# Questions tagged [integrable-systems]

The integrable-systems tag has no usage guidance.

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### 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,...

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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 ...

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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 ...

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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 (...

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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 ...

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votes

**1**answer

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? (...

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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 ...

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**1**answer

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 ...

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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 ...

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votes

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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 ...

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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 ...

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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/...

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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 ...

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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 \...

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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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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 $...

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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 ...

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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 ...

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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 ...

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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^\...

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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 ...

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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 ...

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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,...

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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^...

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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, \...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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},
\...

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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}...

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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 ...

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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, ...

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votes

**1**answer

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 ...

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votes

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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 ...

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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
...

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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(...

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**1**answer

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 ...

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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 (...

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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?...

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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 ...

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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{...

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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 $...

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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 ...

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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 ...

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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 ...

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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_{...

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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 ...

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**1**answer

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 ...