On Isospectral flows related to (dual) cubic strings: Novikov, DP peakons and B, C-Toda lattices

Abstract: The Degasperis--Procesi (DP) equation can be viewed as an isospectral deformation of the boundary value problem for the so-called cubic string, while the Novikov equation can be formally regarded as linked to the dual cubic string. However, their relationships have not been thoroughly investigated. This paper examines various intrinsic connections between these two systems from different perspectives. We uncover a bijective relationship between the DP and Novikov pure peakon trajectories. In particular, this allows us to derive, not previously known, explicit expressions for the constants of motion in the Novikov peakon dynamical system. We also establish a one-to-one correspondence between the corresponding discrete cubic and dual cubic boundary value problems. Furthermore, we propose a new integrable lattice that features bilinear relations involving both determinants and Pfaffians, demonstrating that it can be connected to both the B-Toda and C-Toda lattices, which correspond to isospectral flows, involving positive powers of the spectral parameter, associated with (dual) cubic strings.