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arXiv · 2107.07280

On Lax operators

Alberto De Sole·Victor G. Kac·Daniele Valeri
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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.

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Alberto De Sole, Victor G. Kac, Daniele Valeri. 2021-07-15. On Lax operators. https://arxiv.org/abs/2107.07280

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