On Lax operators

TL;DR

This paper generalizes the theory of KP and KdV hierarchies to arbitrary scalar Lax operators, establishing their consistency and defining properties within the framework of pseudodifferential operators.

Contribution

It extends the classical Lax operator theory to a broader class of scalar operators, ensuring the validity of KP and KdV hierarchies beyond traditional settings.

Findings

Lax equations are consistent for a wide class of pseudodifferential operators.

The traditional KP and KdV hierarchies are valid for arbitrary scalar Lax operators.

The paper establishes the non-zero flow condition for infinitely many k.

Abstract

We define a Lax operator as a monic pseudodifferential operator $L(\partial)$ of order $N\geq 1$, such that the Lax equations $\dfrac{\partial L(\partial)}{\partial t_k}=[(L^{\frac kN}(\partial))_+,L(\partial)]$ are consistent and non-zero for infinitely many positive integers $k$. Consistency of an equation means that its flow is defined by an evolutionary vector field. In the present paper we demonstrate that the traditional theory of the KP and the $N$-th KdV hierarchies holds for arbitrary scalar Lax operators.