Finite-dimensional integrable systems related to the n-wave interaction equations

Abstract

Under a constraint between the potentials and the eigenfunctions, Lax pairs and adjoint Lax pairs of a soliton hierarchy associated with the n×n generalized Zakharov–Shabat eigenvalue problem are transformed into a spatial finite-dimensional Hamiltonian system and a hierarchy of temporal finite-dimensional Hamiltonian systems. The Lax representations, r-matrix structure and integrals of motion are explicitly presented. These integrals of motion are functionally independent and in involution in pairs, which shows that these systems, especially the whole hierarchy of temporal finite-dimensional Hamiltonian systems, are Liouville integrable.