Operator-valued Camassa–Holm systems and their integrability

Abstract

We study a Sturm–Liouville-type operator with operator-valued coefficients and its iso-spectral deformations, related to a new two-component Camassa–Holm-type completely integrable dynamical systems. Based on a specially devised gradient-holonomic scheme, generalizing the one before developed for studying a Sturm–Liouville-type spectral problem on a spatially multidimensional Hilbert–Schmidt operator-valued Hilbert space, we constructed the related two compatible Poisson structures and an infinite hierarchy of commuting to each other conservation laws of the derived two-component Camassa–Holm-type Hamiltonian system. The latter makes it possible to state under some additional constraints its complete integrability, and in particular, to develop the corresponding inverse spectral-type-based method for constructing its exact solutions.