# Questions tagged [integrable-systems]

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### Equivalence problem of classifying heat equations

I have tried to search for references online but I am unable to do so. I am looking for references that uses Cartan's method of moving frames to classify heat equations. Also are there references that ...
293 views

### What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]

. Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link. . . . . . Below is ...
214 views

### Has anyone written down an approach to the Lenard-Magri integrability scheme via algebraic geometry?

I’ve been thinking about the algebro-geometric meaning of the Lenard-Magri scheme of getting an integrable system from a pair of compatible Poisson structures. I think one might be able to prove a ...
208 views

### Intuition for almost periodic solution and Poincaré recurrence theorem

I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer. Suppose that we have a PDE that admit a solution $u$ that can be ...
81 views

### Spectrum of a Lax Pair and conservation laws of a PDE

I would like to ask a question that I had asked a few days ago on the site math.stackexchange and I still have not received an answer. If we have a Lax operator, we know that the spectrum of this ...
68 views

### Coordinates for quasiperiodic motion after reconstruction

Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
1 vote
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### Dynamics of composition of reflections

Let $C$ be a curve defined by $y = f(x)$, and define the vertical reflection over $C$ to be the map $(x,y) \mapsto (x,y')$, where $y' = 2 f(x) - y$. In other words, the vertical distance from $(x,y)$ ...
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### Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)

I'm interested in solutions to the Yang-Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
167 views

### Integrable systems and Lagrangian fibrations

It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...
303 views

### (Super)integrable systems on quiver varieties

In recent papers https://arxiv.org/abs/2101.05520 https://arxiv.org/abs/2001.06911 (super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed. My question: are ...
317 views

### Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"?

The classical Catalan numbers $$C_n = \frac{1}{n+1} \binom{2n}{n},$$ well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...
1 vote
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### Maximal dimension guaranteed for integral manifolds of hyperplane distributions

To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...
187 views

### Designer metric for a vector field

A vector field $V$ on a manifold $M$ admits an invariant metric if there exists a Riemannian metric $g$ with $L_Vg = 0$. How can one characterize the vector fields on $M$ that admit an invariant ...
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### Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form $$x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)}$$ for ...
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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 ...
209 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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### 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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566 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 ...
997 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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### 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 ...
1 vote
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### 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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### 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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### 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 ...
159 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 \...
217 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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### 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 ... 441 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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### 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 ...
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? (...
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 ...