# Questions tagged [integrable-systems]

The integrable-systems tag has no usage guidance.

146
questions

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### Connecting the higher energies of GP and KdV via a Riccati equation

I will describe my set-up and then the problem.
We use the branch of the complex square root where
$$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...

3
votes

1
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393
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### Integrability of Schroedinger's equation

Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...

4
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1
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208
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### Building a geodesic conjugate parameterization on catenoid

I believe that a catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma_{uv} \in\mathrm{span}(\sigma_u, \sigma_v)$) ...

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### Solving a Catalan-like recursion of polynomials, related to the KdV energies

I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are ...

4
votes

1
answer

170
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### A system of linear PDEs with boundary conditions

I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I could simplify one of my geometric problems (in the smooth scenario) into the solutions of a system of linear PDEs ...

4
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121
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### Deformation quantization of an integrable system

What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?

2
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### How to recover a subspace in an infinite-dimensional Grassmannian from its $\tau$-function or $\psi$-function?

Solutions of the KP hierarchy are parametrized by an infinite-dimensional Grassmannian (Sato Grassmannian or Wilson's adelic Grassmannian). I heard somewhere that one can recover a subspace in the ...

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47
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### Is there a relation between symplectic toric orbifolds and semi-toric systems?

So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ ...

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0
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27
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### Question on the proof of doing a nodal trade, almost-toric fibrations

I am trying to understand the details of the proof of lemma $6.3$ of the following notes https://arxiv.org/pdf/math/0210033.pdf, which give us specific conditions of when we can swap a neighborhood of ...

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54
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### Doing a nodal trade in a semi-toric system

Recently I have been studying semi-toric systems and almost toric fibrations. For the purpose of semi-toric fibrations I have been reading these notes https://arxiv.org/pdf/math/0210033.pdf. ...

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34
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### Build up of an external Kalman filter with closed form solution of an ODEs

When using an external Kalman filter, what are the benefits and advantages if, instead of a physical model, e.g. PP-p2D model - described by a system of nonlinear coupled differential equations - a ...

2
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### Zero-curvature formulation of the Camassa-Holm hierarchy

In the book of Gesztesy and Holden (see the following article of the same authors), they state that the (stationary) Camassa-Holm hierarchy may be cast as a zero-curvature equation
\begin{align}
-V_{n,...

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0
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106
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### About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation
$$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$
it has a solution given by the Gauss's Hypergeometric function
$$_2F_1(a,b;c;z)...

1
vote

3
answers

224
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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 ...

2
votes

1
answer

293
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### 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.
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Below is ...

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214
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### 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 ...

4
votes

1
answer

208
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### 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 ...

2
votes

0
answers

81
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### 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 ...

3
votes

0
answers

68
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### 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 ...

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77
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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 $(x,y)$ ...

6
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2
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222
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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 ...

3
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167
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### 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 ...

3
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2
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303
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### (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 ...

6
votes

1
answer

317
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### 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
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0
answers

144
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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 ...

4
votes

1
answer

187
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### 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 ...

4
votes

1
answer

489
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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 ...

2
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0
answers

101
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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 ...

2
votes

1
answer

209
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### 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, ...

6
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1
answer

491
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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(...

5
votes

1
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190
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### 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\,{...

6
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290
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### 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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1
answer

157
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### 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 ...

2
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0
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91
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### 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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1
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548
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### $GL(\infty)$ 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" ...

4
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1
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305
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### 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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### 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 ...

2
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2
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### 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 ...

5
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1
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997
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### 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 ...

6
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164
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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 ...

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0
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98
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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.

2
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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 ...

3
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97
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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 ...

2
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1
answer

159
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### 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 \...

4
votes

2
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217
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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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### 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 ...

5
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441
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### 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 (...

3
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1
answer

274
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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 ...

2
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1
answer

137
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### 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? (...

7
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222
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### 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 ...