# Questions tagged [integrable-systems]

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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 ...
58 views

### 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 ...
90 views

### (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 ...
172 views

### 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 ...
86 views

### 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 ...
112 views

### 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 ...
366 views

### 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 ...
62 views

### 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 ...
160 views

### 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, ...