Lax pairs for BKM hierarchy

TL;DR

This paper constructs Lax pairs for the BKM hierarchy, unifying and extending integrable PDE systems, including many well-known hierarchies, through a Sturm-Liouville operator framework.

Contribution

It provides a unified method to derive Lax pairs for the BKM hierarchy and many related integrable systems, including new ones.

Findings

Lax pairs constructed for BKM equations.

Many known integrable hierarchies are special cases.

Lax pairs involve Sturm-Liouville operators with rational spectral dependence.

Abstract

We construct Lax pairs for the recently (2023) introduced integrable PDE systems known as the BKM equations. As many known and previously studied integrable systems are special cases of the BKM systems, our construction provides Lax pairs for many integrable hierarchies, including previously studied ones such as Camassa-Holm, Dullin-Gottwald-Holm, cKdV, Ito, and Marvan-Pavlov, as well as new ones. The corresponding pair is related to a Sturm-Liouville operator on the real line whose potential depends rationally on the spectral parameter.