Questions tagged [integrable-systems]
The integrable-systems tag has no usage guidance.
154 questions
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Wobbly divisor in the moduli space of rank 2 degree 1semi-stable vector bundles over a curve of genus 2
I am looking at Nigel Hitchin's lecture "Higgs fields in low genus" on the occasion of Oscar Garcia-Prada's 60th birthday. In the rank 2 odd degree case, he mentions a map $f$ from the ...
2
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0
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78
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On reproducing the Poincare section figure in a paper by Sato, Akiyama and Doyne Farmer [closed]
I am trying to reproduce Figure 1 in the paper "Chaos in learning a simple two-person game" (English) Proc. Natl. Acad. Sci. USA 99, No. 7, 4748-4751 (2002) (MR1895748, Zbl 1015.91014), by ...
1
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0
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105
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KdV/KP-II equation with upper semicontinuous initial data and viscosity solutions
In the article "KP governs random growth off a 1-dimensional substrate", they study the KP-II equation: the function $\phi(t,x,r)=\partial_{r}^{2}\log(F)$ satisfies
$$\partial_{t}\phi+\frac{...
- 5,474
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74
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A strange identity between generalized basic hypergeometric series
Calculating matrix elements in some quantum integrable system, I encountered a strange $q$-series identity for non-terminating basic hypergeometric functions $\phantom{i}_3\phi_2$. It comes from the ...
- 21
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A parabolic–hyperbolic in 3d: $\partial_t u(x,y,t)=\frac{1}{2}(\partial_{xx}u(x,y,t)-\partial_{yy}u(x,y,t))$
I was just wondering if somebody can provide some references for the parabolic–hyperbolic pde
$$\partial_t u(x,y,t)=\frac{1}{2}(\partial_{xx}u(x,y,t)-\partial_{yy}u(x,y,t)).$$
Apparently, the IVP ...
- 5,474
5
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1
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80
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Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricted subspaces of $V^{\otimes N}$
An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
where ...
- 727
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148
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Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
4
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1
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293
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Double q-analog of Pochhammer
Has the function
$$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$
been studied in the math literature? For example, does it obey any difference ...
- 533
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117
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Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?
The special form of Riccati equation
$$
\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}
$$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$.
The sufficiency is ...
- 9
2
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118
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Connecting the higher energies of GP and KdV via a Riccati equation
I will describe my set-up and then the problem.
We use the branch of the complex square root where
$$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...
- 203
3
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1
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429
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Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
- 191
4
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1
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Building a geodesic conjugate parameterization on catenoid
I believe that a catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma_{uv} \in\mathrm{span}(\sigma_u, \sigma_v)$) ...
- 245
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92
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Solving a Catalan-like recursion of polynomials, related to the KdV energies
I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are ...
- 203
4
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1
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214
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A system of linear PDEs with boundary conditions
I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I could simplify one of my geometric problems (in the smooth scenario) into the solutions of a system of linear PDEs ...
- 245
5
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149
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Deformation quantization of an integrable system
What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?
- 401
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48
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How to recover a subspace in an infinite-dimensional Grassmannian from its $\tau$-function or $\psi$-function?
Solutions of the KP hierarchy are parametrized by an infinite-dimensional Grassmannian (Sato Grassmannian or Wilson's adelic Grassmannian). I heard somewhere that one can recover a subspace in the ...
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71
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Is there a relation between symplectic toric orbifolds and semi-toric systems?
So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ ...
- 791
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39
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Question on the proof of doing a nodal trade, almost-toric fibrations
I am trying to understand the details of the proof of lemma $6.3$ of the following notes https://arxiv.org/pdf/math/0210033.pdf, which give us specific conditions of when we can swap a neighborhood of ...
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Doing a nodal trade in a semi-toric system
Recently I have been studying semi-toric systems and almost toric fibrations. For the purpose of semi-toric fibrations I have been reading these notes https://arxiv.org/pdf/math/0210033.pdf. ...
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2
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30
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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 (stationary) Camassa-Holm hierarchy may be cast as a zero-curvature equation
\begin{align}
-V_{n,...
- 121
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127
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About writing solutions of linear ODE's: Is this statement correct?
A motivating example: Consider the Hypergeometric equation
$$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$
it has a solution given by the Gauss's Hypergeometric function
$$_2F_1(a,b;c;z)...
- 480
1
vote
3
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247
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Equivalence problem of classifying heat equations
I have tried to search for references online but I am unable to do so.
I am looking for references that uses Cartan's method of moving frames to classify heat equations.
Also are there references that ...
- 135
2
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1
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341
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What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]
.
Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link.
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Below is ...
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Has anyone written down an approach to the Lenard-Magri integrability scheme via algebraic geometry?
I’ve been thinking about the algebro-geometric meaning of the
Lenard-Magri scheme of getting an integrable system from a pair of
compatible Poisson structures. I think one might be able
to prove a ...
- 137
5
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237
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Intuition for almost periodic solution and Poincaré recurrence theorem
I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...
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2
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102
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Spectrum of a Lax Pair and conservation laws of a PDE
I would like to ask a question that I had asked a few days ago on the site math.stackexchange
and I still have not received an answer.
If we have a Lax operator, we know that the spectrum of this ...
- 93
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0
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74
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Coordinates for quasiperiodic motion after reconstruction
Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
- 141
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80
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Dynamics of composition of reflections
Let $C$ be a curve defined by $y = f(x)$, and define the vertical reflection over $C$ to be the map $(x,y) \mapsto (x,y')$, where $y' = 2 f(x) - y$. In other words, the vertical distance from $(x,y)$ ...
- 213
6
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2
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318
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
- 727
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199
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Integrable systems and Lagrangian fibrations
It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...
3
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2
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367
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(Super)integrable systems on quiver varieties
In recent papers
https://arxiv.org/abs/2101.05520
https://arxiv.org/abs/2001.06911
(super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed.
My question: are ...
6
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1
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362
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Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"?
The classical Catalan numbers
$$ C_n = \frac{1}{n+1} \binom{2n}{n}, $$
well-known for their numerous combinatorial interpretations (the second volume of Stanley's Enumerative Combinatorics famously ...
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188
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Maximal dimension guaranteed for integral manifolds of hyperplane distributions
To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...
- 1,225
4
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1
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245
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Designer metric for a vector field
A vector field $V$ on a manifold $M$ admits an invariant metric if there exists a Riemannian metric $g$ with $L_Vg = 0$. How can one characterize the vector fields on $M$ that admit an invariant ...
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Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form
$$
x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)}
$$
for ...
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Is Krichever's constuction "inverse" to finding the spectral curve?
There is Krichever's algebro-geometric construction of solutions to the KP equations starting from a curve X together with extra data.
There is a way to find the spectral curve given a point of the ...
- 2,964
2
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1
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221
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Can every point of Wilson's adelic Grassmannian be obtained by Krichever construction of solutions to KP equations?
Igor Krichever introduced an algebro-geometric construction of solutions of KP equations starting from an algebraic curve with some additional data.
George Wilson introduced the adelic Grassmannian, ...
- 2,964
6
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1
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661
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Lax pair of an integrable non-linear PDE
The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
- 449
5
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1
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Obstruction to the existence of a globally defined integrating factor
Let $U$ be an open subset of $\Bbb{R}^n$ and take $\omega$ to be a nowhere-vanishing smooth $1$-form on $U$. The Frobenius Theorem implies that, near each point of $U$, $\omega$ may be written as $g\,{...
- 3,599
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339
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Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?
If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system
\begin{align*}L\psi&=\lambda\psi\\
\psi_t&=B\psi\end{align*}
has as its integrability condition ...
- 161
4
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1
answer
170
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Quantum Hamiltonian reduction and tensor products
Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.
Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and ...
- 1,077
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93
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Representation theoretic definition of wavefunctions of an integrable hierarchy?
I am reading Kac's book on infinite dimensional lie algebras. In the last chapter, he starts with a highest weight module of an affine lie algebra $\mathfrak{g}(A)$, and uses it to define tau ...
user35360
6
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1
answer
592
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$GL(\infty)$ group action through the boson-fermion correspondence
Every point of the Sato Grassmannian can be used to generate a tau function of the KP hierarchy. In addition, the Sato Grassmannian can be seen as a subset of the "second quantized fermion Fock space" ...
user35360
4
votes
1
answer
334
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Examples of particle systems with higher-order collisions
In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...
- 801
2
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0
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123
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GUE, tau-function of Painlevé II, and an article of Forrester-Witte
Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a ...
- 593
2
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2
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634
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Explanation of definition of George Wilson's adelic Grassmannian
How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop ...
- 2,964
8
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1
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1k
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Any holomorphic vector bundle over a compact Riemann surface can be defined by only one transition function?
It is known that any holomorphic bundle of any rank over a noncompact Riemann surface is trivial. A proof can be found in Forster's "Lectures on Riemann surfaces", section 30.
Let $E$ be a ...
- 91
6
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0
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170
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Introduction to the Adler-van Moerbeke theory
Is there a good introduction to the Adler-van Moerbeke theory of solving completely integrable systems by linearizing the flow on the Jacobian of an algebraic curve, for someone with a background in ...
1
vote
0
answers
117
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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.
- 92
2
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0
answers
80
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Generalized definition of integrable condition on rough complex subbundle
Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...
- 938