Questions tagged [integrable-systems]
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4
votes
2answers
107 views
Inverse image of rational values
I am a postgraduate student of physics. While doing some research on Poincare's work on the integrability of the three body problem, I came up with the following problem (which I feel unable to handle,...
0
votes
0answers
209 views
The Ricci flow and integrable systems
It is well-know that the Ricci flow is defined by the following equation:
$$ \frac{\partial g}{\partial t}=-2R(g)$$ where $g$ is the metric and $R(g)$ is the Ricci curvature. Let $g_0$ be a metric and ...
4
votes
0answers
50 views
Identification of spectral and differential data for integrable difference equations?
Let $X$ be a projective curve and $G$ be a semisimple Lie group. There is a theorem roughly stating that there exists an isomorphism between the moduli space of principal $G$-bundles on $X$ and the ...
3
votes
2answers
195 views
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 (...
1
vote
0answers
173 views
On the connections between Ruijsenaars-Schneider systems and other areas
I found on the literature plenty of articles dealing with connections between rational/trigonometric/elliptic Calogero-Moser systems and their relativistic generalizations (Ruijsenaars-Schneider), and ...
2
votes
1answer
108 views
Two questions on Zuber's “KdV and W-flows”
I'm having difficulty following computations in the paper "KdV and W-flows" by Zuber.
On pg. 2, what would be the conserved quantity $I_4$, related to the conservation laws of the KdV hierarchy? (...
5
votes
0answers
120 views
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 ...
7
votes
1answer
116 views
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 ...
1
vote
0answers
48 views
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 ...
3
votes
1answer
121 views
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 ...
6
votes
0answers
209 views
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 ...
5
votes
1answer
185 views
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/...
7
votes
0answers
180 views
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 ...
0
votes
0answers
70 views
Quasi-triangular bialgebra construction
Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, $r=\sum_i x_i \otimes y_i$, where $x_i, y_i \in \mathfrak{g}$. Define $r_{12}=\sum_i x_i \otimes y_i \otimes 1$, $r_{13}=\sum_i x_i \otimes 1 \...
20
votes
4answers
1k views
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 ...
4
votes
0answers
101 views
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 $...
3
votes
0answers
82 views
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 ...
10
votes
2answers
406 views
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 ...
2
votes
0answers
98 views
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 ...
3
votes
2answers
282 views
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^\...
9
votes
2answers
284 views
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 ...
5
votes
3answers
224 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 ...
6
votes
3answers
326 views
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,...
6
votes
1answer
209 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^...
1
vote
1answer
91 views
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, \...
4
votes
1answer
422 views
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 ...
7
votes
0answers
101 views
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 ...
7
votes
1answer
388 views
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 ...
2
votes
0answers
177 views
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 ...
3
votes
0answers
115 views
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 ...
6
votes
1answer
140 views
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},
\...
1
vote
1answer
115 views
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):
$$
\overline{r}_{23}\overline{r}_{12}P_{23}...
3
votes
1answer
101 views
Solution of the Yang-Baxter equation associated to the $U_q[osp(2n+2|2m)^{(2)}]$ Lie superalgebra
I have a solution (a $R$ matrix) of the Yang-Baxter equation,
\begin{equation}
R_{12}(x_{1})R_{13}(x_{1}x_{2})R_{23}(x_{2})=R_{23}(x_{2})R_{13}(x_{1}x_{2})R_{12}(x_{1})
\end{equation}
that probably ...
4
votes
2answers
252 views
Literature on ZS-AKNS systems with independent potentials
For those with some familiarity with integrable systems, I'll summarize my question as such:
Where can I find literature on ZS-AKNS systems, and their solution via the inverse scattering transform, ...
3
votes
1answer
160 views
Does this PDE have a name?
I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$:
$ u_{xy} = u_x e^u + u_y e^{-u} $
e.g., Does it have a name? Is it known ...
6
votes
1answer
365 views
Integrable systems and Arnol'd - Liouville theorem
A system with a $2n$-dimensional phase space is Liouville-integrable if it admits $n$ independent first intgrals in involution.
Here integrable means that you can, in some way, solve the equations of ...
4
votes
1answer
171 views
Periodicity of KdV equation in relation to zero-curvature equation
In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation
$$
\partial_t U - \partial_x V + [U,V] = 0
$$
which gives rise to the monodromy matrix
...
3
votes
0answers
118 views
Spectrum of Kernel - Discrete orthogonal polynomials
Trying to solve a problem, I encounter a Kernel of the form
$$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(...
2
votes
1answer
98 views
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 ...
4
votes
0answers
227 views
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 (...
6
votes
1answer
212 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?...
1
vote
1answer
125 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 ...
5
votes
0answers
131 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 $\mathfrak{...
1
vote
1answer
147 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 $...
2
votes
0answers
98 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
votes
1answer
124 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
votes
0answers
130 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
vote
0answers
67 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}, r_{...
6
votes
1answer
167 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 ...
8
votes
1answer
139 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 ...
