On Lax operators

A. De Sole, V.G. Kac, D. Valeri

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