Questions tagged [integrable-systems]

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46 views

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 ...
2
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1answer
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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1answer
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\,{...
4
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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 ...
4
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1answer
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 ...
6
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1answer
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" ...
3
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1answer
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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2answers
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{...
4
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1answer
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 ...
5
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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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0answers
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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1answer
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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2answers
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 ...
3
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2answers
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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1answer
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 ...
2
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1answer
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 ...
7
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1answer
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 ...
3
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1answer
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 ...
6
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1answer
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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4answers
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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0answers
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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2answers
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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0answers
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 ...
3
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2answers
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^\...
10
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2answers
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 ...
5
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3answers
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 ...
6
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3answers
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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1answer
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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1answer
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, \...
4
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1answer
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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0answers
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 ...
7
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1answer
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 ...
2
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0answers
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 ...
3
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0answers
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 ...
6
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1answer
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}, \...
1
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1answer
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}...
4
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1answer
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 ...
4
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2answers
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, ...