannian

How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop ...
Yellow Pig's user avatar
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1 vote
0 answers
115 views

About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation $$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$ it has a solution given by the Gauss's Hypergeometric function $$_2F_1(a,b;c;z)...
Diego Santos's user avatar