Lax pairs for BKM hierarchy
Andrey Yu. Konyaev, Vladimir S. Matveev
TL;DR
The paper develops a comprehensive Lax-pair framework for the BKM integrable systems, showing that a 2×2 operator pair depending on a spectral parameter $\mu$ and a hierarchy parameter $\lambda$ encodes the entire BKM family and its hierarchies. By reformulating the BKM equations in a parametric form and exploiting a formal differential-series calculus, the authors derive explicit Lax pairs for four BKM types and their symmetry hierarchies, and demonstrate how conservation laws arise from the Lax representation through diagonalization and spectral expansion. The work unifies numerous classical integrable models (including KdV, Camassa–Holm, Kaup–Boussinesq, Ito, and Marvan–Pavlov) as special cases and provides a pathway toward inverse scattering and spectral analysis for the rational-$\mu$ potentials. It also highlights a deep connection between BKM systems and Nijenhuis geometry, suggesting broader implications for geometric approaches to integrability and finite-dimensional reductions. Overall, the results furnish a universal toolset for generating and analyzing infinite families of conserved quantities and commuting flows within the BKM framework.
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.