A finite-dimensional integrable system associated with the three-wave interaction equations

Abstract

Under a constraint between the potentials and the eigenfunctions, the 3×3 AKNS matrix spectral problem and its adjoint spectral problem associated with the three-wave interaction equations are nonlinearized so as to be a new finite-dimensional Hamiltonian system. A general scheme for generating involutive systems of conserved integrals and their two new generators are proposed, by which the finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Moreover, the involutive solutions of the three-wave interaction equations are given.