10 problems

Let $P=\sum_{e=1}^{d+2} P^{ij}_e {\partial}_x^{d+2-e}$ be a local operator of homogeneous differential order $d+2$, meaning that … where $d\geq 0$. Assume that $P$ defines a Poisso…

A second-order HHO defines a multiparametric family of systems of conservation laws. In the generic situation, the velocity matrix has double eigenvalues, so diagonalizability is e…

A second-order homogeneous Hamiltonian operator is a homogeneous Hamiltonian operator of differential order two, and an associated hydrodynamic type system is the hydrodynamic type…

Gardner normal-form conjecture. $H$ can be transformed into the Gardner operator $D$ by either a transformation from the potentiation normal-form conjecture or a composition thereo…

Potentiation normal-form conjecture. $H$ can be transformed into a quasiconstant coefficient skew-adjoint differential operator using either a contact transformation or a compositi…

De Sole–Kac–Wakimoto conjecture. There exists a contact transformation that brings $H$ to either a quasiconstant coefficient skew-adjoint differential operator, or to a linear comb…

An integrable bi-Hamiltonian equation in one dependent variable $u$ is considered up to contact transformations. The relevant model hierarchies are the linear, KdV, and HD hierarch…

The compatible pairs $(H_{(11,c(x))},H_{(5,0,2c(x))})$ and $(H_{[11,c(x)]},H_{(5,0,14c(x))})$, with $c”(x)=0$, define Lenard–Magri schemes with $\xi_0=1$ and $\bar{\xi}_0=c(x)$. Or…

Consider scalar Hamiltonian operators of odd order $N$, classified up to contact transformations; the preceding discussion identifies four types, including quasiconstant-coefficien…

Let a non-quasiconstant coefficient Poisson $\lambda$-bracket on $\widetilde{\mathcal{R}}_1$ have odd order $N$ and level $m$, where the level is defined by…