# Questions tagged [integrable-systems]

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120
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### Is Krichever's constuction “inverse” to finding the spectral curve?

There is Krichever's algebro-geometric construction of solutions to the KP equations starting from a curve X together with extra data.
There is a way to find the spectral curve given a point of the ...

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95 views

### Can every point of Wilson's adelic Grassmannian be obtained by Krichever construction of solutions to KP equations?

Igor Krichever introduced an algebro-geometric construction of solutions of KP equations starting from an algebraic curve with some additional data.
George Wilson introduced the adelic Grassmannian, ...

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266 views

### Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...

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105 views

### Obstruction to the existence of a globally defined integrating factor

Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The Frobenius Theorem implies that, near each point of $U$, $\omega$ may be written as $g\,{...

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93 views

### Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?

If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system
\begin{align*}L\psi&=\lambda\psi\\
\psi_t&=B\psi\end{align*}
has as its integrability condition ...

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101 views

### Quantum Hamiltonian reduction and tensor products

Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.
Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...

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65 views

### Representation theoretic definition of wavefunctions of an integrable hierarchy?

I am reading Kac's book on infinite dimensional lie algebras. In the last chapter, he starts with a highest weight module of an affine lie algebra $\mathfrak{g}(A)$, and uses it to define tau ...

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350 views

### $GL(\infty)$ group action through the boson-fermion correspondence

Every point of the Sato Grassmannian can be used to generate a tau function of the KP hierarchy. In addition, the Sato Grassmannian can be seen as a subset of the "second quantized fermion Fock space" ...

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207 views

### Examples of particle systems with higher-order collisions

In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...

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72 views

### GUE, tau-function of Painlevé II, and an article of Forrester-Witte

Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a ...

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363 views

### Explanation of definition of George Wilson's adelic Grassmannian

How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop ...

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63 views

### Lax Pair for Kadomtsev-Petviashvili Equation (2-dim KdV)

Is there a simple way to show that the pair
\begin{equation}\begin{cases}\alpha \psi_y+\psi_{xx}+u\psi=0 \\\psi_t + 4\psi_{xxx}+6u\psi_x+3u_x\psi -3\alpha (\partial_x^{-1}u_y)\psi=0\end{cases}\end{...

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389 views

### Any holomorphic vector bundle over a compact Riemann surface can be defined by only one transition function?

It is known that any holomorphic bundle of any rank over a noncompact Riemann surface is trivial. A proof can be found in Forster's "Lectures on Riemann surfaces", section 30.
Let $E$ be a ...

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133 views

### Introduction to the Adler-van Moerbeke theory

Is there a good introduction to the Adler-van Moerbeke theory of solving completely integrable systems by linearizing the flow on the Jacobian of an algebraic curve, for someone with a background in ...

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54 views

### Why is Jacobi Identity equivalent to holonomy of system? [closed]

Or equivalently, why is jacobi identity equivalent to integrability of system? How do I understand it intuitively? Thanks.

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53 views

### Generalized definition of integrable condition on rough complex subbundle

Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...

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87 views

### Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...

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114 views

### What is the expectation/variance of the GOE (Airy-1) point process on a partition of the real line?

Let $\chi^{\mathrm{Ai}}(I)$ denote the GUE (Airy-2) point process on the interval $I \subset \mathbb{R}$.
Soshnikov proved
\begin{align}
\mathbb{E}(\chi^{\mathrm{Ai}}(-T, +\infty)) &\sim \...

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157 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,...

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56 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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259 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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235 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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120 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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155 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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137 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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52 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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125 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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241 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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237 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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214 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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2k 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 ...

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120 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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101 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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487 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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104 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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505 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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321 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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283 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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425 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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244 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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117 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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510 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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111 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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415 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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211 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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128 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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156 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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121 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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116 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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276 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, ...