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1answer
46 views

Importance of a Hamiltonian integrable system be a bi-Hamiltonian system?

Once I know a complete integrable system $(f_1=H,\ldots,f_m):M^{2m}\to\mathbb{R}$, $H$ being the Hamiltonian of the system. What is the importance to know about a second Hamiltonian representation for ...
6
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0answers
83 views

Modular double of elliptic quantum group

By studying dynamical quantum Yang-Baxter equations and corresponding $RLL$ relations, Felder defined an elliptic version of quantum group $E_{\tau, \eta}(sl_2)$, which can be understood as ...
1
vote
1answer
68 views

Lagrangian foliation

Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f_1$, so we find $n = \frac{1}{2}\dim M$ functions ...
3
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0answers
77 views

How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly?

In the paper, Stolin classifies all quasi-Frobenius subalgebras of $sl_3$. How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly using these quasi-Frobenius subalgebras? Thank you ...
3
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1answer
92 views

Are all the Lie bialgebra structure on $sl_n$ coboundary?

In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore ...
2
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0answers
83 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 ...
1
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0answers
55 views

Low-dimensional classical r-matrices

Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties: (1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$. (2) $[r_{12}, ...
6
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1answer
100 views

Classifying Low Dimensional Solutions of the Yang--Baxter Equation

What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions? To make my question more specific, have all solutions for dimension $2$ and $3$ been ...
6
votes
1answer
89 views

Sign problem in a Calogero-Moser system: proof of integrability?

Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has ...
2
votes
1answer
118 views

How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?

I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations. In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...
2
votes
1answer
88 views

Integrability of complex gaussian random matrix model

It is known that the partition function $$ \mathcal{Z}_1=\int dH e^{-N{\rm Tr}(H^2)}e^{-NV(H)},$$ where the integral is over $N\times N$ hermitian matrices $H$, with the potential $$ V(H)=\sum_{j\ge ...
4
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2answers
549 views

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 ...
21
votes
1answer
432 views

Why does McMahon formula look like the inclusion-exclusion principle?

The McMahon formula for the number of tilings of an $a \times b \times c$ hexagon by lozenges: $$ \Big[H(a)H(b)H(c)\Big] \Big[H(a+b)H(b+c)H(c+a)\Big]^{-1} \Big[H(a+b+c)\Big]$$ looks oddly like the ...
1
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0answers
61 views

anomaly polynomial of generalized Hitchin system

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theroy can be described in terms of ...
1
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0answers
81 views

Beauville's Integrable System with singular spectral curves

Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...
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0answers
61 views

Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems. Are there any papers or books that ...
1
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1answer
176 views

Integrability - conditions of lax pairs

I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation: $$ \partial_t U - \partial_x V + [U,V]=0 $$ where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ ...
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0answers
202 views

moduli space of meromorphic $G$-Higgs bundles

I want to clarify with some topics in moduli space of semistable $G$-Higgs bundles on curve $X$ (genus $g$ is large enough) of fixing topological type $d \in \pi_1(G)$. Simpson's construction gives us ...
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0answers
251 views

On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems". Proposition 1.4: When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
2
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0answers
115 views

Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation $$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$ can ...
3
votes
1answer
176 views

Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle. I ...
7
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0answers
155 views

Toda Flow Embeddings

What are strategies for generating the following types of pictures: Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are: ...
1
vote
0answers
123 views

Calogero-Moser eigenfunction

The folllowing function \begin{equation} J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...
0
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0answers
97 views

Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider: It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...
2
votes
0answers
242 views

J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle ...
8
votes
1answer
251 views

Detecting Monodromy in Integrable Systems

Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...
6
votes
1answer
347 views

Calogero-Moser system: relationship between dual variables and the KKS construction

This is a question about the relationship between two ways of viewing the Calogero-Moser system. $$\ddot x_i=2\sum_{j\neq i}\frac{1}{(x_i-x_j)^3}\qquad i=1,\ldots N$$ By introducing the $N$ ...
59
votes
3answers
4k 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 ...
2
votes
0answers
304 views

A modification of Maurer-Cartan equation

In deformation theory of complex structure, the Maurer-Cartan equation takes the form $$\bar{\partial}\varphi(t)+\frac{1}{2}[\varphi(t),\varphi(t)]=0.$$ where ...
1
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0answers
89 views

Divisors, factorisations of matrix valued functions, and the Lorentz group

How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem ...
4
votes
1answer
729 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 ...
4
votes
1answer
615 views

Any applications integrable systems (pde,ode, q-,…) to math. biology (pharmakinetics, pharmadynamics) ?

Question Are there any relations/applications of integrable system theory (take it as broadly as one can: ODE, PDE, quantum, box-ball,...) to mathematical biology (in particular pharmacokinetics, ...
2
votes
1answer
298 views

functions whose average along orbits is zero or a constant

Is there some name in ergodic theory or integrable systems theory for a function whose average value on every orbit is zero? (Of course when I say "every orbit" in the context of ergodic theory I ...
19
votes
1answer
820 views

Integrability of the Cohen map

In the 1990's, Henri Cohen asked whether the map $(x,y) \mapsto (\sqrt{1+x^2}-y,x)$ from $\mathbb{R}^2$ to itself is integrable. In other words, are the orbits confined to the level curves of some ...
4
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0answers
194 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 ...
3
votes
2answers
1k views

What is soliton

I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking.. In literature, i am finding following words:(Wikipedia+ others). ...
3
votes
0answers
175 views

Krichever-Novikov-Dubrovin description for not-algebraic spectral curve

Non-algebraic curves play an increasing role in string theory, sometimes they are known to be related to the integrable systems of the KP/Toda type. Are there any investigated examples of the ...
3
votes
1answer
747 views

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 ...
8
votes
1answer
452 views

Multiple Hodge integrals and integrability

It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function (see Kazarian). ...
4
votes
2answers
313 views

Non-polynomial integrals of motion for polynomial dynamical systems

Does there exist a polynomial Hamiltonian function $H$ on some $\mathbb{R}^{2n}$ such that Any polynomial function $P$ such that $\{P,H\}=0$ is of the form $p(H)$ for some polynomial $p$ in one ...
3
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0answers
531 views

Find a second integral for Arnold's example

Consider Arnold's example for Arnold diffusion 1964. $$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t)) $$ We can first make it a system of three degrees of freedom. Then we ...
1
vote
1answer
675 views

About the geometry of completely integrable systems

During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of ...
6
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0answers
403 views

Weakest condition for an integrable, almost-symplectic manifold?

I was recently speaking with someone who works in Computational Chemistry and they mentioned that in a lot of the computational simulations they do, they have systems that are not symplectic but still ...
2
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1answer
258 views

Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?

It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog ...
18
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1answer
3k views

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 ...
4
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0answers
147 views

Uniqueness for solution of a d-dbar system related to Davey-Stewartson Solitons

This question concerns a system of equations that arise in the study of one-soliton solutions to the Davey-Stewartson equation. In what follows, $f(z)$ denotes a function which depends smoothly (but ...
7
votes
1answer
1k views

Gromov-Witten and integrability.

The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are ...
8
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3answers
395 views

Solitary waves and their symmetries

This is probably a very naive question from a field that I don't have much background from, but a combination of curiosity and the fact that conceptual questions get very good answers here on MO ...
7
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1answer
386 views

Two interacting bodies in an external field

Hope, MO is the right place for this question (if not so: where would you pose it?). Consider a two-body system in classical mechanics. As long as the interaction depends only on the distance of the ...
6
votes
0answers
419 views

Differential equation of line tangent to caustics

This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most ...