17 problems

Nekhoroshev stability conjecture. Nekhoroshev's theorem actually holds on the set $\mathcal{P}^{\bullet}$.

Let $f(a_1,p_1,\ldots,a_\kappa,p_\kappa)$ be a density of support $\kappa$, and define its translates by $f_j(a,p)=f(a_{j+1},p_{j+1},\ldots,a_{j+\kappa},p_{j+\kappa})$. For a perio…

Venakides–Deift–Oba conjecture. The solution is asymptotically close to a modulated single-phase quasi-periodic solution.

Let $P_i$, $A_i$, $r$, and $s$ be the parameters entering the ultra-discrete two-dimensional Toda lattice solution … where … An $(N_+,N_-)$-soliton is a resonant soliton solution o…

Let $\tau_{l,m,n}^{(N)}$ be the positive-definite Hankel-determinant $\tau$-function constructed from $M$ parameters $0<p_1<\cdots<p_M$, with $1\leq N\leq M-1$. An $(N_+,N_-)$-soli…

N=(2|2) superspace formulation conjecture. These equations are consistent with the chirality constraints

The odd-dimensional open Volterra system, the even-dimensional periodic Volterra system, the full Toda lattice, the multidimensional Euler top, and the regular case of the cited bi…

The Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bihamiltonian Lie coalgebra are the systems under consideration, together with the kno…

Degree–singularity–root conjecture. The degree $\eta(w_*)$ of $p(q)$ is the multiplicity $d$ at $p_o$, where $d$ is given by the minimal degree of the product of the $\tau$-functio…

Let $Z_{N_f,-N_b}(\\{x_f\\}|\\{y_b\\})$ be the partition function with $N_f$ fermionic masses $\\{x_f\\}$ and $N_b$ bosonic masses $\\{y_b\\}$. For fermionic mass $x_i$ and bosonic…

Let Z_{N_f,-N_b}(\\{x_f\}|\\{y_b\\}) be the partition function with integer numbers $N_f$ of fermionic flavors and $N_b$ of bosonic flavors. For masses $x_i$ and $x_j$, let $m$ a…

Let $Z_{N_f,-N_b}$ denote the QCD partition function with $N_f$ fermionic flavors and $N_b$ bosonic flavors, and let the fermionic and bosonic partition functions form their respec…

Let $n$ be the number of points in the configuration, and let a solution mean a solution of the polynomial system defining the Toda lattice configuration described in the paper. Tr…

Linearity conjecture for local Toda conservation laws. For fixed $\kappa$ and $N$ sufficiently large, there exist coefficients $c_0,c_1,\ldots,c_\kappa$ such that

Toda asymptotic-expansion conjecture. The next term in the asymptotic expansion of the solution is of order

Toda modulation-region asymptotic conjecture. The next term in the asymptotic expansion of the shock wave in the modulation region is of order

Consider initial data that are the $x$-derivative of a rapidly decreasing smooth function in $L_{2}(\mathbb{R}^{2})$ with a single maximum. Write the two-dimensional dispersionless…