Questions tagged [integrable-systems]

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glueing Pfaffian equation

QUESTION: Can we regard integrable pfaffin equation as a section of certain vector bundle with some connection? DETAIL: Let U and V be open sets of a manifold, with integrable pfaffin equaitons on ...
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22 views

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 Camassa-Holm hierarchy may be casted as a zero-curvature equation \begin{align} -V_{n,x}+\left[U,...
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37 views

On integrability of a distribution in $\mathbb R^3.$

$\mathbf {The \ Problem \ is}:$ Let, $P,Q,R$ be smooth real-valued maps on $M=\mathbb R^{3}$ with $P^{2}+Q^{2}+R^{2} \neq 0.$ Let, $E(a)$ denote the orthogonal complement of the subspace $S$ in $T_a(M)...
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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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52 views

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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41 views

Does every Hamiltonian equation can be written in a some system of coordinates as a canonical Hamiltonian system

Does every Hamiltonian equation can be written in a some system of coordinates as a canonical Hamiltonian system i.e. $\exists(q,p)$ such that $$\begin{equation*} \begin{cases}\dfrac{dq}{dt}=\...
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35 views

Spectrum of a Lax Pair and conservation laws of a PDE

If we have a Lax operator, we know that the spectrum of this operator is conserved over time. Suppose that we have an equation that admits $L_u$ as a Lax operator, then can we say that all the ...
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36 views

Find Lax pair for solid body equations

Consider the system $$ \begin{align*} \dot{u_1}&=u_2u_3\\ \dot{u_2}&=u_3u_1\\ \dot{u_3}&=u_1u_2. \end{align*} $$ In the book "Integrable systems" by Hitchin et al., it is said, ...
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135 views

How to write an ODE with a Darboux polynomial?

Question: Given a polynomial ODE $\dot{x}=f(x)\in\mathbb{R}^n$ that possesses a Darboux polynomial$^*$ $p(x)$ satisfying $\dot{p}(x)=c(x)p(x)$ for some function $c(x)$ (called the cofactor) how can ...
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31 views

ODEs with rational first-integrals

I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ ...
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72 views

What is the relationship between these two definitions of generating functions?

I'm doing my bachelor's thesis on Integrable Hamiltonian Systems, and one important part of the thesis will be proving the Liouville theorem. For this theorem I'm using the book by Arnold "...
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73 views

Finding Lie point symmetries of ODE $\frac{du}{dx}=xG(\frac{u}{x^2})$

I have been tasked with finding all the Lie point symmetries of the following ODE: $$\frac{du}{dx}=xG\left(\frac{u}{x^2}\right)$$ I have found by applying the prolongation formula that a symmetry ...
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1answer
58 views

Camassa-Holm functional derivative

I'm a physicist, and I am reading the Wikipedia article of the well known integrable system Camassa-Holm equation (original article) \begin{align} \dot{u}+2\kappa u'-\dot{u}''+3u u'=2u'u''+uu''', \end{...
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96 views

Explicit solutions for Lax pair using Jacobian

Given a Lax pair $(L(\lambda),M(\lambda))$ i.e a pair of matrices that depends on time $t$ and on a parameter $\lambda$ such that $$\frac{\mathrm{d} L(\lambda)}{\mathrm{d}t} = [M(\lambda), L(\lambda)]$...
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1answer
43 views

Proof functions in involution for R matrix Lie-Poisson bracket

I'm reading the well-known article of Reyman and Semenov-Tian-Shansky called "Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems". Suppose $\mathfrak{g}$ is a ...
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123 views

Proving an Identity involving sums related to the $Z(N)$-Ising model

Background: I am trying to construct meromorphic functions satisfying a number of axioms, so-called form factors which are important objects in integrable quantum models, following this paper. ...
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28 views

bi-Hamiltonian structure of the Sine-Gordon equation

It is well known that the sine-Gordon equation in laboratory (or light-cone) coordinates, \begin{align} \ddot\phi-\phi''=\sin\phi,\\ \dot\varphi'=\sin\varphi, \end{align} respectively, is an ...
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43 views

Bäcklund transformation Nonlinear Schrödinger equation

I am studying the work of M. Boiti and F. Pempinelli, "Nonlinear Schrödinger Equation, Bäcklund Transformations and Painlevé Transcendent" (1980). In the nonlinear Schrödinger equation \...
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1answer
56 views

Trigonometric identity (for two-soliton solution in SG)

Context: I am studying the construction of the two-soliton solution of the Sine-Gordon equation. Following this presentation, they obtain \begin{align} a_1\left[\sin\left(\frac{\phi_2+\phi_1}{2}\right)...
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192 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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57 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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50 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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1answer
96 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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87 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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1answer
303 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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45 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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56 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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81 views

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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1answer
63 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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137 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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69 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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173 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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57 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
56 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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426 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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58 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 ...
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1answer
185 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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120 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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120 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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1answer
142 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
741 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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88 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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147 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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42 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
122 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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80 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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117 views

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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55 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 ...