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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Moment equations of linear PDE

I am studying an Fokker-Planck equation for $u(x,t)$ of the form $\frac{\partial{u}}{\partial t} = a(t) \frac{\partial^2 Q_b(x) u}{\partial x^2} + b(t) \frac{\partial R_c(x) u}{\partial x},$ where $...
4 votes
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Find all conserved quantities in a Lagrangian system

Given a Lagrangian system $(M,\mathcal{L})$, the equation of motion is given by the Euler-Lagrange equation. Usually it's hard to solve this equation directly, so one may try to find some conserved ...
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Prove that $\int_a^bf(x)dx=\int_{a+c}^{b+c}g(x)dx$

I am having trouble with this problem. f be integrable on $[a,b]$. Suppose $c\in R$ and $g: [a+c,b+c] \to R$ such that $g(x)=f(x-c), x\in[a+c,b+c]$ $$\int_a^bf(x)dx=\int_{a+c}^{b+c}g(x)dx$$ I am ...
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Integrability of Euler top

I'm taking an integrability course, it's a bit too fresh and my physics a bit rusty. I'm stuggling with an exercise about the Euler top. I'd appreciate a hand with some questions, or a least some tips....
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Recommendation of an Introduction to Integrable Probability

I am interested in the topic of Integrable Probability with topics like KPZ Universality and Tracy-Widom Distribution. What are the prerequisites for this subject? I have standard Undergraduate level ...
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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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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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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 ...
3 votes
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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8 votes
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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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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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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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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 ...
3 votes
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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 ...