7 problems

Classification conjectures. (1) Every Hamiltonian operator is equivalent to one whose leading coefficient is $1$. For $N=2n+3\geq 3$, this is equivalent to an operator with leading…

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…

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…

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…