Lax-Sato formulation of the Novikov-Veselov Hierarchy

Abstract

We construct a hierarchy of pairwise commuting flows $d/d{t}_{i,n}$ indexed by $i\in \{1,2\}$ and $n\in {\mathbb{Z}}_{\ge 0}$ on triples $({\mathcal{L}}_1,{\mathcal{L}}_2,\mathcal{H})$ where ${\partial }_1$ and ${\partial }_2$ are two commuting derivations, ${\partial }_i{\mathcal{L}}_i$ is a self-adjoint pseudodifferential operator in ${\partial }_i$ and $\mathcal{H}$ is the formal Schrödinger operator $\mathcal{H}={\partial }_1{\partial }_2+u$. ${\mathcal{L}}_1,{\mathcal{L}}_2$ and $\mathcal{H}$ are coupled by the relations $\mathcal{H}{\mathcal{L}}_i+{\mathcal{L}}_i^{\ast }\mathcal{H}=0$. We show that the flows $d/d{t}_{1,n}+d/d{t}_{2,n}$ commute with the involution $({\mathcal{L}}_1,{\mathcal{L}}_2,\mathcal{H})\mapsto ({\mathcal{L}}_2,{\mathcal{L}}_1,\mathcal{H})$ and that the first equation of this reduced hierarchy is the Novikov-Veselov equation.