# Questions tagged [integrable-systems]

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### Understanding what is meant by “Integrability” with regards to the Euler Top

I am reading Discrete Systems and Integrability by F.W. Nijhoff, J. Hietarinta, and N. Joshi. Currently, I am investigating Chapter 6 and in particular the Euler Top. The book says the following: ...
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### Understanding the discretization of the Euler Top with Hirota-Kimura

The Euler Top is the system $$\begin{cases}x_1' = \alpha_1 x_2 x_3 \\ x_2' = \alpha_2 x_3 x_1 \\ x_3' = \alpha_3 x_1 x_2\end{cases}$$ The HK discretization scheme is given by: HK I am reading this ...
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### Are levelsets of a scalar field without critical points a foliation?

Let $f$ be a differentiable scalar field on an $n$-dimensional Riemannian manifold $X$ without critical points, i.e. $\nabla f \neq 0$ everywhere on $X$. (Assuming $X$ has properties as required to ...
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### What is the definition of integrable tensor?

I just googled 'integrable tensor' but I haven't found any stuff about it. I know this question is silly but the rigid definition is also important. Thanks in advance. I saw this word here : ...
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### Finding action-angle variables for integrable maps

Suppose I have an Liouville-Arnold integrable area preserving map $(\bar{x}, \bar{y}) = F(x,y)$ with a first integral $I(x,y)$. How does one seek appropriate action-angle variables such that the map ...
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### Inverse scattering transform and GLM (Gel'fand-Levitan-Marchenko) equation

Can anyone please explain how to derive the GLM equation and why one can recover the potential using just the scattering data?
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### The Double Affine Hecke algebra & Hyperbolic Ruijsenaars-Schneider systems

First of all, I am wondering what makes double Affine Hecke Algebras in a close relationship with quantum integrable systems. In the examples that I saw, the generators of the spherical subalgebra in ...
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### Action-angle variables in non-compact level sets

I am currently studying the Arnold-Liouville theorem, more precisely the construction of action-angle coordinates. I am following mainly the books "Physics for mathematicians I: Mechanics" by Spivak ...
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### Flow on a Torus is Transitive iff it is Incommensurate

Consider the system $$\dot{\mathbf{x}}(t)=\mathbf{a}$$ where $t\in\mathbb{R},\mathbf{x}\in\mathbb{R}^n,$ and $\mathbf{a}\in\mathbb{R}^n.$ It is well known that the flow on the n-torus $\mathbb{T}^n$ ...
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### A few general questions on the Penrose transform

Let us consider the Bateman or Whittaker's pioneering examples of a Penrose transform. Starting from a holomorphic function on an open subset of twistor space, they constructed a solution to the ...
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### How can I show whether or not the Dirichlet function is integrable using the definition of upper and lower integrals?

If $f$ is a version of the Dirichlet function: $f(x) = 1$ when $x$ is an irrational in $[0,1]$ and $f(x) = 0$ otherwise How can I determine whether or not $f$ is integrable using the definition of ...
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### Integrability condition for a vector field on $\mathbb{R}^2$

The Frobenius theorem says that for two vector fields $X,Y$ to be compatible it is sufficient to show that for some functions $\alpha ,\beta$ the following equation holds true \begin{equation} \alpha ...
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### How to prove non-integrability of PDE

Given a certain PDE, we have some well-known methods for proving integrability. For example, if we have a Hamiltonian system and 'sufficiently many' first integrals, we can reduce the system to ...
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### Proof of relationship between lax pairs and zero curvature

I was going through a proof in some integrable systems lecture notes about the relationship between lax pairs and zero curvature. The proof starts as follows: Let $Lf = \lambda f$, where $L$ is a ...
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### Example of non integrable systems and consequences

I'm a PhD student in maths and I am intersted in classical integrable systems (integrable hierarchies). I am currently preparing an introduction to classical integrable systems and I was asking myself ...
### Closed forms of $\int_A^B \sin(\sin(ax))dx$ and $\int_A^B \sin(\sin(ax)) \cdot \sin(\sin(bx))dx$
I would like to know if: $\int_A^B \sin(\sin(ax))dx$ has a closed-form? The solution of Maple requires the presence of Struve functions in its expression. But at least Maple is able to solve it, so ...