On Lax operators

Abstract

We define a Lax operator as a monic pseudodifferential operator $L(\partial )$ of order $N\ge 1$, such that the Lax equations $\frac{\partial L(\partial )}{\partial t_k}=[(L^{\frac{k}{N}}(\partial ){)}_{+},L(\partial )]$ 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.

39 pages, v2: minor editing and corrections following the referee's suggestions