Questions tagged [integrable-systems]
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Explicit form of raising and lowering operators in spherical gl(n) DAHA
I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$.
Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
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Exceptional Quantum Groups as FRT-Algebras
Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
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Deriving the time evolution of the reflection coefficient for 1d cubic NLS
Update: I have found that the detailed answer to my questions is contained in the book "Solitons: an introduction" by P.G. Drazin and R.S. Johnson. Generally speaking, this seems to be a great book ...
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Integrating matrix maps
This (which is a follow-up to Lifting a determinant map) must be standard, and yet I am failing to find a reference. Consider a map $f:\mathbb{R}^n \to M^{n\times n}.$ You can pick its degree of ...
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Connection between integrable systems and group actions
An integrable system can be defined as a symplectic manifold together with the maxiumum possible number of Poisson commuting functions on the manifold which are almost everywhere independent. By the ...
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Toda Hierarchy and Quantum Cohomology of $\mathbb{P}^1$ Frobenius manifolds
People usually say that the quantum cohomology of $\mathbb{P}^1$ Frobenius manifold $QH^*(\mathbb{P}^1)$, corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/...
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Lax pairs in an abstract formalism
I am reading Integrals of Nonlinear Equations of
Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
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What is an "integrable hierarchy"? (to a mathematician)
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 ...
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Integrable systems with Fano phase space?
What are some known examples of finite-dimensional integrable systems with symplectic Fano phase space?
Here by integrable system we mean a symplectic manifold $(X,
\omega)$ of dimension $2n$ with $...
user74900
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Spectral and bispectral problems in quantum integrable systems
I am recently interested in the concept of bispectrality (or self-duality) in quantum integrable systems, but some concepts are not clear to me. I may have a (big) lack of precision and rigor in my ...
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Relation between affine flag and Grassmannian Steinberg variety
Let $\mathcal{K}=\mathbb{C}((t))$ be the field of formal Laurent series over $\mathbb{C}$, and by $\mathcal{O}=\mathbb{C}[[t]]$ the ring of formal power series over $\mathbb{C}$.
Given a semi-simple ...
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Embeddings of the configuration space into the phase space of integrable systems
As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of ...
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Integrability conditions for differential equations on $J^\infty$
Is there any result on the existence of solutions of differential equations of the form
$$
D_\alpha\Phi([u])=U_\alpha([u])\Phi([u]),
$$
where $[u]$ is an element of an infinite dimensional bundle $J^\...
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Proving a system is nonintegrable /not solvable with Inverse Scattering Transform
Question:
Given a PDE, is there a general method to show that it is not solvable using the inverse scattering transform?
Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it ...
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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 ...
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References for infinite-dimensional integrable systems?
There are lots of papers on say, W-algebras, that relate them to integrable systems like KdV, the KP hierarchy, etc. Algebraically this is done just by writing down infinitely many commuting operating,...
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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^...
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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, \...
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Why is every Hamiltonian system locally integrable?
It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional ...
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Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold
Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...
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Cohomology of a projective variety with points removed
Take the variety $X$ to be $\mathbb{C}_\infty \times\mathbb{C}_\infty $ with the points $(0,0)$ and $(\infty,\infty)$ removed. Use coordinates $(z,w)\in\mathbb{C}\times \mathbb{C}$ for one chart of ...
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KP tau-functions and $GL(\infty)$
It is usually assumed, that some version of the central extended $GL(\infty)$ group acts transitively on the space of tau-function of the KP integrable hierarchy. It means that any tau-function can be ...
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Classical Yang-Baxter equation for Lie algebras and Lie superalgebras
The classical Yang-Baxter equation is
\begin{align}
[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. \quad (1)
\end{align}
What are the differences between this equation in the case of Lie ...
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How can I verify that a given solution of the Quantum Yang-Baxter equation is associated to a given Lie algebra?
Take, for instance, the $R$ matrix,
\begin{equation}
R(u)=\begin{pmatrix}u+1 & 0 & 0 & 0\\0 & u & 1 & 0\\0 & 1 & u & 0\\0 & 0 & 0 & u+1\end{pmatrix},
\...
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How to obtain the classical Yang-Baxter equation from a related equation
I have a question about the equation (1.24) in the paper about classical r-matrices.
It is said that when we put $\overline{r} = Pr$ in the equation (1.24):
$$
\overline{r}_{23}\overline{r}_{12}P_{23}...
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Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra
I have a solution (a $R$ matrix) of the Yang-Baxter equation,
\begin{equation}
R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1})
\end{equation}
that probably ...
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Literature on ZS-AKNS systems with independent potentials
For those with some familiarity with integrable systems, I'll summarize my question as such:
Where can I find literature on ZS-AKNS systems, and their solution via the inverse scattering transform, ...
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Does this PDE have a name?
I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$:
$ u_{xy} = u_x e^u + u_y e^{-u} $
e.g., Does it have a name? Is it known ...
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Integrable systems and Arnol'd - Liouville theorem
A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.
Here integrable means that you can, in some way, solve the equations of ...
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Periodicity of KdV equation in relation to zero-curvature equation
In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation
$$
\partial_t U - \partial_x V + [U,V] = 0
$$
which gives rise to the monodromy matrix
...
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Spectrum of Kernel - Discrete orthogonal polynomials
Trying to solve a problem, I encounter a Kernel of the form
$$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
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multiplicity free actions - Guillemin&Sternbergy collective integrability
In this post I already ask a similar question.
Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^*}$ the corresponding ...
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A soft question on Gauge Equivalence in Integrable Systems
I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
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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?...
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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 ...
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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 $\mathfrak{...
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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 $...
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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 ...
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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 ...
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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 ...
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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}, r_{...
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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 ...
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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 ...
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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 ...
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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 $$ V(H)=\sum_{j\ge 1}...
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What does it mean for a differential equation "to be integrable"? [duplicate]
What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...
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Why does McMahon formula look like the inclusion-exclusion principle?
The McMahon formula for the number of tilings of an $a \times b \times c$ hexagon by lozenges:
$$ \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 ...
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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 ...
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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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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 ...
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