The integrable-systems tag has no wiki summary.
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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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103 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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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 ...
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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
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1answer
153 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 ...
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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:
...
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115 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} ...
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71 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 ...
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211 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 ...
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126 views
Equivariant J-function of Laumon space
Let $G=SL(n)$ and $B$ be the Borel subgroup of $G$. $G/B$ is the complete flags variety $0\subset V_1 \subset \cdots \subset V_n=\mathbb{C}^n$. The Laumon space is the space ...
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1answer
189 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 ...
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1answer
291 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 $N$ ...
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3answers
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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 ...
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285 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 ...
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85 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 ...
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1answer
515 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 ...
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1answer
518 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, ...
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1answer
280 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 ...
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1answer
688 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 ...
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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 ...
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2answers
928 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).
...
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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 ...
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1answer
606 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 ...
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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). ...
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2answers
297 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 ...
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517 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 ...
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1answer
637 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 ...
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379 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 ...
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1answer
242 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 ...
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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 ...
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1answer
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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 ...
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3answers
381 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 ...
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1answer
377 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 ...
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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 ...
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1answer
859 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 ...
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1answer
507 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 ...
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1answer
678 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 ...
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2answers
487 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 ...
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1answer
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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 ...
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2answers
802 views
Integrable dynamical system - relation to elliptic curves
From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system):
...
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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 "integrable" ...
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3answers
561 views
Equations for Integrable Systems
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 ...
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What is the relationship between integrable systems and toric degenerations?
Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?
An integrable system ...
