# Questions tagged [integrable-systems]

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### What's the definition of Lie bracket of two tensor in "Introduction to Classical Integrable Systems" pp. 14

I'm reading "Introduction to Classical Integrable Systems". In pp. 14 the proposition of r-matrices there's fomula like $$\{L_1, L_2\} = [r_{12}, L_1] − [r_{21}, L_2]$$. If my understand is ...
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### Is the domain of a leaf of a foliation arising from an involutive distribution convex?

I have a $C^1$ plane distribution $\mathscr{D}$ on $P^n$ (here P is the strictly positive reals, so $P^n$ is the strictly positive quadrant of $R^n$) which is involutive and gives rise to a ...
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### Symplectic form on spheres

I have an exercise that says: calculate the symplectic form of 2-dimensional spheres with ratio r and let be H the height function, calculate the integrable system I understand that for the ...
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### What functions do we need to solve linear second order differential equations with polynomial coeficients?

Disclaimer: I have posted this question on mathoverflow.net following the instructions of this topic. I'm now trying to understand how can a ordinary differential equation be tested to decide if it's ...
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### Intuition for almost periodic solution and Poincaré recurrence theorem

Suppose that we have a PDE that admit a solution $u$ that can be expressed in a certain system of coordinates (angle-action variables) as advection with constant velocity on tori. And suppose also ...
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### Lax pairs and Nonlinear Schrodinger Equation's components

Briefly: I have two Lax pairs in matrix form and using compatibility condition I have found a nonlinear partial differential equation system. I have searched this system very much seems the nonlinear ...
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### Definition of chaos for a bounded orbit

Let $\rho : \mathbb{R} \times \mathbb{R}^{2n} \longrightarrow \mathbb{R}^{2n}$ be a hamiltonian flow, i.e. a 1-parameter family of symplectomorphisms with respect to $t$ obtained from integrating a ...
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### A Tricky sum to evaluate (Haldane)

I'm trying to find a way to evaluate this sum (found by Haldane in Phys. Rev. Lett. 60, 635 (1988): $$S_{pq}=\sum_{n=1}^{N-1} z^{nJ} (1-z^{n})^{p-1}(1-z^{-n})^{q-1}$$ with $z= e^{\frac{2i\pi}{N}}$ ...
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### Constants of motion for a system of two points moving on spherical surface with a force depending only on their relative distance

Consider two points that moves only on a spherical surface of radius $R$. There is only a force between them that has a potential $U(d)$ where $d$ is the distance between the two points. What is the ...
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### unsure how to expand a LAX pair for KDV equation

This is my first post so im not sure how to make it all mathsy so im going to write it on here, I know that to find the lax equation you find [LM-ML]=0 but im struggling to follow the expansion, for ... 196 views

### Understanding the notation when finding action-angle coordinates

I'm trying to learn the basics of KAM theory and I wanted to first get to grips with Liouville integrability for Hamiltonian systems but I'm getting rather confused by the notation which seems to be ...
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### integrability of system of SDE relative to Langevin dynamics

Similar to the system in this question and a follow up to here, consider the system (S) defined as \begin{align} {\rm d}X_t&=V_t\,{\rm d}t+\sigma_x\,{\rm d}W_t^1\\ {\rm d}V_t&=F_t\,{\rm d}t+\...
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### Quantizing solutions to the reflection algebra

I am trying to find the quantum analogues to classical solutions of Sklyanin's reflection algebra (RE). I have a solution to the classical Poisson bracket for known r-matrix $r(\mu)$ \begin{equation}\...
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### there seems to be a statement that is provable in integrable system, please let me know where can I learn about this statement.

the statement is: given $f,g \in C^{\infty}(M)$ the it's true that $\xi_{\{f,g\}}=[\xi_{f},\xi_{g}]$ M should be a symplectic poisson manifold
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Suppose $\dot{x} = f(x)$ is a dynamical system on the state space $X$. My notes define a conservative system as one where there exists a (nontrivial) function $H: X \rightarrow \mathbb{R}$ such that $$... 1 vote 0 answers 61 views ### Finding action-angle variables for integrable maps Suppose I have an Liouville-Arnold integrable area preserving map (\bar{x}, \bar{y}) = F(x,y) with a first integral I(x,y). How does one seek appropriate action-angle variables such that the map ... • 1,512 1 vote 1 answer 230 views ### Inverse scattering transform and GLM (Gel'fand-Levitan-Marchenko) equation Can anyone please explain how to derive the GLM equation and why one can recover the potential using just the scattering data? • 11 5 votes 0 answers 133 views ### Action-angle variables in non-compact level sets I am currently studying the Arnold-Liouville theorem, more precisely the construction of action-angle coordinates. I am following mainly the books "Physics for mathematicians I: Mechanics" by Spivak ... • 456 3 votes 0 answers 140 views ### Flow on a Torus is Transitive iff it is Incommensurate Consider the system$$ \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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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 ...