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Questions tagged [integrable-systems]

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25 views

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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20 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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64 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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0answers
21 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
23 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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18 views

Integrable system vs lagrangian fibration

Every complete integrable system $I:M \rightarrow \mathbb{R}^n$ is a regular langrangian fibration on a dense subset of the symplectic manifold $M$. It is also known that locally every lagrangian ...
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36 views

How to gain the exact solution of the partial discrete equation :$u(n,m+1)=u(n-1,m)+u(n,m)(u(n+1,m)-u(n-1,m))$

I made a mathematical discrete model of one dimensional flow. Then I achieved this equation: $u(n,m+1)=u(n-1,m)+u(n,m)(u(n+1,m)-u(n-1,m))$ Where n ,m are integer and $0\leq u \leq 1 $ for all n, m. ...
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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
2
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1answer
97 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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26 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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1answer
65 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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32 views

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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56 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 ...
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77 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 $\mathbb{T}^n$ ...
2
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1answer
72 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 ...
2
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1answer
200 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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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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70 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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38 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
110 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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63 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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41 views

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
66 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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0answers
55 views

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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347 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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66 views

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 ...
3
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120 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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0answers
98 views

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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115 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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1answer
41 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
56 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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0answers
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 ...
3
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0answers
50 views

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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1answer
60 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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1answer
861 views

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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0answers
66 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(...
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0answers
148 views

Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
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81 views

A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (...
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0answers
101 views

Hirota's Bilinear Form

My query is related to conversion of PDE into bilinear form using some transformation. The detailed description about bilinear forms can be seen in book of Yoshimasa Matsuno . The bilinear operator ...
2
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1answer
94 views

Show that $f$ is integrable [closed]

I am learning elementary analysis and I am doing some exercises. Need help with this one. Thanks! Suppose $f: [0,1] → \Bbb R$ such that $f(x)=0$ except at finitely many points ${x_1, ... x_k}$. Show ...
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1answer
135 views

Proving the existence of parallel sections in Chern's Lectures on Differential Geometry

Here is a quotation from page 110, section 4-1, of Chern's Lectures on Differential Geometry: If a section $s$ of a vector bundle $E$ satisfies the condition $$ Ds = 0\ \ \ \ \ \ \ \ \ \ \ (1.36),...
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123 views

What is the relation between solutions of classical Yang-Baxter equations and solutions of modified Yang-Baxter equations.

Let $g$ be a Lie algebra. The classical Yang-Baxter equation (CYBE) is: $$ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0. $$ The modified classical Yang-Baxter equation (MCYBE) is: $$ [r_{...
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1answer
70 views

Analytic solution to a second order nonlinear ODE involving $\operatorname{sech}^2(x)$?

I am trying to look for an analytic solution to the following equation $\frac{1}{4D} (y')^2 -\frac{1}{2}y'' = -n(n+1)A\operatorname{sech}^2(\frac{x}{b})$ with $A>0$, $D>0$ and $b>0$ and $n&...
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212 views

Frobenius Condition for Singular Integrable Distributions

A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ ...
2
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1answer
400 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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2answers
112 views

Second order nonlinear differential equation $x''+Hx =A(1-J/(2x^2))$

I have arrived at a differential equation and I need to solve for $x$. $$ \frac{\mathrm{d}^2x}{\mathrm{d}E^2}+Hx =A\left(1-\frac{J}{2x^2}\right) $$ where $H$, $A$, and $J$ are constants. I know ...
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1answer
1k views

Proving that the integrals of two functions are the same if they are equal everywhere except a point [closed]

Let $f(x)$ and $g(x)$ be integrable functions over $[a,b]$ and let $∂$ be a point on $[a,b]$. If $f(x) = g(x)$ for all $x≠∂$, then $$\int_a^b f(x)dx=\int_a^b g(x)dx$$