Questions tagged [integrable-systems]
The integrable-systems tag has no usage guidance.
154 questions
2
votes
1
answer
236
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 ...
- 6,201
3
votes
1
answer
164
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 $$ V(H)=\sum_{j\ge 1}...
- 2,552
10
votes
2
answers
8k
views
What does it mean for a differential equation "to be integrable"? [duplicate]
What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...
- 201
33
votes
1
answer
1k
views
Why does McMahon formula look like the inclusion-exclusion principle?
The McMahon formula for the number of tilings of an $a \times b \times c$ hexagon by lozenges:
$$ \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]$$
looks oddly like the ...
- 22.8k
1
vote
0
answers
97
views
anomaly polynomial of generalized Hitchin system
I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theroy can be described in terms of ...
- 171
2
votes
0
answers
216
views
Beauville's Integrable System with singular spectral curves
Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...
- 171
2
votes
0
answers
1k
views
Proof of Arnold-Liouville theorem in classical mechanics [closed]
I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...
- 181
1
vote
0
answers
78
views
Singularity Confinement For Differential-Difference Systems
This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems.
Are there any papers or books that ...
- 111
1
vote
2
answers
1k
views
Integrability - conditions of lax pairs
I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
$$
\partial_t U - \partial_x V + [U,V]=0
$$
where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ ...
- 215
2
votes
0
answers
329
views
moduli space of meromorphic $G$-Higgs bundles
I want to clarify with some topics in moduli space of semistable $G$-Higgs bundles on curve $X$ (genus $g$ is large enough) of fixing topological type $d \in \pi_1(G)$. Simpson's construction gives us ...
- 171
1
vote
0
answers
580
views
On the Hitchin fibration
I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
- 111
2
votes
0
answers
165
views
Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom
Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...
- 1,038
4
votes
1
answer
395
views
Weinstein's local classification of Lagrangian foliations
In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...
- 783
3
votes
1
answer
1k
views
How to find a Lax Pair for the modified KdV equation
Question
I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as:
$
X = \begin{...
- 131
7
votes
0
answers
267
views
Toda Flow Embeddings
What are strategies for generating the following types of pictures:
Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are:
$$\frac{d}{dt}a_k=2(b_k^2-b_{k-1}^2),$$
...
- 4,952
1
vote
0
answers
157
views
Calogero-Moser eigenfunction
The folllowing function
\begin{equation}
J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} e^{-\frac{a_1t_1+a_2t_2+a_3t_3}{h}}\sum_{k_{1,1},k_{2,1},k_{2,2}\ge0}e^{(t_1-t_2)k_{1,...
- 2,273
0
votes
0
answers
188
views
Do principally polarized abelian varieties enjoy a genus expansion?
This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...
- 2,029
1
vote
1
answer
103
views
Reduction along an Orbit for C.-M. systems
I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf
where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on ...
- 121
3
votes
0
answers
373
views
J-function of cotangent bundle of complete flag variety
Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle $T^*...
- 2,273
9
votes
1
answer
502
views
Detecting Monodromy in Integrable Systems
Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...
- 401
7
votes
1
answer
554
views
Calogero-Moser system: relationship between dual variables and the KKS construction
This is a question about the relationship between two ways of viewing the Calogero-Moser system.
$$\ddot x_i=2\sum_{j\neq i}\frac{1}{(x_i-x_j)^3}\qquad i=1,\ldots N$$
By introducing the $N$ ...
- 1,038
75
votes
4
answers
6k
views
What is the amplituhedron?
The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
- 24.7k
2
votes
0
answers
382
views
A modification of Maurer-Cartan equation
In deformation theory of complex structure, the Maurer-Cartan equation takes the form
$$\bar{\partial}\varphi(t)+\frac{1}{2}[\varphi(t),\varphi(t)]=0.$$
where $\varphi(t)\in\Gamma_{smooth}(X,\bigwedge^...
- 29
1
vote
0
answers
114
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 -...
- 1,143
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 ...
- 1,719
10
votes
1
answer
1k
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
1
answer
361
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 ...
- 19.7k
8
votes
1
answer
1k
views
is the geodesic flow on Hyperbolic Plane completely integrable?
I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\...
- 345
22
votes
1
answer
1k
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 ...
- 16.8k
4
votes
0
answers
408
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 ...
- 41
9
votes
3
answers
3k
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).
...
- 541
4
votes
1
answer
255
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 ...
- 1,343
5
votes
1
answer
1k
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 ...
- 1,343
8
votes
1
answer
566
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). ...
- 1,343
5
votes
2
answers
397
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 ...
- 478
3
votes
0
answers
559
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 ...
- 197
1
vote
1
answer
826
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 ...
- 4,306
11
votes
1
answer
713
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 ...
- 515
2
votes
1
answer
355
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 http://blog.djalil.chafai.net/2010/11/02/aspects-of-the-...
- 4,466
26
votes
1
answer
6k
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 ...
- 40.2k
4
votes
0
answers
182
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 ...
- 41
10
votes
2
answers
2k
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 ...
- 1,343
10
votes
4
answers
518
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 ...
- 85.6k
8
votes
1
answer
432
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 ...
- 9,720
6
votes
0
answers
450
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 ...
- 281
2
votes
2
answers
1k
views
Plotting path between sphere or ellipsoid points?
Hi, my apologies if this is not the right place to ask this- I am not a mathematician (I'm a software engineer) and Im working on some 3D applications.
My situation is this- given an origin of 0,0,0 ...
- 131
8
votes
1
answer
1k
views
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 ...
- 948
4
votes
1
answer
818
views
Spectral curve of Elliptic Calogero-Moser systems
First, why all the coefficients in the characteristic polynomial of L are elliptic functions, since the diagonal entries of the matrix L are the momentums?
second, how to understand the ramification ...
- 77
10
votes
3
answers
864
views
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 ...
- 13k
7
votes
1
answer
3k
views
Connection between bi-Hamiltonian systems and complete integrability
As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.
Two Poisson brackets
$\{ \cdot,\cdot \}...
- 85.6k