Lax-Sato formulation of the Novikov-Veselov Hierarchy

TL;DR

This paper develops a Lax-Sato formulation for the Novikov-Veselov hierarchy, introducing a commuting flow structure on operator triples and revealing the hierarchy's symmetry and connection to the Novikov-Veselov equation.

Contribution

It constructs a new hierarchy of commuting flows for coupled pseudodifferential and Schrödinger operators, extending the integrable systems framework for the Novikov-Veselov equation.

Findings

Established a hierarchy of commuting flows on operator triples.

Proved the hierarchy's symmetry under an involution exchanging operators.

Identified the first equation of the hierarchy as the Novikov-Veselov equation.

Abstract

We construct a hierarchy of pairwise commuting flows $d/dt_{i,n}$ indexed by $i \in \{1,2 \}$ and $n \in \mathbb{Z}_{\geq 0}$ 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 Schr\"{o}dinger operator $\mathcal{H}=\partial_1 \partial_2 +u$. $\mathcal{L}_1, \mathcal{L}_2$ and $\mathcal{H}$ are coupled by the relations $\mathcal{H} \mathcal{L}_i+\mathcal{L}_i^* \mathcal{H}=0$. We show that the flows $d/dt_{1,n}+d/dt_{2,n}$ 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.