Questions tagged [integrable-systems]

Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated to the invariants of the foliation span the tangent distribution: there exists a maximal set of Poisson-commuting invariants in phase space. May be used more broadly for systems possessing simple analytic solutions.

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Confusion about the action variable definition

Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set $$M_f = \{ (p,q)...
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Discretization of derivative of delta function and affine Kac-Moody algebra

In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented: $$ \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
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Thermodynamic Bethe ansatz validity

The thermodynamic Bethe ansatz (TBA) involves introducing a chemical potential $h$ conjugate to some charge $Q$, and considering the hamiltonian $H-hQ$. Particles with the smallest value of the mass ...
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Is the Hamiltonian of interacting systems integrable if the interaction is linear?

Suppose we allow two integrable systems with Hamiltonians $H_1$ and $H_2$ to interact. Then their combined dynamics can be described by a joint Hamiltonian, $$H = H_1(\mathbf{q}_1,\mathbf{p}_1) + H_2(\...
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How many constants of motion are there in a 2D two body problem?

A system consists of two masses interacting with gravitational force, rotating around their centre of mass. If we only consider the $xy$ plane where the masses rotate, the system has 8 degrees of ...
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Numerical solution of Bethe ansatz equation

In this paper https://arxiv.org/abs/2003.14202, the authors plot the solution of coupled non-linear algebraic equation (Bethe Ansatz Equations): \begin{equation} k_j L=2\pi I_j -\sum_{\beta=1}^M \...
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Differentiation of an operator equation in paper by Chen, Lee, Pereira 1979

This 1979 paper by Chen, Lee, and Pereira gives an operator $L$ satisfying $$\dot L = [A, L],\tag{1}$$ where $A$ is another operator, and the dot denotes time differentiation. They then define $I_n = \...
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Semi-classical spinning strings and AdS-CFT

I'm trying to understand how the AdS/CFT correspondence is precisely formulated when on the bulk side people are working with the string theory as a sigma model on the worldsheet expanded about some ...
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Hilbert transform in soliton paper

I asked this question over at the Mathematics SE, see here, but have not gotten any responses, so I figured I might as well try here as well. While the question is mathematical, it does appear in a ...
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Correspondence between quantum and classical integrability

I'm looking for connections between quantum and classical integrability. I know quantum integrability is not well-defined, but let us just take one of the popular definitions by promoting the Poisson ...
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Phase space of Ising minimal model + deformations

Consider the Ising field theory, a conformal field theory in 2 dimensions which corresponds to the minimal model $\mathcal{M}_{4,3}$ and it's perturbations by the relevant operators $\epsilon, \sigma$ ...
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What is the $XXX_s$ Hamiltonian in terms of $\vec{S}_i \cdot \vec{S}_{i+1}$?

Faddeev, Takhtajan, and others united and discovered many integrable models through the Algebraic Bethe Ansatz. For example, the integrable spin-1/2 Heisenberg model $$H_{1/2} = \sum_{i=1}^L \vec{S}_i ...
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How local are the conserved charges in a quantum integrable model?

For the purposes of this question, let us define a quantum integrable model as one solvable by the Bethe Ansatz. That structure endows the model with a set of conserved charges $\{H^{(n)}\}$ whose ...
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$XXX_s$ spin chains as constrained $XXX_\frac{1}{2}$ spin chains

The eigenenergies of an $XXX_{s}$ spin chain are found by solving the Bethe Ansatz. For a closed chain of length $L$ and $N$ rapidities, $u_i$, encoding excitations from the ground state moving along ...
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Is the motion of a particle in the surface of a torus always periodic?

I am trying to see if there are ballistic trajectories in the surface of the torus that are not periodic and to what extent. Maybe it is not only quasiperiodic but chaotic. I guess there are ...
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Is 2D rectangular billiard an integrable system? What's the form of explicit solution?

Suppose the free particle moving inside 2D box. $$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+V$$ where the potential is zero inside the box and infinite outside the box. It's clear that $p_x,p_y$ are not ...
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Parity of XYZ model ground state

I am considering the XYZ Hamiltonian (with PBC) $$\widehat{H}_{\mathrm{XYZ}}=\sum_{i=1}^{N} \left(\hat{\sigma}_{i}^{x} \hat{\sigma}_{i+1}^{x}+J_{y}\hat{\sigma}_{i}^{y} \hat{\sigma}_{i+1}^{y}+J_{z}\hat{...
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What are the symmetries of circular billiards that makes it integrable?

I have often heard that integrability in is equivalent to extensively many conserved quantities $A_i$, i.e. the Poisson bracket $\{H,A_i\}=0$ or in quantum mechanics $[H,A_i]=0$. What are the ...
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Why do Action-Angle Variables form an invariant Torus?

I've been casually reading up on Hamiltonian Mechanics and integrable systems and one term that is used a lot of "invariant torus" where bounded orbits live. KAM theory is also mentioned as ...
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Why is parameter $q$ of $U_q(\mathfrak{sl}_2)$ being root of unity a problem?

I am recently studying the quantum groups and the exactly solvable models. I understand the solution of the Yang-Baxter equation is elements in $U_q(\mathfrak{sl}_2)$. I am informed that when $q$ is ...
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Explanation of some physical terms related to hydodynamic limit to a non-physicist & Integrable system

I'm a mathematician, and I'm reading an article and am struggling with some of the terminology related to physics. [...] This equation is formally obtained from the hydrodynamic limit of the ...
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Unphysical Solution of the Bethe Ansatz

I actually want to ask an elementary question regarding the algebraic Bethe-Ansatz. Say I have constructed the Bethe Ansatz Equations (BAE) in the algebraic framework with pseudovacuum $\phi$, $B(u)$ ...
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ODEs with rational first integrals [closed]

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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Bidirectional Benjamin-Ono vs Benjamin-ono

I am a mathematician and not a physicist, and I am trying to physically understand the difference between the model described by the Benjamin-Ono equation and the model described by the bidirectional ...
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Derivation of the Zakharov-Shabat System and Lax Pair for the Gross–Pitaevskii Equation

Question: In addition to showing that the nonlinear Schrodinger equation $i \Psi_t + \Psi_{xx} - 2|\Psi|^2 \Psi = 0$ (without a potential) is integrable and isospectral, the existence of a Lax pair ...
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Nonlinear Schrödinger equation in a potential

I've recently become interested in the integrability of nonlinear PDEs while reading these lecture notes. Question 1: Would the equation $i\Psi_t + \Psi_{xx} - (2|\Psi|^2 + V) \Psi = 0$ for a ...
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CFTs of Integrable models XXX, XXZ, Hubbard

I want to know if someone knows the answer to the following question (or can explain to me why the question makes no sense): Consider the standard integrable models, like the XXX-Chain, the XXZ-Chain, ...
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Do you know about any book which discusses solitons in Benjamin-Ono Equation?

Benjamin-Ono equation is an integrable equation with soliton solutions. There are many books on solitons. The ones I know about mainly discuss solitons in Korteweg de-Vries and related equations. Do ...
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Integrable Lattice Lieb Liniger Model

Is there a model of the 1D Bose gas (aka the Lieb Liniger Model) on a lattice that retains integrability? As a naive guess, I had thought that the Bose Hubbard model would fit this criterion, since ...
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No-interaction theorem in classical relativistic mechanics

In classical relativistic Hamiltonian mechanics there is a so-called "no-interaction theorem" (see, for example, this article for a proof). Roughly, it states that if we have an $N$-body ...
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About the diagonalization of non-linear (in fields) actions

Suppose we have some interacting theory with the action: $$ S = \int d^{D} x \left(\partial_\mu \phi \partial^\mu \phi + V(\phi)\right) $$ Where $V(\phi)$ is a potential (some polynomial of degree $&...
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2D harmonic oscillator trajectory

Consider the Hamiltonian for the classic planar harmonic oscillator: $$H = H_x + H_y$$ where $$H_x~=~\frac{1}{2}(p_x^2+x^2), \qquad H_y~=~\frac{1}{2}(p_y^2+y^2).$$ So it is possible to obtain a set of ...
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How to compute the connected spectral form factor for integrable models?

Given a spectrum of $N$ real eigenvalues, $\{E_m \}$ of some Hermitian operator, the connected spectral form factor is defined as follows: \begin{align} K_c(t) = \langle \sum_{m,n=1}^N e^{it (E_m-E_n)}...
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Is harmonic oscillator a free particle in disguise?

Let me start with an example. If one considers a free particle motion on two-dimensional plane and projects it onto the radial coordinate, one gets the following Hamiltonian $$H = \frac{p^2}{2}+g^2 \...
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Constants of motion [duplicate]

For any system performing any kind of motion with $n$ degrees of freedom, are $2n-1$ integrals of motion and also $2n$ constants of motion always present? If yes, then is there always a symmetry for ...
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Time derivative of path ordered monodromy matrix

I am currently gettting familiar with integrably systems and came the following statement in my literature: $U=U(x,\lambda,t)$ some matrix (Lax component) we define $$T(\lambda,t) = \mathcal{P} \exp \...
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The planar limit, self-duality and their relation to two dimensions

In the lecture notes by Beisert on integrability, it is stated that integrability is a property mainly in two-dimensional field theories, with some higher-dimensional examples. As higher-dimensional ...
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WZW model - affine Kac-Moody current algebra from Quantum Group exchange algebra

In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model. ...
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XXZ chain exact ground state energy

I would like to know the analytical expression of the ground state energy of the XXZ model, if such formula exists (probably from a Bethe Ansatz solution) and if it is valid in all parameter regimes.
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Are partial derivatives in the context of Action-Angle variables different from partial derivatives of functions?

Let's say I have a system with two degrees of freedom and I can find two independent action variables. One action variable is total energy expression, such as is often used in classical mechanics. $$...
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Conserved Quantities and Integrability in the $N$-Body Problem

Under my understanding of integrability, a system with $2n$-dimensional phase space is integrable when there are at least $n$ constants of motion satisfying some conditions (e.g., they are in ...
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Grand-canonical partition function for an integrable system

When speaking about the grand-canonical ensemble of a statistical system, one usually works with a case, when there are several conserved quantities - total number of particles $N$, angular momentum $...
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Completely Integrable Frustrated Lattice Systems

The Toda lattice is a prime example of a lattice system that is completely integrable, in the sense that it admits a Lax pair, https://doi.org/10.1143/PTP.51.703, making it easy to find soliton ...
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Itzykson Zuber Quantum Field Theory: meaning of integrable system

Here is a part of the book Quantum Field Theory by Itzykson and Zuber: I have two questions: what does the author mean that equation (1-30) form and integrable system, and why? what is the ...
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Non-Integrable models in 1+1D

Is it possible to have a non-integrable system in (1+1)D in Classical Physics? For some reason, I get the intuition that there shouldn't be any such systems. What if we consider (1+1)D systems in ...
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The physical meaning of maximal non-integrability of the contact structure

So, basically integrability is equivalent to the existence of an integral manifold of the distribution and I guess, the integral manifold is like a plane of motion where state moves in physical sense. ...
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Doubt on Lax formulation of Korteweg–de Vries equation

The Korteweg–de Vries equation is given by: $$\frac{\partial u(x,t)}{\partial t}-6u\frac{\partial u(x,t)}{\partial x}+\frac{\partial^3 u(x,t)}{\partial x^3}=0$$ This equation can be formulated using ...
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Definition of 'Antichiral' in the Thirring Model

This is my first post on this website so please excuse any poor formatting. In the lecture series on quantum integrable models, much attention is focused towards the Thirring model, $$\mathscr{L} = \...
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Question about a 2D Harmonic Oscillator with incommensurate frequencies and Integrability

In Classical Dynamics by José & Saletan [section 4.2.2] they give the example of a 2D Harmonic Oscillator whose equations of motion are \begin{equation} \ddot{x}_i+\omega_i^2x_i=0 \ \ \ \ \ \text{...
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Does the Hamilton-Jacobi equation imply that there are always $N$ conserved quantities for any system with $N$ degrees of freedom? [duplicate]

I'm reviewing the Hamilton-Jacobi equation because I'm working on a research project about Kerr black holes and the geodesics of particles gravitating them (This is not really relevant to the question,...
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