Questions tagged [integrable-systems]

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Symplectic quotient involving $SL(n,\mathbb C)$

The following corresponds to Hitchin Integrable Systems: Let $X_1,\dots,X_r\in \mathfrak{sl}(n,\mathbb C)$ with different eigenvalues and consider $$ M= SL(n,\mathbb C)X_1\times \dots \times SL(n,\...
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60 views

Is $f(x)$ Riemann integrable?

$$f(x) = \left\{ \begin{array}{11} x & \mbox{when $x$ is rational} \\ (-x) & \mbox{when $x$ is irrational} \end{array} \right. $$ Prove that $f(x)$ is not integrable over $...
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168 views

Commuting vector fields with common first integrals

Let $M$ be a $n-$dimensional smooth manifold and $X\in\chi(M)$ be a smooth vector field defined on it. Let $f_1,...,f_{n-2}:M\rightarrow \mathbb{R}$ be functionally independent first integrals of $X$, ...
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36 views

Hypersurfaces are coisotropic

In the book "Introduction to symplectic topology" by McDuff and Salomon, it says that Hypersurfaces of symplectic manifolds are coisotropic, so for $S \subset M$ it holds $\forall x \in S, (T_xS)^{\...
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25 views

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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38 views

Flow map of sum of commuting vector fields

my question concerns this exercise. First two parts are straightforward. In particular the second can be done by looking at the linear term of the series expansion in $s$. Where I am stuck is the ...
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1answer
57 views

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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1answer
64 views

Korteweg-de Vries equation: Term has evaporated

In the original paper of Korteweg and de Vries, they derive the equation $$ \frac{\partial \eta}{\partial t} = \frac{3}{2}\sqrt{\frac{g}{l}} \partial_{x}(\eta^2/2 + 2\alpha \eta/3 + \sigma \partial_x^...
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267 views

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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41 views

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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46 views

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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Conserved quntities from Lax pair in case of differential operators

It is known that in integrable systems existence of a Lax pair $$ \dot{L} = [M,L]$$ leads to the existence of conserved quantities $$ \frac{d}{dt}\mathrm{Tr}(L^k) = \sum \mathrm{Tr}(L^i [M,L] L^{k-i})...
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57 views

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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97 views

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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36 views

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 ...
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96 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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38 views

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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1answer
42 views

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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238 views

A conserved quantity reduces the dimension of the system?

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 $$...
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50 views

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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111 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?
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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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104 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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102 views

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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1answer
424 views

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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72 views

Classical Yang-Baxter Equation

Let $\mathfrak{g}$ be a Lie algebra, $r = \sum_i x_i \otimes y_i \in \mathfrak{g} \otimes \mathfrak{g}.$ Define $r_{12} = \sum_i x_i \otimes y_i \otimes 1,$ $r_{13} = \sum_i x_i \otimes 1 \otimes y_i,...
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115 views

Decomposing rational function with poles

Given a matrix valued rational function $f(\lambda)$ with poles of order $n_k$ at points $\lambda_k$, not at infinity, a book I am following says we can decompose as $$f(\lambda)=f_0+\sum_k f_k(\...
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40 views

Integrable Functions

Let f and g be $2π-periodic$ integrable functions on $\mathbb{R} $ and let $f ∗ g$ denote their convolution on $[−π, π]$. Prove that $\widehat{f∗g(n)} = \hat{f}(n)\hat{g}(n)$. I'm stuck with how to ...
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2answers
118 views

How do you solve this functional equation?

$$f(x)f(y)=f(x+y)(f(x)+f(y)-2\cos\gamma)+1.$$ I want to obtain the solution $f(x)=\sin(x+\gamma)/\sin x$, instead of the one which is well defined at 0, which I have posted a question before. Is ...
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1answer
15 views

Equations of motion for certain representation of the force

This might seem like an easy question, I would appreciate any help you can provide. For the force F can be factored as below, is the equation of motion (EOM) integrable? a) $F\left({x}_{i},t\right)=...
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72 views

First Integrals of the Following Hamiltonian

I have a Hamiltonian $$ H = \sum_{1\leq i < j \leq n} p_i p_j \frac{\sinh({\lambda} (q_i - q_j))}{\lambda^2}, $$ where ($\textbf{q},\textbf{p}) \in \mathbb{R}^{2n}$, and $\lambda$ is a free ...
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Action-angle variables in a 2 degree of freedom integrable Hamiltonian?

Suppose I have the following Hamiltonian $$H(q, p, x,y) = \frac{p^{2}}{2} - \frac{q^{2}}{2} \left(\left(\frac{y^{2}}{2} + \omega^{2}x^{2}\right)^{2} - \frac{q^{2}}{2}\right)$$ Where $(q, p, x,y) \in \...
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Existence of solutions to the Non-linear Schrödinger equation with compact support

Consider the non-linear Schrödinger equation (NLS) in the following version: \begin{align} u_{xx} + i u_t +2|u|^2 u = 0, \text{ where }u:\mathbb{R}\times [0,\infty) \to \mathbb{R}. \end{align} We ...
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1answer
70 views

Laplace Transform of Final integral Function

I want to prove that if signal $u(t)$ satisfies: $$ \lim_{t \to \infty} \int_0^t u(r) \, dr=c<\infty $$ then the Laplace Transform of the signal the following: $$ \lim_{t \to \infty} \int_0^t \...
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An example of slow perturbation of an integrable Hamiltonian system?

Consider the following (classical) $2 \frac{1}{2}$ degree of freedom Hamiltonian system: $H(u,v,p,q,τ)$, where $(u,v)$ and $(p,q)$ are conjugated variables and $τ=ϵt$ is a slowly varying parameter, $0&...
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494 views

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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173 views

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 ...
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1answer
134 views

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 ...
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64 views

How to derive the formula $\{t_{ij}, t_{kl}\} = \sum_{a,b} (r^{ajbl} t_{ia}t_{kb} - r^{iakb} t_{aj}t_{bl})$?

I am reading the book. Let $G$ be a Poisson-Lie group and $r = \sum_{s,t} r^{st} X_s \otimes X_t \in g \wedge g$, where $g$ is the Lie algebra of $G$. In the end of page 60, the bracket on $\mathbb{...
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1answer
62 views

Integrating factor for a differential equation that contains an arbitray function

I have the following differential equation: $y(t)dx+(f(t)-x(t))dy=0$ It is suppose to be non integrable for a differentiable but arbitrary $f(t)$. How do I know this is true? This is part of the ...
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39 views

How to construct a Poisson bracket for a PDE? [closed]

I'm working with a PDE, and I would like to find one or more Poisson brackets. Is there any algorithm for this? And if not, do you have any other suggestions to find them (perhaps a book or notes that ...
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What integrability of Hamiltonian system says about non compact orbits? [duplicate]

Suppose that we have an $n$-dimensional manifold $M$ and consider a Hamiltonian system on cotangent bundle which is integrable in the following way: there exist $n$ functions $f_1,...,f_n$, which are ...
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61 views

Prove that $\int g\ {\rm d} \mu<\infty$?

Let $(S,\mathcal{A},\mu)$ be aprobability space and $g\ge 0$ with $\int g\ln^+\ln^+ g {\rm d} \mu<\infty$. Can you help me prove that $\int g\ {\rm d} \mu<\infty$?
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On the Liouville-Arnold theorem

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to ...
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73 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...